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vector-toc-level-3"> <a class="vector-toc-link" href="#Example_Game_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.1</span> <span>Example Game 2</span> </div> </a> <ul id="toc-Example_Game_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combined_Game" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Combined_Game"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.2</span> <span>Combined Game</span> </div> </a> <ul id="toc-Combined_Game-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Equivalence</span> </div> </a> <ul id="toc-Equivalence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-First_Lemma" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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href="https://es.wikipedia.org/wiki/Teorema_de_Sprague-Grundy" title="Teorema de Sprague-Grundy – Spanish" lang="es" hreflang="es" data-title="Teorema de Sprague-Grundy" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D8%A7%D8%B3%D9%BE%D8%B1%D8%A7%DA%AF%E2%80%93%DA%AF%D8%B1%D8%A7%D9%86%D8%AF%DB%8C" title="قضیه اسپراگ–گراندی – Persian" lang="fa" hreflang="fa" data-title="قضیه اسپراگ–گراندی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Sprague-Grundy" title="Théorème de Sprague-Grundy – French" lang="fr" hreflang="fr" data-title="Théorème de Sprague-Grundy" data-language-autonym="Français" 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href="https://pl.wikipedia.org/wiki/Twierdzenie_Sprague%E2%80%99a-Grundy%E2%80%99ego" title="Twierdzenie Sprague’a-Grundy’ego – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie Sprague’a-Grundy’ego" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%8F_%D0%A8%D0%BF%D1%80%D0%B0%D0%B3%D0%B0_%E2%80%94_%D0%93%D1%80%D0%B0%D0%BD%D0%B4%D0%B8" title="Функция Шпрага — Гранди – Russian" lang="ru" hreflang="ru" data-title="Функция Шпрага — Гранди" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>This article has multiple issues.</b> Please help <b><a href="/wiki/Special:EditPage/Sprague%E2%80%93Grundy_theorem" title="Special:EditPage/Sprague–Grundy theorem">improve it</a></b> or discuss these issues on the <b><a href="/wiki/Talk:Sprague%E2%80%93Grundy_theorem" title="Talk:Sprague–Grundy theorem">talk page</a></b>. <small><i>(<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove these messages</a>)</i></small> <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, 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href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">June 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> </div> </div><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">combinatorial game theory</a>, the <b>Sprague–Grundy theorem</b> states that every <a href="/wiki/Impartial_game" title="Impartial game">impartial game</a> under the <a href="/wiki/Normal_play_convention" title="Normal play convention">normal play convention</a> is equivalent to a one-heap game of <a href="/wiki/Nim" title="Nim">nim</a>, or to an infinite generalization of nim. It can therefore be represented as a <a href="/wiki/Natural_number" title="Natural number">natural number</a>, the size of the heap in its equivalent game of nim, as an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> in the infinite generalization, or alternatively as a <a href="/wiki/Nimber" title="Nimber">nimber</a>, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. </p><p>The <b>Grundy value</b> or <b>nim-value</b> of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the <b>nim-sequence</b> of the game. </p><p>The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently by <a href="/wiki/Roland_Sprague" title="Roland Sprague">R. P. Sprague</a> (1936)<sup id="cite_ref-SpraguePaper_1-0" class="reference"><a href="#cite_note-SpraguePaper-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Patrick_Michael_Grundy" title="Patrick Michael Grundy">P. M. Grundy</a> (1939).<sup id="cite_ref-GrundyPaper_2-0" class="reference"><a href="#cite_note-GrundyPaper-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the purposes of the Sprague–Grundy theorem, a <i><b>game</b></i> is a two-player <a href="/wiki/Sequential_game" title="Sequential game">sequential game</a> of <a href="/wiki/Perfect_information" title="Perfect information">perfect information</a> satisfying the <i>ending condition</i> (all games come to an end: there are no infinite lines of play) and the <i>normal play condition</i> (a player who cannot move loses). </p><p>At any given point in the game, a player's <i><b>position</b></i> is the set of <i><b>moves</b></i> they are allowed to make. As an example, we can define the <i>zero game</i> to be the two-player game where neither player has any legal moves. Referring to the two players as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> (for Alice) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> (for Bob), we would denote their positions as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,B)=(\{\},\{\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A,B)=(\{\},\{\})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21402c32f65ffc1026afd4f270e888cfc77ee7d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.942ex; height:2.843ex;" alt="{\displaystyle (A,B)=(\{\},\{\})}"></span>, since the set of moves each player can make is empty. </p><p>An <i><b><a href="/wiki/Impartial_game" title="Impartial game">impartial game</a></b></i> is one in which at any given point in the game, each player is allowed exactly the same set of moves. Normal-play <a href="/wiki/Nim" title="Nim">nim</a> is an example of an impartial game. In nim, there are one or more heaps of objects, and two players (we'll call them Alice and Bob), take turns choosing a heap and removing 1 or more objects from it. The winner is the player who removes the final object from the final heap. The game is <i><b>impartial</b></i> because for any given configuration of pile sizes, the moves Alice can make on her turn are exactly the same moves Bob would be allowed to make if it were his turn. In contrast, a game such as <a href="/wiki/Checkers" title="Checkers">checkers</a> is not impartial because, supposing Alice were playing red and Bob were playing black, for any given arrangement of pieces on the board, if it were Alice's turn, she would only be allowed to move the red pieces, and if it were Bob's turn, he would only be allowed to move the black pieces. </p><p>Note that any configuration of an impartial game can therefore be written as a single position, because the moves will be the same no matter whose turn it is. For example, the position of the <i>zero game</i> can simply be written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}"></span>, because if it's Alice's turn, she has no moves to make, and if it's Bob's turn, he has no moves to make either. A move can be associated with the position it leaves the next player in. </p><p>Doing so allows positions to be defined recursively. For example, consider the following game of Nim played by Alice and Bob. </p> <div class="mw-heading mw-heading3"><h3 id="Example_Nim_Game"><span class="anchor" id="firstExample"></span>Example Nim Game</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=2" title="Edit section: Example Nim Game"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1195917819">.mw-parser-output .pre-borderless{border:none}</style><pre class="pre">Sizes of heaps Moves A B C   1 2 2 Alice takes 1 from A 0 2 2 Bob takes 1 from B 0 1 2 Alice takes 1 from C 0 1 1 Bob takes 1 from B 0 0 1 Alice takes 1 from C 0 0 0 Bob has no moves, so Alice wins</pre> <ul><li>At step 6 of the game (when all of the heaps are empty) the position is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}"></span>, because Bob has no valid moves to make. We name this position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caad18176de507e1ea2661b225b329ff2676f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *0}"></span>.</li> <li>At step 5, Alice had exactly one option: to remove one object from heap C, leaving Bob with no moves. Since her <i>move</i> leaves Bob in position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caad18176de507e1ea2661b225b329ff2676f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *0}"></span>, her <i>position</i> is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{*0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{*0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9478d62614ddfaed7764935899a4dc9b97c839ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \{*0\}}"></span>. We name this position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aa42130e7f99f9bd7a314a8fb5331a061d456f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *1}"></span>.</li> <li>At step 4, Bob had two options: remove one from B or remove one from C. Note, however, that it didn't really matter which heap Bob removed the object from: Either way, Alice would be left with exactly one object in exactly one pile. So, using our recursive definition, Bob really only has one move: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aa42130e7f99f9bd7a314a8fb5331a061d456f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *1}"></span>. Thus, Bob's position is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{*1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{*1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28188735abb6dc93cfdb5f8eb9b02e41d57fad2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \{*1\}}"></span>.</li> <li>At step 3, Alice had 3 options: remove two from C, remove one from C, or remove one from B. Removing two from C leaves Bob in position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aa42130e7f99f9bd7a314a8fb5331a061d456f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *1}"></span>. Removing one from C leaves Bob with two piles, each of size one, i.e., position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{*1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{*1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28188735abb6dc93cfdb5f8eb9b02e41d57fad2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \{*1\}}"></span>, as described in step 4. However, removing 1 from B would leave Bob with two objects in a single pile. <i>His</i> moves would then be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caad18176de507e1ea2661b225b329ff2676f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aa42130e7f99f9bd7a314a8fb5331a061d456f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *1}"></span>, so <i>her</i> move would result in the position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{*0,*1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>0</mn> <mo>,</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{*0,*1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a296243c83c3b0f0c2e0b1905232a550cd104ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.009ex; height:2.843ex;" alt="{\displaystyle \{*0,*1\}}"></span>. We call this position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6121f423a7b99b6ceefc8ae2f3b6762783ea1cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *2}"></span>. Alice's position is then the set of all her moves: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big \{}*1,\{*1\},*2{\big \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big \{}*1,\{*1\},*2{\big \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2dd8443e159bb04585983b5ad9835d5561b3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.11ex; height:3.176ex;" alt="{\displaystyle {\big \{}*1,\{*1\},*2{\big \}}}"></span>.</li> <li>Following the same recursive logic, at step 2, Bob's position is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big \{}\{*1,\{*1\},*2\},*2{\big \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big \{}\{*1,\{*1\},*2\},*2{\big \}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52909a36652142d579d6a2628772e6c5fab1b87f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.409ex; height:3.176ex;" alt="{\displaystyle {\big \{}\{*1,\{*1\},*2\},*2{\big \}}.}"></span></li> <li>Finally, at step 1, Alice's position is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo>∗</mo> <mn>2</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ea9b844d8d3f921d424ef47a80cd9602649816" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:71.815ex; height:4.843ex;" alt="{\displaystyle {\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Nimbers">Nimbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=3" title="Edit section: Nimbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The special names <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caad18176de507e1ea2661b225b329ff2676f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aa42130e7f99f9bd7a314a8fb5331a061d456f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *1}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6121f423a7b99b6ceefc8ae2f3b6762783ea1cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *2}"></span> referenced in our example game are called <i><b><a href="/wiki/Nimber" title="Nimber">nimbers</a></b></i>. In general, the nimber <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e63d2a1bafd7f5391afce8819255d0120193263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:1.676ex;" alt="{\displaystyle *n}"></span> corresponds to the position in a game of nim where there are exactly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> objects in exactly one heap. Formally, nimbers are defined inductively as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caad18176de507e1ea2661b225b329ff2676f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *0}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1=\{*0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1=\{*0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a835a95eb3da50cc1cf4eeb4ff7aa6509af6c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.073ex; height:2.843ex;" alt="{\displaystyle *1=\{*0\}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *2=\{*0,*1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>2</mn> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>0</mn> <mo>,</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *2=\{*0,*1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44edba79f69fd42fe502b5a1b5781331f47f8d17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle *2=\{*0,*1\}}"></span> and for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *(n+1)=*n\cup \{*n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∗</mo> <mi>n</mi> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *(n+1)=*n\cup \{*n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ba53e58d6cc71e38d135c2f691d67e5f9c512b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.49ex; height:2.843ex;" alt="{\displaystyle *(n+1)=*n\cup \{*n\}}"></span>. </p><p>While the word <i>nim</i>ber comes from the game <i>nim</i>, nimbers can be used to describe the positions of any finite, impartial game, and in fact, the Sprague–Grundy theorem states that every instance of a finite, impartial game can be associated with a <i>single</i> nimber. </p> <div class="mw-heading mw-heading3"><h3 id="Combining_Games">Combining Games</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=4" title="Edit section: Combining Games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two games can be combined by <i><b>adding</b></i> their positions together. For example, consider another game of nim with heaps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f732043ebc78ca911b9d7801fd8787b1291b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.509ex;" alt="{\displaystyle C'}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Example_Game_2"><span class="anchor" id="secondExample"></span>Example Game 2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=5" title="Edit section: Example Game 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1195917819"><pre class="pre">Sizes of heaps Moves   A' B' C' 1 1 1 Alice takes 1 from A' 0 1 1 Bob takes one from B' 0 0 1 Alice takes one from C' 0 0 0 Bob has no moves, so Alice wins.</pre> <p>We can combine it with our <a href="#firstExample">first example</a> to get a combined game with six heaps: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f732043ebc78ca911b9d7801fd8787b1291b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.509ex;" alt="{\displaystyle C'}"></span>: </p> <div class="mw-heading mw-heading4"><h4 id="Combined_Game"><span class="anchor" id="thirdExample"></span>Combined Game</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=6" title="Edit section: Combined Game"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1195917819"><pre class="pre">Sizes of heaps Moves A B C A' B' C'   1 2 2 1 1 1 Alice takes 1 from A 0 2 2 1 1 1 Bob takes 1 from A' 0 2 2 0 1 1 Alice takes 1 from B' 0 2 2 0 0 1 Bob takes 1 from C' 0 2 2 0 0 0 Alice takes 2 from B 0 0 2 0 0 0 Bob takes 2 from C 0 0 0 0 0 0 Alice has no moves, so Bob wins.</pre> <p>To differentiate between the two games, for the <a href="#firstExample">first example game</a>, we'll label its starting position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {blue}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="#0000ff"> <mi>S</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {blue}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb043eb0d2c3a317cf75f7dc33da30b842f4f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle \color {blue}S}"></span>, and color it blue: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {blue}S={\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="blue"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo>∗</mo> <mn>2</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {blue}S={\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/230bb092f414121da139de9284868cecfbe7aeca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:75.766ex; height:4.843ex;" alt="{\displaystyle \color {blue}S={\Big \{}{\big \{}*1,\{*1\},*2{\big \}},{\big \{}*2,\{*1,\{*1\},*2\}{\big \}},{\big \{}\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}{\big \}}{\Big \}}}"></span> </p><p>For the <a href="#secondExample">second example game</a>, we'll label the starting position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {red}S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {red}S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d916744c8c488bdfb3a17ea30c73664cd4d504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle \color {red}S'}"></span> and color it red: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {red}S'={\Big \{}\{*1\}{\Big \}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {red}S'={\Big \{}\{*1\}{\Big \}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07aad73000d5884776caacf83a722e2c0a163b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.702ex; height:4.843ex;" alt="{\displaystyle \color {red}S'={\Big \{}\{*1\}{\Big \}}.}"></span> </p><p>To compute the starting position of the <a href="#thirdExample">combined game</a>, remember that a player can either make a move in the first game, leaving the second game untouched, or make a move in the second game, leaving the first game untouched. So the combined game's starting position is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {blue}S\color {black}+\color {red}S'\color {black}={\Big \{}\color {blue}S\color {black}+\color {red}\{*1\}\color {black}{\Big \}}\cup {\Big \{}\color {red}S'\color {black}+\color {blue}\{*1,\{*1\},*2\}\color {black},\color {red}S'\color {black}+\color {blue}\{*2,\{*1,\{*1\},*2\}\}\color {black},\color {red}S'\color {black}+\color {blue}\{\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}\}\color {black}{\Big \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="blue"> <mi>S</mi> <mstyle mathcolor="black"> <mo>+</mo> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mstyle mathcolor="black"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mstyle mathcolor="blue"> <mi>S</mi> <mstyle mathcolor="black"> <mo>+</mo> <mstyle mathcolor="red"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mstyle mathcolor="black"> <mo>+</mo> <mstyle mathcolor="blue"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mstyle mathcolor="black"> <mo>,</mo> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mstyle mathcolor="black"> <mo>+</mo> <mstyle mathcolor="blue"> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>2</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mstyle mathcolor="black"> <mo>,</mo> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> <mstyle mathcolor="black"> <mo>+</mo> <mstyle mathcolor="blue"> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>∗</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {blue}S\color {black}+\color {red}S'\color {black}={\Big \{}\color {blue}S\color {black}+\color {red}\{*1\}\color {black}{\Big \}}\cup {\Big \{}\color {red}S'\color {black}+\color {blue}\{*1,\{*1\},*2\}\color {black},\color {red}S'\color {black}+\color {blue}\{*2,\{*1,\{*1\},*2\}\}\color {black},\color {red}S'\color {black}+\color {blue}\{\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}\}\color {black}{\Big \}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83299edbc19be65e46e0072ee7067e7e140bd819" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:107.404ex; height:4.843ex;" alt="{\displaystyle \color {blue}S\color {black}+\color {red}S'\color {black}={\Big \{}\color {blue}S\color {black}+\color {red}\{*1\}\color {black}{\Big \}}\cup {\Big \{}\color {red}S'\color {black}+\color {blue}\{*1,\{*1\},*2\}\color {black},\color {red}S'\color {black}+\color {blue}\{*2,\{*1,\{*1\},*2\}\}\color {black},\color {red}S'\color {black}+\color {blue}\{\{*1\},\{\{*1\}\},\{*1,\{*1\},*2\}\}\color {black}{\Big \}}}"></span> </p><p>The explicit formula for adding positions is: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+S'=\{S+s'\mid s'\in S'\}\cup \{s+S'\mid s\in S\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>+</mo> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>S</mi> <mo>+</mo> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>∣</mo> <msup> <mi>s</mi> <mo>′</mo> </msup> <mo>∈</mo> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mi>s</mi> <mo>+</mo> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>∣</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S+S'=\{S+s'\mid s'\in S'\}\cup \{s+S'\mid s\in S\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea94b089fe8676cab07b33d6aa3e9afae7f49de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.256ex; height:3.009ex;" alt="{\displaystyle S+S'=\{S+s'\mid s'\in S'\}\cup \{s+S'\mid s\in S\}}"></span>, which means that addition is both commutative and associative. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalence">Equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=7" title="Edit section: Equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Positions in impartial games fall into two <i><b>outcome classes</b></i>: either the next player (the one whose turn it is) wins (an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mathcal {N}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic-bold" mathvariant="bold-script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mathcal {N}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab373433e001cf268863259fcd395f83378701e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.055ex; width:2.622ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\mathcal {N}}}}"></span><i><b>- position</b></i>), or the previous player wins (a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mathcal {P}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic-bold" mathvariant="bold-script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mathcal {P}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de1f4df116dd8cab41805d45e241dd57fb85b639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.966ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\mathcal {P}}}}"></span><i><b>- position</b></i>). So, for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caad18176de507e1ea2661b225b329ff2676f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *0}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position, while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aa42130e7f99f9bd7a314a8fb5331a061d456f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle *1}"></span> is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>-position. </p><p>Two positions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span> are <i><b>equivalent</b></i> if, no matter what position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is added to them, they are always in the same outcome class. Formally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563df25fb28f8751d25d14ba7a07b251b7edd73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.356ex; height:2.176ex;" alt="{\displaystyle \forall H}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> is in the same outcome class as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf44881c004b73802103537e49bc332622f2a5ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.415ex; height:2.676ex;" alt="{\displaystyle G'+H}"></span>. </p><p>To use our running examples, notice that in both the <a href="#firstExample">first</a> and <a href="#secondExample">second</a> games above, we can show that on every turn, Alice has a move that forces Bob into a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. Thus, both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {blue}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="#0000ff"> <mi>S</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {blue}S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb043eb0d2c3a317cf75f7dc33da30b842f4f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle \color {blue}S}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {red}S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {red}S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d916744c8c488bdfb3a17ea30c73664cd4d504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle \color {red}S'}"></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>-positions. (Notice that in the combined game, <i>Bob</i> is the player with the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>-positions. In fact, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {blue}S\color {black}+\color {red}S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="#0000ff"> <mi>S</mi> <mstyle mathcolor="black"> <mo>+</mo> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {blue}S\color {black}+\color {red}S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d3a9971a99a310a5bd1c185dc3417f1050b16ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.546ex; height:2.676ex;" alt="{\displaystyle \color {blue}S\color {black}+\color {red}S'}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position, which as we will see in Lemma 2, means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {blue}S\color {black}\approx \color {red}S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="#0000ff"> <mi>S</mi> <mstyle mathcolor="black"> <mo>≈</mo> <mstyle mathcolor="red"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {blue}S\color {black}\approx \color {red}S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491785241152e4e01973e530879b4115d1916536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.804ex; height:2.509ex;" alt="{\displaystyle \color {blue}S\color {black}\approx \color {red}S'}"></span>.) </p> <div class="mw-heading mw-heading2"><h2 id="First_Lemma">First Lemma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=8" title="Edit section: First Lemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As an intermediate step to proving the main theorem, we show that for every position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> and every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, the equivalence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx A+G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <mi>A</mi> <mo>+</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx A+G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94de9d8ca602017e0b3cf3f110e1b461253f829b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.335ex; height:2.343ex;" alt="{\displaystyle G\approx A+G}"></span> holds. By the above definition of equivalence, this amounts to showing that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef00aa87430bb04d85a33985d998e802e5a5515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.314ex; height:2.343ex;" alt="{\displaystyle A+G+H}"></span> share an outcome class for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. </p><p>Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. Then the previous player has a winning strategy for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef00aa87430bb04d85a33985d998e802e5a5515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.314ex; height:2.343ex;" alt="{\displaystyle A+G+H}"></span>: respond to moves in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> according to their winning strategy for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> (which exists by virtue of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> being a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position), and respond to moves in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> according to their winning strategy for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> (which exists for the analogous reason). So <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef00aa87430bb04d85a33985d998e802e5a5515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.314ex; height:2.343ex;" alt="{\displaystyle A+G+H}"></span> must also be a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. </p><p>On the other hand, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> is an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>-position, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef00aa87430bb04d85a33985d998e802e5a5515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.314ex; height:2.343ex;" alt="{\displaystyle A+G+H}"></span> is also an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>-position, because the next player has a winning strategy: choose a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position from among the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> options, and we conclude from the previous paragraph that adding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> to that position is still a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. Thus, in this case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef00aa87430bb04d85a33985d998e802e5a5515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.314ex; height:2.343ex;" alt="{\displaystyle A+G+H}"></span> must be a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>-position, just like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span>. </p><p>As these are the only two cases, the lemma holds. </p> <div class="mw-heading mw-heading2"><h2 id="Second_Lemma">Second Lemma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=9" title="Edit section: Second Lemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a further step, we show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span> if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6acd5a00bcd223d227542e4ccaa022d0f5ed83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.179ex; height:2.676ex;" alt="{\displaystyle G+G'}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. </p><p>In the forward direction, suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span>. Applying the definition of equivalence with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d695258de13213a2c8c4a63706e2d0a74ef27f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.989ex; height:2.176ex;" alt="{\displaystyle H=G}"></span>, we find that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'+G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'+G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdfb1d308adfd943ae33889671af513b8d70d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.179ex; height:2.676ex;" alt="{\displaystyle G'+G}"></span> (which is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6acd5a00bcd223d227542e4ccaa022d0f5ed83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.179ex; height:2.676ex;" alt="{\displaystyle G+G'}"></span> by <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> of addition) is in the same outcome class as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd1852db34f2ccb88786676cc7c437d808bd486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.494ex; height:2.343ex;" alt="{\displaystyle G+G}"></span>. But <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd1852db34f2ccb88786676cc7c437d808bd486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.494ex; height:2.343ex;" alt="{\displaystyle G+G}"></span> must be a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position: for every move made in one copy of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, the previous player can respond with the same move in the other copy, and so always make the last move. </p><p>In the reverse direction, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=G+G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>G</mi> <mo>+</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=G+G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61588d745bc1dead05abf40d8b74e613f1e571a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.02ex; height:2.676ex;" alt="{\displaystyle A=G+G'}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position by hypothesis, it follows from the first lemma, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G+A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <mi>G</mi> <mo>+</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G+A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7650e163354ff44cb742aa42fcc03a13ec96983a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.335ex; height:2.343ex;" alt="{\displaystyle G\approx G+A}"></span>, that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G+(G+G')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <mi>G</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo>+</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G+(G+G')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262785661938b919e44bfa8b24f7b777d99da794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.58ex; height:3.009ex;" alt="{\displaystyle G\approx G+(G+G')}"></span>. Similarly, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=G+G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>G</mi> <mo>+</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=G+G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d98cdfcb3e2e895743743ef72f72cb1470fb807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.356ex; height:2.343ex;" alt="{\displaystyle B=G+G}"></span> is also a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position, it follows from the first lemma in the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'\approx G'+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'\approx G'+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf82b3d1b45cbea76c69b3f702184c0b3a70f621" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.726ex; height:2.676ex;" alt="{\displaystyle G'\approx G'+B}"></span> that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'\approx G'+(G+G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo>+</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'\approx G'+(G+G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72d8fac48db04b537198666d84d30c4c1a4050a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.265ex; height:3.009ex;" alt="{\displaystyle G'\approx G'+(G+G)}"></span>. By <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> and commutativity, the right-hand sides of these results are equal. Furthermore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span> is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> because equality is an equivalence relation on outcome classes. Via the <a href="/wiki/Transitive_relation" title="Transitive relation">transitivity</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span>, we can conclude that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=10" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We prove that all positions are equivalent to a nimber by <a href="/wiki/Structural_induction" title="Structural induction">structural induction</a>. The more specific result, that the given game's initial position must be equivalent to a nimber, shows that the game is itself equivalent to a nimber. </p><p>Consider a position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\{G_{1},G_{2},\ldots ,G_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\{G_{1},G_{2},\ldots ,G_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12951ed39b979d33537691cfeccf1ab5dda157a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.14ex; height:2.843ex;" alt="{\displaystyle G=\{G_{1},G_{2},\ldots ,G_{k}\}}"></span>. By the <a href="/wiki/Mathematical_induction" title="Mathematical induction"> induction hypothesis</a>, all of the options are equivalent to nimbers, say <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{i}\approx *n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≈</mo> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{i}\approx *n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe3fcca07410d0271dd5769d55f3b6cdd06b345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.082ex; height:2.509ex;" alt="{\displaystyle G_{i}\approx *n_{i}}"></span>. So let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'=\{*n_{1},*n_{2},\ldots ,*n_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'=\{*n_{1},*n_{2},\ldots ,*n_{k}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72be32d3ddee85932dcf2cd9f693fdd591704e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.016ex; height:3.009ex;" alt="{\displaystyle G'=\{*n_{1},*n_{2},\ldots ,*n_{k}\}}"></span>. We will show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f3dd239f8c14409a47c0f77b49659ad5b98686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.128ex; height:2.176ex;" alt="{\displaystyle G\approx *m}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the <a href="/wiki/Mex_(mathematics)" title="Mex (mathematics)">mex (minimum exclusion)</a> of the numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1},n_{2},\ldots ,n_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1},n_{2},\ldots ,n_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589b9d5e1bc9bdaf5cb775e046db6d2a11b79ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.594ex; height:2.009ex;" alt="{\displaystyle n_{1},n_{2},\ldots ,n_{k}}"></span>, that is, the smallest non-negative integer not equal to some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}"></span>. </p><p>The first thing we need to note is that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span>, by way of the second lemma. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is zero, the claim is trivially true. Otherwise, consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6acd5a00bcd223d227542e4ccaa022d0f5ed83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.179ex; height:2.676ex;" alt="{\displaystyle G+G'}"></span>. If the next player makes a move to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd9fe8d455762608cc4e0a946b452492790ee5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.626ex; height:2.509ex;" alt="{\displaystyle G_{i}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, then the previous player can move to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0460e7c68434057e52aafa0493eb6969db08493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.357ex; height:2.009ex;" alt="{\displaystyle *n_{i}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span>, and conversely if the next player makes a move in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span>. After this, the position is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position by the lemma's forward implication. Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6acd5a00bcd223d227542e4ccaa022d0f5ed83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.179ex; height:2.676ex;" alt="{\displaystyle G+G'}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position, and, citing the lemma's reverse implication, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span>. </p><p>Now let us show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'+*m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>+</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'+*m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f105777c3ece61bdba7686fd982c7ef5dc851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.555ex; height:2.676ex;" alt="{\displaystyle G'+*m}"></span> is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position, which, using the second lemma once again, means that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'\approx *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>≈</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'\approx *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c58f842933ffd746d179f518e5081b23a6be34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.813ex; height:2.509ex;" alt="{\displaystyle G'\approx *m}"></span>. We do so by giving an explicit strategy for the previous player. </p><p>Suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cd1da000e968d7e519e10468c96d77b9b82fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:1.676ex;" alt="{\displaystyle *m}"></span> are empty. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'+*m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>+</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'+*m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f105777c3ece61bdba7686fd982c7ef5dc851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.555ex; height:2.676ex;" alt="{\displaystyle G'+*m}"></span> is the null set, clearly a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. </p><p>Or consider the case that the next player moves in the component <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cd1da000e968d7e519e10468c96d77b9b82fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:1.676ex;" alt="{\displaystyle *m}"></span> to the option <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df154ee008843a7d415f66e0ba23e1b4bcea44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.888ex; height:2.509ex;" alt="{\displaystyle *m'}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m'<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m'<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b938155755d2a56cb5b1a9531ce0036fffa74796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.864ex; height:2.509ex;" alt="{\displaystyle m'<m}"></span>. Because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> was the <i>minimum</i> excluded number, the previous player can move in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df154ee008843a7d415f66e0ba23e1b4bcea44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.888ex; height:2.509ex;" alt="{\displaystyle *m'}"></span>. And, as shown before, any position plus itself is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position. </p><p>Finally, suppose instead that the next player moves in the component <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span> to the option <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0460e7c68434057e52aafa0493eb6969db08493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.357ex; height:2.009ex;" alt="{\displaystyle *n_{i}}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f8340bf3eac493e1ddd8a37f6a817a165767b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.333ex; height:2.176ex;" alt="{\displaystyle n_{i}<m}"></span> then the previous player moves in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cd1da000e968d7e519e10468c96d77b9b82fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:1.676ex;" alt="{\displaystyle *m}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0460e7c68434057e52aafa0493eb6969db08493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.357ex; height:2.009ex;" alt="{\displaystyle *n_{i}}"></span>; otherwise, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}>m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>></mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}>m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24371da867045d9d431118f514019d9afb4e9806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.333ex; height:2.176ex;" alt="{\displaystyle n_{i}>m}"></span>, the previous player moves in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0460e7c68434057e52aafa0493eb6969db08493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.357ex; height:2.009ex;" alt="{\displaystyle *n_{i}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cd1da000e968d7e519e10468c96d77b9b82fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:1.676ex;" alt="{\displaystyle *m}"></span>; in either case the result is a position plus itself. (It is not possible that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}=m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}=m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17580b83ca50797f99491384015a807128294c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.333ex; height:2.009ex;" alt="{\displaystyle n_{i}=m}"></span> because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> was defined to be different from all the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}"></span>.) </p><p>In summary, we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569a9075aa4456920ab50421a73530e1234b77d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle G\approx G'}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'\approx *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>≈</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'\approx *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c58f842933ffd746d179f518e5081b23a6be34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.813ex; height:2.509ex;" alt="{\displaystyle G'\approx *m}"></span>. By transitivity, we conclude that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f3dd239f8c14409a47c0f77b49659ad5b98686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.128ex; height:2.176ex;" alt="{\displaystyle G\approx *m}"></span>, as desired. </p> <div class="mw-heading mw-heading2"><h2 id="Development">Development</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=11" title="Edit section: Development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is a position of an impartial game, the unique integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\approx *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≈</mo> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\approx *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f3dd239f8c14409a47c0f77b49659ad5b98686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.128ex; height:2.176ex;" alt="{\displaystyle G\approx *m}"></span> is called its Grundy value, or Grundy number, and the function that assigns this value to each such position is called the Sprague–Grundy function. R. L. Sprague and P. M. Grundy independently gave an explicit definition of this function, not based on any concept of equivalence to nim positions, and showed that it had the following properties: </p> <ul><li>The Grundy value of a single nim pile of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> (i.e. of the position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cd1da000e968d7e519e10468c96d77b9b82fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:1.676ex;" alt="{\displaystyle *m}"></span>) is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>;</li> <li>A position is a loss for the next player to move (i.e. a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>-position) if and only if its Grundy value is zero; and</li> <li>The Grundy value of the sum of a finite set of positions is just the <a href="/wiki/Nim-sum" class="mw-redirect" title="Nim-sum">nim-sum</a> of the Grundy values of its summands.</li></ul> <p>It follows straightforwardly from these results that if a position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> has a Grundy value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6f7af1f8e569c854fdb3f7011d441bc4ffbcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.731ex; height:2.343ex;" alt="{\displaystyle G+H}"></span> has the same Grundy value as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *m+H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> <mi>m</mi> <mo>+</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *m+H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d72ebeb4814249a8d9e0fbef080f4c351b34d5b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.107ex; height:2.343ex;" alt="{\displaystyle *m+H}"></span>, and therefore belongs to the same outcome class, for any position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. Thus, although Sprague and Grundy never explicitly stated the theorem described in this article, it follows directly from their results and is credited to them.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> These results have subsequently been developed into the field of <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">combinatorial game theory</a>, notably by <a href="/wiki/Richard_K._Guy" title="Richard K. Guy">Richard Guy</a>, <a href="/wiki/E._R._Berlekamp" class="mw-redirect" title="E. R. Berlekamp">Elwyn Berlekamp</a>, <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Horton Conway</a> and others, where they are now encapsulated in the Sprague–Grundy theorem and its proof in the form described here. The field is presented in the books <i><a href="/wiki/Winning_Ways_for_your_Mathematical_Plays" class="mw-redirect" title="Winning Ways for your Mathematical Plays">Winning Ways for your Mathematical Plays</a></i> and <i><a href="/wiki/On_Numbers_and_Games" title="On Numbers and Games">On Numbers and Games</a></i>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Genus_theory" title="Genus theory">Genus theory</a></li> <li><a href="/wiki/Indistinguishability_quotient" title="Indistinguishability quotient">Indistinguishability quotient</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-SpraguePaper-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-SpraguePaper_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSprague1936" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Roland_Sprague" title="Roland Sprague">Sprague, R. P.</a> (1936). <a rel="nofollow" class="external text" href="http://www.jstage.jst.go.jp/article/tmj1911/41/0/41_0_438/_article">"Über mathematische Kampfspiele"</a>. <i><a href="/wiki/Tohoku_Mathematical_Journal" title="Tohoku Mathematical Journal">Tohoku Mathematical Journal</a></i> (in German). <b>41</b>: 438–444. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:62.1070.03">62.1070.03</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0013.29004">0013.29004</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Tohoku+Mathematical+Journal&rft.atitle=%C3%9Cber+mathematische+Kampfspiele&rft.volume=41&rft.pages=438-444&rft.date=1936&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0013.29004%23id-name%3DZbl&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A62.1070.03%23id-name%3DJFM&rft.aulast=Sprague&rft.aufirst=R.+P.&rft_id=http%3A%2F%2Fwww.jstage.jst.go.jp%2Farticle%2Ftmj1911%2F41%2F0%2F41_0_438%2F_article&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASprague%E2%80%93Grundy+theorem" class="Z3988"></span></span> </li> <li id="cite_note-GrundyPaper-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-GrundyPaper_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrundy1939" class="citation journal cs1"><a href="/wiki/Patrick_Michael_Grundy" title="Patrick Michael Grundy">Grundy, P. M.</a> (1939). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070927192024/http://www.archim.org.uk/eureka/27/games.html">"Mathematics and games"</a>. <i><a href="/wiki/Eureka_(University_of_Cambridge_magazine)" title="Eureka (University of Cambridge magazine)">Eureka</a></i>. <b>2</b>: 6–8. Archived from <a rel="nofollow" class="external text" href="http://www.archim.org.uk/eureka/27/games.html">the original</a> on 2007-09-27.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Eureka&rft.atitle=Mathematics+and+games&rft.volume=2&rft.pages=6-8&rft.date=1939&rft.aulast=Grundy&rft.aufirst=P.+M.&rft_id=http%3A%2F%2Fwww.archim.org.uk%2Feureka%2F27%2Fgames.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASprague%E2%80%93Grundy+theorem" class="Z3988"></span> Reprinted, 1964, <b>27</b>: 9–11.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1960" class="citation cs2">Smith, Cedric A.B. (1960), "Patrick Michael Grundy, 1917–1959", <i>Journal of the Royal Statistical Society, Series A</i>, <b>123</b> (2): 221–22</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Royal+Statistical+Society%2C+Series+A&rft.atitle=Patrick+Michael+Grundy%2C+1917%E2%80%931959&rft.volume=123&rft.issue=2&rft.pages=221-22&rft.date=1960&rft.aulast=Smith&rft.aufirst=Cedric+A.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASprague%E2%80%93Grundy+theorem" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchleicher,_DierkStoll,_Michael2006" class="citation journal cs1">Schleicher, Dierk; Stoll, Michael (2006). "An introduction to Conway's games and numbers". <i>Moscow Mathematical Journal</i>. <b>6</b> (2): 359–388. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.CO/0410026">math.CO/0410026</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17323%2F1609-4514-2006-6-2-359-388">10.17323/1609-4514-2006-6-2-359-388</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7175146">7175146</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Moscow+Mathematical+Journal&rft.atitle=An+introduction+to+Conway%27s+games+and+numbers&rft.volume=6&rft.issue=2&rft.pages=359-388&rft.date=2006&rft_id=info%3Aarxiv%2Fmath.CO%2F0410026&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7175146%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.17323%2F1609-4514-2006-6-2-359-388&rft.au=Schleicher%2C+Dierk&rft.au=Stoll%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASprague%E2%80%93Grundy+theorem" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sprague%E2%80%93Grundy_theorem&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Games/Grundy.shtml">Grundy's game</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li> <li><a rel="nofollow" class="external text" href="https://www.math.ucla.edu/~tom/Game_Theory/comb.pdf">Easily readable, introductory account from the UCLA Math Department</a></li> <li><a rel="nofollow" class="external text" href="http://sputsoft.com/blog/2009/04/the-game-of-nim.html">The Game of Nim</a> at <a rel="nofollow" class="external text" href="http://sputsoft.com">sputsoft.com</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilvang-Jensen,_Brit_C._A.2000" class="citation cs2">Milvang-Jensen, Brit C. A. (2000), <a rel="nofollow" class="external text" href="http://www.itu.dk/people/brit/Brits%20thesis.pdf"><i>Combinatorial Games, Theory and Applications</i></a> <span class="cs1-format">(PDF)</span>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.89.805">10.1.1.89.805</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+Games%2C+Theory+and+Applications&rft.date=2000&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.89.805%23id-name%3DCiteSeerX&rft.au=Milvang-Jensen%2C+Brit+C.+A.&rft_id=http%3A%2F%2Fwww.itu.dk%2Fpeople%2Fbrit%2FBrits%2520thesis.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASprague%E2%80%93Grundy+theorem" class="Z3988"></span></li></ul> <div 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class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Congestion_game" title="Congestion game">Congestion game</a></li> <li><a href="/wiki/Cooperative_game_theory" title="Cooperative game theory">Cooperative game</a></li> <li><a href="/wiki/Determinacy" title="Determinacy">Determinacy</a></li> <li><a href="/wiki/Escalation_of_commitment" title="Escalation of commitment">Escalation of commitment</a></li> <li><a href="/wiki/Extensive-form_game" title="Extensive-form game">Extensive-form game</a></li> <li><a href="/wiki/First-player_and_second-player_win" title="First-player and second-player win">First-player and second-player win</a></li> <li><a href="/wiki/Game_complexity" title="Game complexity">Game complexity</a></li> <li><a href="/wiki/Graphical_game_theory" title="Graphical game theory">Graphical game</a></li> <li><a href="/wiki/Hierarchy_of_beliefs" title="Hierarchy of beliefs">Hierarchy of beliefs</a></li> <li><a href="/wiki/Information_set_(game_theory)" title="Information set (game theory)">Information set</a></li> <li><a href="/wiki/Normal-form_game" title="Normal-form game">Normal-form game</a></li> <li><a href="/wiki/Preference_(economics)" title="Preference (economics)">Preference</a></li> <li><a href="/wiki/Sequential_game" title="Sequential game">Sequential game</a></li> <li><a href="/wiki/Simultaneous_game" title="Simultaneous game">Simultaneous game</a></li> <li><a href="/wiki/Simultaneous_action_selection" title="Simultaneous action selection">Simultaneous action selection</a></li> <li><a href="/wiki/Solved_game" title="Solved game">Solved game</a></li> <li><a href="/wiki/Succinct_game" title="Succinct game">Succinct game</a></li> <li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Economic_equilibrium" title="Economic equilibrium">Equilibrium</a><br /><a href="/wiki/Solution_concept" title="Solution concept">concepts</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayes_correlated_equilibrium" title="Bayes correlated equilibrium">Bayes correlated equilibrium</a></li> <li><a href="/wiki/Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a></li> <li><a href="/wiki/Berge_equilibrium" title="Berge equilibrium">Berge equilibrium</a></li> <li><a href="/wiki/Core_(game_theory)" title="Core (game theory)"> Core</a></li> <li><a href="/wiki/Correlated_equilibrium" title="Correlated equilibrium">Correlated equilibrium</a></li> <li><a href="/wiki/Coalition-proof_Nash_equilibrium" title="Coalition-proof Nash equilibrium">Coalition-proof Nash equilibrium</a></li> <li><a href="/wiki/Epsilon-equilibrium" title="Epsilon-equilibrium">Epsilon-equilibrium</a></li> <li><a href="/wiki/Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">Evolutionarily stable strategy</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs equilibrium</a></li> <li><a href="/wiki/Mertens-stable_equilibrium" title="Mertens-stable equilibrium">Mertens-stable equilibrium</a></li> <li><a href="/wiki/Markov_perfect_equilibrium" title="Markov perfect equilibrium">Markov perfect equilibrium</a></li> <li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a></li> <li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li> <li><a href="/wiki/Perfect_Bayesian_equilibrium" title="Perfect Bayesian equilibrium">Perfect Bayesian equilibrium</a></li> <li><a href="/wiki/Proper_equilibrium" title="Proper equilibrium">Proper equilibrium</a></li> <li><a href="/wiki/Quantal_response_equilibrium" title="Quantal response equilibrium">Quantal response equilibrium</a></li> <li><a href="/wiki/Quasi-perfect_equilibrium" title="Quasi-perfect equilibrium">Quasi-perfect equilibrium</a></li> <li><a href="/wiki/Risk_dominance" title="Risk dominance">Risk dominance</a></li> <li><a href="/wiki/Satisfaction_equilibrium" title="Satisfaction equilibrium">Satisfaction equilibrium</a></li> <li><a href="/wiki/Self-confirming_equilibrium" title="Self-confirming equilibrium">Self-confirming equilibrium</a></li> <li><a href="/wiki/Sequential_equilibrium" title="Sequential equilibrium">Sequential equilibrium</a></li> <li><a href="/wiki/Shapley_value" title="Shapley value">Shapley value</a></li> <li><a href="/wiki/Strong_Nash_equilibrium" title="Strong Nash equilibrium">Strong Nash equilibrium</a></li> <li><a href="/wiki/Subgame_perfect_equilibrium" title="Subgame perfect equilibrium">Subgame perfection</a></li> <li><a href="/wiki/Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">Trembling hand equilibrium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Strategy_(game_theory)" title="Strategy (game theory)">Strategies</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Appeasement" title="Appeasement">Appeasement</a></li> <li><a href="/wiki/Backward_induction" title="Backward induction">Backward induction</a></li> <li><a href="/wiki/Bid_shading" title="Bid shading">Bid shading</a></li> <li><a href="/wiki/Collusion" title="Collusion">Collusion</a></li> <li><a href="/wiki/Cheap_talk" title="Cheap talk">Cheap talk</a></li> <li><a href="/wiki/De-escalation" title="De-escalation">De-escalation</a></li> <li><a href="/wiki/Deterrence_theory" title="Deterrence theory">Deterrence</a></li> <li><a href="/wiki/Conflict_escalation" title="Conflict escalation">Escalation</a></li> <li><a href="/wiki/Forward_induction" class="mw-redirect" title="Forward induction">Forward induction</a></li> <li><a href="/wiki/Grim_trigger" title="Grim trigger">Grim trigger</a></li> <li><a href="/wiki/Markov_strategy" title="Markov strategy">Markov strategy</a></li> <li><a href="/wiki/Pairing_strategy" title="Pairing strategy">Pairing strategy</a></li> <li><a href="/wiki/Strategic_dominance" title="Strategic dominance">Dominant strategies</a></li> <li><a href="/wiki/Strategy_(game_theory)" title="Strategy (game theory)">Pure strategy</a></li> <li><a href="/wiki/Strategy_(game_theory)#Mixed_strategy" title="Strategy (game theory)">Mixed strategy</a></li> <li><a href="/wiki/Strategy-stealing_argument" title="Strategy-stealing argument">Strategy-stealing argument</a></li> <li><a href="/wiki/Tit_for_tat" title="Tit for tat">Tit for tat</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Game_theory_game_classes" title="Category:Game theory game classes">Classes<br />of games</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Auction" title="Auction">Auction</a></li> <li><a href="/wiki/Bargaining_problem" class="mw-redirect" title="Bargaining problem">Bargaining problem</a></li> <li><a href="/wiki/Global_game" title="Global game">Global game</a></li> <li><a href="/wiki/Intransitive_game" title="Intransitive game">Intransitive game</a></li> <li><a href="/wiki/Mean-field_game_theory" title="Mean-field game theory">Mean-field game</a></li> <li><a href="/wiki/N-player_game" title="N-player game"><i>n</i>-player game</a></li> <li><a href="/wiki/Perfect_information" title="Perfect information">Perfect information</a></li> <li><a href="/wiki/Poisson_games" class="mw-redirect" title="Poisson games">Large Poisson game</a></li> <li><a href="/wiki/Potential_game" title="Potential game">Potential game</a></li> <li><a href="/wiki/Repeated_game" title="Repeated game">Repeated game</a></li> <li><a href="/wiki/Screening_game" title="Screening game">Screening game</a></li> <li><a href="/wiki/Signaling_game" title="Signaling game">Signaling game</a></li> <li><a href="/wiki/Strictly_determined_game" title="Strictly determined game">Strictly determined game</a></li> <li><a href="/wiki/Stochastic_game" title="Stochastic game">Stochastic game</a></li> <li><a href="/wiki/Symmetric_game" title="Symmetric game">Symmetric game</a></li> <li><a href="/wiki/Zero-sum_game" title="Zero-sum game">Zero-sum game</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_games_in_game_theory" title="List of games in game theory">Games</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Go_(game)" title="Go (game)">Go</a></li> <li><a href="/wiki/Chess" title="Chess">Chess</a></li> <li><a href="/wiki/Infinite_chess" title="Infinite chess">Infinite chess</a></li> <li><a href="/wiki/Draughts" class="mw-redirect" title="Draughts">Checkers</a></li> <li><a href="/wiki/All-pay_auction" title="All-pay auction">All-pay auction</a></li> <li><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a href="/wiki/Gift-exchange_game" title="Gift-exchange game">Gift-exchange game</a></li> <li><a href="/wiki/Optional_prisoner%27s_dilemma" title="Optional prisoner's dilemma">Optional prisoner's dilemma</a></li> <li><a href="/wiki/Traveler%27s_dilemma" title="Traveler's dilemma">Traveler's dilemma</a></li> <li><a href="/wiki/Coordination_game" title="Coordination game">Coordination game</a></li> <li><a href="/wiki/Chicken_(game)" title="Chicken (game)">Chicken</a></li> <li><a href="/wiki/Centipede_game" title="Centipede game">Centipede game</a></li> <li><a href="/wiki/Lewis_signaling_game" title="Lewis signaling game">Lewis signaling game</a></li> <li><a href="/wiki/Volunteer%27s_dilemma" title="Volunteer's dilemma">Volunteer's dilemma</a></li> <li><a href="/wiki/Dollar_auction" title="Dollar auction">Dollar auction</a></li> <li><a href="/wiki/Battle_of_the_sexes_(game_theory)" title="Battle of the sexes (game theory)">Battle of the sexes</a></li> <li><a href="/wiki/Stag_hunt" title="Stag hunt">Stag hunt</a></li> <li><a href="/wiki/Matching_pennies" title="Matching pennies">Matching pennies</a></li> <li><a href="/wiki/Ultimatum_game" title="Ultimatum game">Ultimatum game</a></li> <li><a href="/wiki/Electronic_mail_game" title="Electronic mail game">Electronic mail game</a></li> <li><a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a></li> <li><a href="/wiki/Pirate_game" title="Pirate game">Pirate game</a></li> <li><a href="/wiki/Dictator_game" title="Dictator game">Dictator game</a></li> <li><a href="/wiki/Public_goods_game" title="Public goods game">Public goods game</a></li> <li><a href="/wiki/Blotto_game" title="Blotto game">Blotto game</a></li> <li><a href="/wiki/War_of_attrition_(game)" title="War of attrition (game)">War of attrition</a></li> <li><a href="/wiki/El_Farol_Bar_problem" title="El Farol Bar problem">El Farol Bar problem</a></li> <li><a href="/wiki/Fair_division" title="Fair division">Fair division</a></li> <li><a href="/wiki/Fair_cake-cutting" title="Fair cake-cutting">Fair cake-cutting</a></li> <li><a href="/wiki/Bertrand_competition" title="Bertrand competition">Bertrand competition</a></li> <li><a href="/wiki/Cournot_competition" title="Cournot competition">Cournot competition</a></li> <li><a href="/wiki/Stackelberg_competition" title="Stackelberg competition">Stackelberg competition</a></li> <li><a href="/wiki/Deadlock_(game_theory)" title="Deadlock (game theory)">Deadlock</a></li> <li><a href="/wiki/Unscrupulous_diner%27s_dilemma" title="Unscrupulous diner's dilemma">Diner's dilemma</a></li> <li><a href="/wiki/Guess_2/3_of_the_average" title="Guess 2/3 of the average">Guess 2/3 of the average</a></li> <li><a href="/wiki/Kuhn_poker" title="Kuhn poker">Kuhn poker</a></li> <li><a href="/wiki/Bargaining_problem" class="mw-redirect" title="Bargaining problem">Nash bargaining game</a></li> <li><a href="/wiki/Induction_puzzles" title="Induction puzzles">Induction puzzles</a></li> <li><a href="/wiki/Dictator_game#Trust_game" title="Dictator game">Trust game</a></li> <li><a href="/wiki/Princess_and_monster_game" title="Princess and monster game">Princess and monster game</a></li> <li><a href="/wiki/Rendezvous_problem" title="Rendezvous problem">Rendezvous problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aumann%27s_agreement_theorem" title="Aumann's agreement theorem">Aumann's agreement theorem</a></li> <li><a href="/wiki/Folk_theorem_(game_theory)" title="Folk theorem (game theory)">Folk theorem</a></li> <li><a href="/wiki/Minimax" title="Minimax">Minimax theorem</a></li> <li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash's theorem</a></li> <li><a href="/wiki/Negamax" title="Negamax">Negamax theorem</a></li> <li><a href="/wiki/Purification_theorem" title="Purification theorem">Purification theorem</a></li> <li><a href="/wiki/Revelation_principle" title="Revelation principle">Revelation principle</a></li> <li><a class="mw-selflink selflink">Sprague–Grundy theorem</a></li> <li><a href="/wiki/Zermelo%27s_theorem_(game_theory)" title="Zermelo's theorem (game theory)">Zermelo's theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key<br />figures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a></li> <li><a href="/wiki/Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li> <li><a href="/wiki/Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Antoine Augustin Cournot</a></li> <li><a href="/wiki/Ariel_Rubinstein" title="Ariel Rubinstein">Ariel Rubinstein</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a></li> <li><a href="/wiki/David_K._Levine" title="David K. Levine">David K. Levine</a></li> <li><a href="/wiki/David_M._Kreps" title="David M. Kreps">David M. Kreps</a></li> <li><a href="/wiki/Donald_B._Gillies" title="Donald B. Gillies">Donald B. Gillies</a></li> <li><a href="/wiki/Drew_Fudenberg" title="Drew Fudenberg">Drew Fudenberg</a></li> <li><a href="/wiki/Eric_Maskin" title="Eric Maskin">Eric Maskin</a></li> <li><a href="/wiki/Harold_W._Kuhn" title="Harold W. Kuhn">Harold W. Kuhn</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a></li> <li><a href="/wiki/Herv%C3%A9_Moulin" title="Hervé Moulin">Hervé Moulin</a></li> <li><a href="/wiki/John_Conway" class="mw-redirect" title="John Conway">John Conway</a></li> <li><a href="/wiki/Jean_Tirole" title="Jean Tirole">Jean Tirole</a></li> <li><a href="/wiki/Jean-Fran%C3%A7ois_Mertens" title="Jean-François Mertens">Jean-François Mertens</a></li> <li><a href="/wiki/Jennifer_Tour_Chayes" title="Jennifer Tour Chayes">Jennifer Tour Chayes</a></li> <li><a href="/wiki/John_Harsanyi" title="John Harsanyi">John Harsanyi</a></li> <li><a href="/wiki/John_Maynard_Smith" title="John Maynard Smith">John Maynard Smith</a></li> <li><a href="/wiki/John_Forbes_Nash_Jr." title="John Forbes Nash Jr.">John Nash</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a></li> <li><a href="/wiki/Kenneth_Binmore" title="Kenneth Binmore">Kenneth Binmore</a></li> <li><a href="/wiki/Leonid_Hurwicz" title="Leonid Hurwicz">Leonid Hurwicz</a></li> <li><a href="/wiki/Lloyd_Shapley" title="Lloyd Shapley">Lloyd Shapley</a></li> <li><a href="/wiki/Melvin_Dresher" title="Melvin Dresher">Melvin Dresher</a></li> <li><a href="/wiki/Merrill_M._Flood" title="Merrill M. Flood">Merrill M. Flood</a></li> <li><a href="/wiki/Olga_Bondareva" title="Olga Bondareva">Olga Bondareva</a></li> <li><a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a></li> <li><a href="/wiki/Paul_Milgrom" title="Paul Milgrom">Paul Milgrom</a></li> <li><a href="/wiki/Peyton_Young" title="Peyton Young">Peyton Young</a></li> <li><a href="/wiki/Reinhard_Selten" title="Reinhard Selten">Reinhard Selten</a></li> <li><a href="/wiki/Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li> <li><a href="/wiki/Robert_Aumann" title="Robert Aumann">Robert Aumann</a></li> <li><a href="/wiki/Robert_B._Wilson" title="Robert B. Wilson">Robert B. Wilson</a></li> <li><a href="/wiki/Roger_Myerson" title="Roger Myerson">Roger Myerson</a></li> <li><a href="/wiki/Samuel_Bowles_(economist)" title="Samuel Bowles (economist)"> Samuel Bowles</a></li> <li><a href="/wiki/Suzanne_Scotchmer" title="Suzanne Scotchmer">Suzanne Scotchmer</a></li> <li><a href="/wiki/Thomas_Schelling" title="Thomas Schelling">Thomas Schelling</a></li> <li><a href="/wiki/William_Vickrey" title="William Vickrey">William Vickrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Search optimizations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li> <li><a href="/wiki/Aspiration_window" title="Aspiration window">Aspiration window</a></li> <li><a href="/wiki/Principal_variation_search" title="Principal variation search">Principal variation search</a></li> <li><a href="/wiki/Max%5En_algorithm" title="Max^n algorithm">max^n algorithm</a></li> <li><a href="/wiki/Paranoid_algorithm" title="Paranoid algorithm">Paranoid algorithm</a></li> <li><a href="/wiki/Lazy_SMP" title="Lazy SMP">Lazy SMP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li> <li><a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a></li> <li><a href="/wiki/Confrontation_analysis" title="Confrontation analysis">Confrontation analysis</a></li> <li><a href="/wiki/Coopetition" title="Coopetition">Coopetition</a></li> <li><a href="/wiki/Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a></li> <li><a href="/wiki/Glossary_of_game_theory" title="Glossary of game theory">Glossary of game theory</a></li> <li><a href="/wiki/List_of_game_theorists" title="List of game theorists">List of game theorists</a></li> <li><a href="/wiki/List_of_games_in_game_theory" title="List of games in game theory">List of games in game theory</a></li> <li><a href="/wiki/No-win_situation" title="No-win situation">No-win situation</a></li> <li><a href="/wiki/Topological_game" title="Topological game">Topological game</a></li> <li><a href="/wiki/Tragedy_of_the_commons" title="Tragedy of the commons">Tragedy of the commons</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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