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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Enumeration_of_polyominoes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enumeration_of_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Enumeration of polyominoes</span> </div> </a> <button aria-controls="toc-Enumeration_of_polyominoes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Enumeration of polyominoes subsection</span> </button> <ul id="toc-Enumeration_of_polyominoes-sublist" class="vector-toc-list"> <li id="toc-Free,_one-sided,_and_fixed_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free,_one-sided,_and_fixed_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Free, one-sided, and fixed polyominoes</span> </div> </a> <ul id="toc-Free,_one-sided,_and_fixed_polyominoes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetries_of_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetries_of_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Symmetries of polyominoes</span> </div> </a> <ul id="toc-Symmetries_of_polyominoes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithms_for_enumeration_of_fixed_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algorithms_for_enumeration_of_fixed_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Algorithms for enumeration of fixed polyominoes</span> </div> </a> <ul id="toc-Algorithms_for_enumeration_of_fixed_polyominoes-sublist" class="vector-toc-list"> <li id="toc-Inductive_algorithms" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inductive_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.1</span> <span>Inductive algorithms</span> </div> </a> <ul id="toc-Inductive_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transfer-matrix_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transfer-matrix_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.2</span> <span>Transfer-matrix method</span> </div> </a> <ul id="toc-Transfer-matrix_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Asymptotic_growth_of_the_number_of_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotic_growth_of_the_number_of_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Asymptotic growth of the number of polyominoes</span> </div> </a> <ul id="toc-Asymptotic_growth_of_the_number_of_polyominoes-sublist" class="vector-toc-list"> <li id="toc-Fixed_polyominoes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fixed_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4.1</span> <span>Fixed polyominoes</span> </div> </a> <ul id="toc-Fixed_polyominoes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_polyominoes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Free_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4.2</span> <span>Free polyominoes</span> </div> </a> <ul id="toc-Free_polyominoes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Special_classes_of_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_classes_of_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Special classes of polyominoes</span> </div> </a> <ul id="toc-Special_classes_of_polyominoes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tiling_with_polyominoes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tiling_with_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tiling with polyominoes</span> </div> </a> <button aria-controls="toc-Tiling_with_polyominoes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Tiling with polyominoes subsection</span> </button> <ul id="toc-Tiling_with_polyominoes-sublist" class="vector-toc-list"> <li id="toc-Tiling_regions_with_sets_of_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tiling_regions_with_sets_of_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Tiling regions with sets of polyominoes</span> </div> </a> <ul id="toc-Tiling_regions_with_sets_of_polyominoes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tiling_regions_with_copies_of_a_single_polyomino" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tiling_regions_with_copies_of_a_single_polyomino"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Tiling regions with copies of a single polyomino</span> </div> </a> <ul id="toc-Tiling_regions_with_copies_of_a_single_polyomino-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tiling_the_plane_with_copies_of_a_single_polyomino" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tiling_the_plane_with_copies_of_a_single_polyomino"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Tiling the plane with copies of a single polyomino</span> </div> </a> <ul id="toc-Tiling_the_plane_with_copies_of_a_single_polyomino-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tiling_a_common_figure_with_various_polyominoes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tiling_a_common_figure_with_various_polyominoes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Tiling a common figure with various polyominoes</span> </div> </a> <ul id="toc-Tiling_a_common_figure_with_various_polyominoes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polyominoes_in_puzzles_and_games" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polyominoes_in_puzzles_and_games"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Polyominoes in puzzles and games</span> </div> </a> <ul id="toc-Polyominoes_in_puzzles_and_games-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Etymology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Etymology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Etymology</span> </div> </a> <ul id="toc-Etymology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Polyomino</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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title="Polyomino – German" lang="de" hreflang="de" data-title="Polyomino" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Poliomin%C3%B3" title="Poliominó – Spanish" lang="es" hreflang="es" data-title="Poliominó" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Plurkvadrato" title="Plurkvadrato – Esperanto" lang="eo" hreflang="eo" data-title="Plurkvadrato" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polyomino" title="Polyomino – French" lang="fr" hreflang="fr" data-title="Polyomino" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%B4%EB%A6%AC%EC%98%A4%EB%AF%B8%EB%85%B8" title="폴리오미노 – Korean" lang="ko" hreflang="ko" data-title="폴리오미노" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Polimino" title="Polimino – Italian" lang="it" hreflang="it" data-title="Polimino" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%90%D7%95%D7%9E%D7%99%D7%A0%D7%95" title="פוליאומינו – Hebrew" lang="he" hreflang="he" data-title="פוליאומינו" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Polimino" title="Polimino – Latvian" lang="lv" hreflang="lv" data-title="Polimino" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Polyomino" title="Polyomino – Dutch" lang="nl" hreflang="nl" data-title="Polyomino" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9D%E3%83%AA%E3%82%AA%E3%83%9F%E3%83%8E" title="ポリオミノ – Japanese" lang="ja" hreflang="ja" data-title="ポリオミノ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Polimin%C3%B3" title="Poliminó – Portuguese" lang="pt" hreflang="pt" data-title="Poliminó" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Poliomino" title="Poliomino – Romanian" lang="ro" hreflang="ro" data-title="Poliomino" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BC%D0%B8%D0%BD%D0%BE" title="Полимино – Russian" lang="ru" hreflang="ru" data-title="Полимино" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Poliomina" title="Poliomina – Slovenian" lang="sl" hreflang="sl" data-title="Poliomina" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D1%96%D0%BC%D1%96%D0%BD%D0%BE" title="Поліміно – Ukrainian" lang="uk" hreflang="uk" data-title="Поліміно" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%9A%E6%A0%BC%E9%AA%A8%E7%89%8C" title="多格骨牌 – Chinese" lang="zh" hreflang="zh" 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free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Geometric shapes formed from squares</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Polyominoes" redirects here. For the book by Solomon Golomb, see <a href="/wiki/Polyominoes:_Puzzles,_Patterns,_Problems,_and_Packings" title="Polyominoes: Puzzles, Patterns, Problems, and Packings">Polyominoes: Puzzles, Patterns, Problems, and Packings</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:All_18_Pentominoes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/All_18_Pentominoes.svg/220px-All_18_Pentominoes.svg.png" decoding="async" width="220" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/All_18_Pentominoes.svg/330px-All_18_Pentominoes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/All_18_Pentominoes.svg/440px-All_18_Pentominoes.svg.png 2x" data-file-width="850" data-file-height="300" /></a><figcaption>The 18 one-sided <a href="/wiki/Pentomino" title="Pentomino">pentominoes</a>, including 6 mirrored pairs.</figcaption></figure> <p>A <b>polyomino</b> is a <a href="/wiki/Shape" title="Shape">plane geometric figure</a> formed by joining one or more equal <a href="/wiki/Square" title="Square">squares</a> edge to edge. It is a <a href="/wiki/Polyform" title="Polyform">polyform</a> whose cells are squares. It may be regarded as a finite <a href="/wiki/Subset" title="Subset">subset</a> of the regular <a href="/wiki/Square_tiling" title="Square tiling">square tiling</a>. </p><p>Polyominoes have been used in popular <a href="/wiki/Puzzle" title="Puzzle">puzzles</a> since at least 1907, and the enumeration of <a href="/wiki/Pentomino" title="Pentomino">pentominoes</a> is dated to antiquity.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Many results with the pieces of 1 to 6 squares were first published in <i><a href="/wiki/Fairy_Chess_Review" title="Fairy Chess Review">Fairy Chess Review</a></i> between the years 1937 and 1957, under the name of "dissection problems." The name <i>polyomino</i> was invented by <a href="/wiki/Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a> in 1953,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and it was popularized by <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> in a November 1960 "<a href="/wiki/Mathematical_Games_(column)" class="mw-redirect" title="Mathematical Games (column)">Mathematical Games</a>" column in <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>.<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> </p><p>Related to polyominoes are <a href="/wiki/Polyiamond" title="Polyiamond">polyiamonds</a>, formed from <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a>; <a href="/wiki/Polyhex_(mathematics)" title="Polyhex (mathematics)">polyhexes</a>, formed from regular <a href="/wiki/Hexagon" title="Hexagon">hexagons</a>; and other plane <a href="/wiki/Polyform" title="Polyform">polyforms</a>. Polyominoes have been generalized to higher <a href="/wiki/Dimension" title="Dimension">dimensions</a> by joining <a href="/wiki/Cube_(geometry)" class="mw-redirect" title="Cube (geometry)">cubes</a> to form <a href="/wiki/Polycube" title="Polycube">polycubes</a>, or <a href="/wiki/Hypercube" title="Hypercube">hypercubes</a> to form polyhypercubes. </p><p>In <a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">statistical physics</a>, the study of polyominoes and their higher-dimensional analogs (which are often referred to as <b>lattice animals</b> in this literature) is applied to problems in physics and chemistry. Polyominoes have been used as models of <a href="/wiki/Branching_(polymer_chemistry)" title="Branching (polymer chemistry)">branched polymers</a> and of <a href="/wiki/Percolation" title="Percolation">percolation</a> clusters.<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> </p><p>Like many puzzles in <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>, polyominoes raise many <a href="/wiki/Combinatorial" class="mw-redirect" title="Combinatorial">combinatorial</a> problems. The most basic is <a href="/wiki/Enumeration" title="Enumeration">enumerating</a> polyominoes of a given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for calculating them. </p><p>Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> polyominoes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Enumeration_of_polyominoes">Enumeration of polyominoes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=1" title="Edit section: Enumeration of polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional"> <li class="gallerycaption">Free polyominoes (<i>n</i>=2 to 6)</li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Domino_green.svg" class="mw-file-description" title="One free domino"><img alt="One free domino" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Domino_green.svg/68px-Domino_green.svg.png" decoding="async" width="68" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Domino_green.svg/103px-Domino_green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Domino_green.svg/137px-Domino_green.svg.png 2x" data-file-width="48" data-file-height="84" /></a></span></div> <div class="gallerytext">One free <a href="/wiki/Domino_(mathematics)" title="Domino (mathematics)">domino</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Trominoes.svg" class="mw-file-description" title="Two free trominoes"><img alt="Two free trominoes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Trominoes.svg/120px-Trominoes.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Trominoes.svg/180px-Trominoes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Trominoes.svg/240px-Trominoes.svg.png 2x" data-file-width="210" data-file-height="140" /></a></span></div> <div class="gallerytext">Two free <a href="/wiki/Tromino" title="Tromino">trominoes</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Free_tetrominoes.svg" class="mw-file-description" title="Five free tetrominoes"><img alt="Five free tetrominoes" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Free_tetrominoes.svg/120px-Free_tetrominoes.svg.png" decoding="async" width="120" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Free_tetrominoes.svg/180px-Free_tetrominoes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Free_tetrominoes.svg/240px-Free_tetrominoes.svg.png 2x" data-file-width="512" data-file-height="158" /></a></span></div> <div class="gallerytext">Five free <a href="/wiki/Tetromino" title="Tetromino">tetrominoes</a></div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Free_pentominos_001.svg" class="mw-file-description" title="12 free pentominoes, colored according to their symmetry"><img alt="12 free pentominoes, colored according to their symmetry" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Free_pentominos_001.svg/120px-Free_pentominos_001.svg.png" decoding="async" width="120" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Free_pentominos_001.svg/180px-Free_pentominos_001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Free_pentominos_001.svg/240px-Free_pentominos_001.svg.png 2x" data-file-width="406" data-file-height="202" /></a></span></div> <div class="gallerytext">12 free <a href="/wiki/Pentomino" title="Pentomino">pentominoes</a>, colored according to their symmetry</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:All_35_free_hexominoes.svg" class="mw-file-description" title="35 free hexominoes, colored according to their symmetry"><img alt="35 free hexominoes, colored according to their symmetry" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/All_35_free_hexominoes.svg/117px-All_35_free_hexominoes.svg.png" decoding="async" width="117" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/All_35_free_hexominoes.svg/175px-All_35_free_hexominoes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/02/All_35_free_hexominoes.svg/233px-All_35_free_hexominoes.svg.png 2x" data-file-width="360" data-file-height="370" /></a></span></div> <div class="gallerytext">35 free <a href="/wiki/Hexomino" title="Hexomino">hexominoes</a>, colored according to their symmetry</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Free,_one-sided,_and_fixed_polyominoes"><span id="Free.2C_one-sided.2C_and_fixed_polyominoes"></span>Free, one-sided, and fixed polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=2" title="Edit section: Free, one-sided, and fixed polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are three common ways of distinguishing polyominoes for enumeration:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <ul><li><i>free</i> polyominoes are distinct when none is a rigid transformation (<a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a>, <a href="/wiki/Rotation" title="Rotation">rotation</a>, <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> or <a href="/wiki/Glide_reflection" title="Glide reflection">glide reflection</a>) of another (pieces that can be picked up and flipped over). Translating, rotating, reflecting, or glide reflecting a free polyomino does not change its shape.</li> <li><i>one-sided polyominoes</i> are distinct when none is a translation or rotation of another (pieces that cannot be flipped over). Translating or rotating a one-sided polyomino does not change its shape.</li> <li><i>fixed</i> polyominoes are distinct when none is a translation of another (pieces that can be neither flipped nor rotated). Translating a fixed polyomino will not change its shape.</li></ul> <p>The following table shows the numbers of polyominoes of various types with <i>n</i> cells. </p> <table class="wikitable"> <tbody><tr> <th rowspan="2"><i>n</i> </th> <th rowspan="2">name </th> <th colspan="3">free </th> <th rowspan="2">one-sided </th> <th rowspan="2">fixed </th></tr> <tr> <th>total </th> <th>with holes </th> <th>without holes </th></tr> <tr align="right"> <td>1</td> <td align="left">monomino</td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td>2</td> <td align="left"><a href="/wiki/Domino_(mathematics)" title="Domino (mathematics)">domino</a></td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td>2 </td></tr> <tr align="right"> <td>3</td> <td align="left"><a href="/wiki/Tromino" title="Tromino">tromino</a></td> <td>2</td> <td>0</td> <td>2</td> <td>2</td> <td>6 </td></tr> <tr align="right"> <td>4</td> <td align="left"><a href="/wiki/Tetromino" title="Tetromino">tetromino</a></td> <td>5</td> <td>0</td> <td>5</td> <td>7</td> <td>19 </td></tr> <tr align="right"> <td>5</td> <td align="left"><a href="/wiki/Pentomino" title="Pentomino">pentomino</a></td> <td>12</td> <td>0</td> <td>12</td> <td>18</td> <td>63 </td></tr> <tr align="right"> <td>6</td> <td align="left"><a href="/wiki/Hexomino" title="Hexomino">hexomino</a></td> <td>35</td> <td>0</td> <td>35</td> <td>60</td> <td>216 </td></tr> <tr align="right"> <td>7</td> <td align="left"><a href="/wiki/Heptomino" title="Heptomino">heptomino</a></td> <td>108</td> <td>1</td> <td>107</td> <td>196</td> <td>760 </td></tr> <tr align="right"> <td>8</td> <td align="left"><a href="/wiki/Octomino" title="Octomino">octomino</a></td> <td>369</td> <td>6</td> <td>363</td> <td>704</td> <td>2,725 </td></tr> <tr align="right"> <td>9</td> <td align="left"><a href="/wiki/Nonomino" title="Nonomino">nonomino</a></td> <td>1,285</td> <td>37</td> <td>1,248</td> <td>2,500</td> <td>9,910 </td></tr> <tr align="right"> <td>10</td> <td align="left"><a href="/wiki/Decomino" title="Decomino">decomino</a></td> <td>4,655</td> <td>195</td> <td>4,460</td> <td>9,189</td> <td>36,446 </td></tr> <tr align="right"> <td>11</td> <td align="left">undecomino</td> <td>17,073</td> <td>979</td> <td>16,094</td> <td>33,896</td> <td>135,268 </td></tr> <tr align="right"> <td>12</td> <td align="left">dodecomino</td> <td>63,600</td> <td>4,663</td> <td>58,937</td> <td>126,759</td> <td>505,861 </td></tr> <tr align="right"> <td colspan="2"><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequence </td> <td><a href="//oeis.org/A000105" class="extiw" title="oeis:A000105">A000105</a> </td> <td><a href="//oeis.org/A001419" class="extiw" title="oeis:A001419">A001419</a> </td> <td><a href="//oeis.org/A000104" class="extiw" title="oeis:A000104">A000104</a> </td> <td><a href="//oeis.org/A000988" class="extiw" title="oeis:A000988">A000988</a> </td> <td><a href="//oeis.org/A001168" class="extiw" title="oeis:A001168">A001168</a> </td></tr></tbody></table> <p>Fixed polyominoes were enumerated in 2004 up to <span class="nowrap"><i>n</i> = 56</span> by Iwan Jensen,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and in 2024 up to <span class="nowrap"><i>n</i> = 70</span> by Gill Barequet and Gil Ben-Shachar.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Free polyominoes were enumerated in 2007 up to <span class="nowrap"><i>n</i> = 28</span> by Tomás Oliveira e Silva,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in 2012 up to <span class="nowrap"><i>n</i> = 45</span> by Toshihiro Shirakawa,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and in 2023 up to <span class="nowrap"><i>n</i> = 50</span> by John Mason.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The above OEIS sequences, with the exception of A001419, include the count of 1 for the number of null-polyominoes; a null-polyomino is one that is formed of zero squares. </p> <div class="mw-heading mw-heading3"><h3 id="Symmetries_of_polyominoes">Symmetries of polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=3" title="Edit section: Symmetries of polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> <i>D</i><sub>4</sub> is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a> (<a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a>) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the <i>x</i>-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of <i>D</i><sub>4</sub>. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the fewer distinct fixed counterparts it has. Therefore, a free polyomino that is invariant under some or all non-trivial symmetries of <i>D</i><sub>4</sub> may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of fixed polyominoes under the group <i>D</i><sub>4</sub>. </p><p>Polyominoes have the following possible symmetries;<sup id="cite_ref-Redelmeier,_section_3_13-0" class="reference"><a href="#cite_note-Redelmeier,_section_3-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> the least number of squares needed in a polyomino with that symmetry is given in each case: </p> <ul><li>8 fixed polyominoes for each free polyomino: <ul><li>no symmetry (4)</li></ul></li> <li>4 fixed polyominoes for each free polyomino: <ul><li>mirror symmetry with respect to one of the grid line directions (4)</li> <li>mirror symmetry with respect to a diagonal line (3)</li> <li>2-fold rotational symmetry: <i>C</i><sub>2</sub> (4)</li></ul></li> <li>2 fixed polyominoes for each free polyomino: <ul><li>symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: <i>D</i><sub>2</sub> (2) (also known as the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a>)</li> <li>symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: <i>D</i><sub>2</sub> (7)</li> <li>4-fold rotational symmetry: <i>C</i><sub>4</sub> (8)</li></ul></li> <li>1 fixed polyomino for each free polyomino: <ul><li>all symmetry of the square: <i>D</i><sub>4</sub> (1).</li></ul></li></ul> <p>In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows: </p> <ul><li>2 one-sided polyominoes for each free polyomino: <ul><li>no symmetry</li> <li>2-fold rotational symmetry: <i>C</i><sub>2</sub></li> <li>4-fold rotational symmetry: <i>C</i><sub>4</sub></li></ul></li> <li>1 one-sided polyomino for each free polyomino: <ul><li>all symmetry of the square: <i>D</i><sub>4</sub></li> <li>mirror symmetry with respect to one of the grid line directions</li> <li>mirror symmetry with respect to a diagonal line</li> <li>symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: <i>D</i><sub>2</sub></li> <li>symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: <i>D</i><sub>2</sub>.</li></ul></li></ul> <p>The following table shows the numbers of polyominoes with <i>n</i> squares, sorted by symmetry groups. </p> <table class="wikitable"> <tbody><tr> <th><i>n</i> </th> <th>none </th> <th>mirror<br />90° </th> <th>mirror<br />45° </th> <th><i>C</i><sub>2</sub> </th> <th><i>D</i><sub>2</sub><br />90° </th> <th><i>D</i><sub>2</sub><br />45° </th> <th><i>C</i><sub>4</sub> </th> <th><i>D</i><sub>4</sub> </th></tr> <tr align="right"> <td>1</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>1 </td></tr> <tr align="right"> <td>2</td> <td>0</td> <td>0</td> <td>0</td> <td>0</td> <td>1</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr align="right"> <td>3</td> <td>0</td> <td>0</td> <td>1</td> <td>0</td> <td>1</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr align="right"> <td>4</td> <td>1</td> <td>1</td> <td>0</td> <td>1</td> <td>1</td> <td>0</td> <td>0</td> <td>1 </td></tr> <tr align="right"> <td>5</td> <td>5</td> <td>2</td> <td>2</td> <td>1</td> <td>1</td> <td>0</td> <td>0</td> <td>1 </td></tr> <tr align="right"> <td>6</td> <td>20</td> <td>6</td> <td>2</td> <td>5</td> <td>2</td> <td>0</td> <td>0</td> <td>0 </td></tr> <tr align="right"> <td>7</td> <td>84</td> <td>9</td> <td>7</td> <td>4</td> <td>3</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr align="right"> <td>8</td> <td>316</td> <td>23</td> <td>5</td> <td>18</td> <td>4</td> <td>1</td> <td>1</td> <td>1 </td></tr> <tr align="right"> <td>9</td> <td>1,196</td> <td>38</td> <td>26</td> <td>19</td> <td>4</td> <td>0</td> <td>0</td> <td>2 </td></tr> <tr align="right"> <td>10</td> <td>4,461</td> <td>90</td> <td>22</td> <td>73</td> <td>8</td> <td>1</td> <td>0</td> <td>0 </td></tr> <tr align="right"> <td>11</td> <td>16,750</td> <td>147</td> <td>91</td> <td>73</td> <td>10</td> <td>2</td> <td>0</td> <td>0 </td></tr> <tr align="right"> <td>12</td> <td>62,878</td> <td>341</td> <td>79</td> <td>278</td> <td>15</td> <td>3</td> <td>3</td> <td>3 </td></tr> <tr> <td><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> sequence </td> <td><a href="//oeis.org/A006749" class="extiw" title="oeis:A006749">A006749</a> </td> <td><a href="//oeis.org/A006746" class="extiw" title="oeis:A006746">A006746</a> </td> <td><a href="//oeis.org/A006748" class="extiw" title="oeis:A006748">A006748</a> </td> <td><a href="//oeis.org/A006747" class="extiw" title="oeis:A006747">A006747</a> </td> <td><a href="//oeis.org/A056877" class="extiw" title="oeis:A056877">A056877</a> </td> <td><a href="//oeis.org/A056878" class="extiw" title="oeis:A056878">A056878</a> </td> <td><a href="//oeis.org/A144553" class="extiw" title="oeis:A144553">A144553</a> </td> <td><a href="//oeis.org/A142886" class="extiw" title="oeis:A142886">A142886</a> </td></tr></tbody></table> <p><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Algorithms_for_enumeration_of_fixed_polyominoes">Algorithms for enumeration of fixed polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=4" title="Edit section: Algorithms for enumeration of fixed polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Inductive_algorithms">Inductive algorithms</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=5" title="Edit section: Inductive algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each polyomino of size <i>n</i>+1 can be obtained by adding a square to a polyomino of size <i>n</i>. This leads to algorithms for generating polyominoes inductively. </p><p>Most simply, given a list of polyominoes of size <i>n</i>, squares may be added next to each polyomino in each possible position, and the resulting polyomino of size <i>n</i>+1 added to the list if not a duplicate of one already found; refinements in ordering the enumeration and marking adjacent squares that should not be considered reduce the number of cases that need to be checked for duplicates.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This method may be used to enumerate either free or fixed polyominoes. </p><p>A more sophisticated method, described by Redelmeier, has been used by many authors as a way of not only counting polyominoes (without requiring that all polyominoes of size <i>n</i> be stored in size to enumerate those of size <i>n</i>+1), but also proving upper bounds on their number. The basic idea is that we begin with a single square, and from there, recursively add squares. Depending on the details, it may count each <i>n</i>-omino <i>n</i> times, once from starting from each of its <i>n</i> squares, or may be arranged to count each once only. </p><p>The simplest implementation involves adding one square at a time. Beginning with an initial square, number the adjacent squares, clockwise from the top, 1, 2, 3, and 4. Now pick a number between 1 and 4, and add a square at that location. Number the unnumbered adjacent squares, starting with 5. Then, pick a number larger than the previously picked number, and add that square. Continue picking a number larger than the number of the current square, adding that square, and then numbering the new adjacent squares. When <i>n</i> squares have been created, an <i>n</i>-omino has been created. </p><p>This method ensures that each fixed polyomino is counted exactly <i>n</i> times, once for each starting square. It can be optimized so that it counts each polyomino only once, rather than <i>n</i> times. Starting with the initial square, declare it to be the lower-left square of the polyomino. Simply do not number any square that is on a lower row, or left of the square on the same row. This is the version described by Redelmeier. </p><p>If one wishes to count free polyominoes instead, then one may check for symmetries after creating each <i>n</i>-omino. However, it is faster<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> to generate symmetric polyominoes separately (by a variation of this method)<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and so determine the number of free polyominoes by <a href="/wiki/Burnside%27s_lemma" title="Burnside's lemma">Burnside's lemma</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Transfer-matrix_method">Transfer-matrix method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=6" title="Edit section: Transfer-matrix method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Currently, the most effective algorithms belong to the transfer-matrix paradigm. They may be called transfer matrix algorithms (TMAs) for short. Andrew Conway<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> first implemented a TMA in the 90s, and calculated 25 terms of the fixed polyomino sequence (<a href="//oeis.org/A001419" class="extiw" title="oeis:A001419">A001419</a> in the OEIS). Iwan Jensen refined Conway's methods and implemented a TMA in parallel for the first time in a pair of papers in the early 2000s.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> He calculated 56 terms. Because of this work, any TMA is sometimes also called Jensen's Algorithm. In 2024, Gill Barequet and his student Gil Ben-Shachar made another improvement by running a TMA on 45° rotation of the square grid, which is an equivalent problem but computationally easier.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> This approach holds the polyomino-counting record, with 70 terms. </p><p>As a rule, TMAs are much faster than the previous methods, but still run in time that is exponential in <i>n</i>. Roughly, this is achieved by fixing a width (in the diagonal case, a diagonal width), and counting polyominoes that fit in rectangles of that width. If this is done, it is only necessary to keep track of a polyomino's boundary, and since multiple polyominoes can correspond to a single boundary, this approach is faster than one generating every polyomino. Repeating this for every width gives every polyomino. </p><p>Although it has excellent running time, the tradeoff is that this algorithm uses exponential amounts of memory (many <a href="/wiki/Gigabyte" title="Gigabyte">gigabytes</a> of memory are needed for <i>n</i> above 50), is much harder to program than the other methods, and cannot currently be used to count free polyominoes. </p> <div class="mw-heading mw-heading3"><h3 id="Asymptotic_growth_of_the_number_of_polyominoes">Asymptotic growth of the number of polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=7" title="Edit section: Asymptotic growth of the number of polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Fixed_polyominoes">Fixed polyominoes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=8" title="Edit section: Fixed polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n </p> <dl><dd><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_{n}\sim {\frac {c\lambda ^{n}}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}\sim {\frac {c\lambda ^{n}}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0794724f011ef10115db4e68dd02006a6325a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.477ex; height:5.343ex;" alt="{\displaystyle A_{n}\sim {\frac {c\lambda ^{n}}{n}}}"></span></dd></dl> <p>where <i>λ</i> = 4.0626 and <i>c</i> = 0.3169.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> However, this result is not proven and the values of <i>λ</i> and <i>c</i> are only estimates. </p><p>The known theoretical results are not nearly as specific as this estimate. It has been proven that </p> <dl><dd><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 \lim _{n\rightarrow \infty }(A_{n})^{\frac {1}{n}}=\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }(A_{n})^{\frac {1}{n}}=\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e0b9c3346e5d670788ae46a07675c7ddaed750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.366ex; height:4.843ex;" alt="{\displaystyle \lim _{n\rightarrow \infty }(A_{n})^{\frac {1}{n}}=\lambda }"></span></dd></dl> <p>exists. In other words, <i>A<sub>n</sub></i> <a href="/wiki/Exponential_growth" title="Exponential growth">grows exponentially</a>. The best known lower bound for <i>λ</i>, found in 2016, is 4.00253.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The best known upper bound is <span class="nowrap"><i>λ</i> < 4.5252</span>.<sup id="cite_ref-:0_24-0" class="reference"><a href="#cite_note-:0-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>To establish a lower bound, a simple but highly effective method is concatenation of polyominoes. Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Then, the upper-right square of any polyomino of size <i>n</i> can be attached to the bottom-left square of any polyomino of size <i>m</i> to produce a unique (<i>n</i>+<i>m</i>)-omino. This proves <span class="nowrap"><i>A<sub>n</sub>A<sub>m</sub></i> ≤ <i>A</i><sub><i>n</i>+<i>m</i></sub></span>. Using this equation, one can show <span class="nowrap"><i>λ</i> ≥ (<i>A<sub>n</sub></i>)<sup>1/<i>n</i></sup></span> for all <i>n</i>. Refinements of this procedure combined with data for <i>A<sub>n</sub></i> produce the lower bound given above. </p><p>The upper bound is attained by generalizing the inductive method of enumerating polyominoes. Instead of adding one square at a time, one adds a cluster of squares at a time. This is often described as adding <i>twigs</i>. By proving that every <i>n</i>-omino is a sequence of twigs, and by proving limits on the combinations of possible twigs, one obtains an upper bound on the number of <i>n</i>-ominoes. For example, in the algorithm outlined above, at each step we must choose a larger number, and at most three new numbers are added (since at most three unnumbered squares are adjacent to any numbered square). This can be used to obtain an upper bound of 6.75. Using 2.8 million twigs, <a href="/wiki/David_A._Klarner" title="David A. Klarner">Klarner</a> and <a href="/wiki/Ron_Rivest" title="Ron Rivest">Rivest</a> obtained an upper bound of 4.65,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> which was subsequently improved by Barequet and Shalah to 4.5252.<sup id="cite_ref-:0_24-1" class="reference"><a href="#cite_note-:0-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Free_polyominoes">Free polyominoes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=9" title="Edit section: Free polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a> (rotation or reflection) corresponds to 8 distinct fixed polyominoes, and for large <i>n</i>, most <i>n</i>-ominoes have no symmetries. Therefore, the number of fixed <i>n</i>-ominoes is approximately 8 times the number of free <i>n</i>-ominoes. Moreover, this approximation is exponentially more accurate as <i>n</i> increases.<sup id="cite_ref-Redelmeier,_section_3_13-1" class="reference"><a href="#cite_note-Redelmeier,_section_3-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Special_classes_of_polyominoes">Special classes of polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=10" title="Edit section: Special classes of polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Exact formulas are known for enumerating polyominoes of special classes, such as the class of <i>convex</i> polyominoes and the class of <i>directed</i> polyominoes. </p><p>The definition of a <i>convex</i> polyomino is different from the usual definition of <a href="/wiki/Convex_set" title="Convex set">convexity</a>, but is similar to the definition used for the <a href="/wiki/Orthogonal_convex_hull" title="Orthogonal convex hull">orthogonal convex hull</a>. A polyomino is said to be <i>vertically</i> or <i>column convex</i> if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be <i>horizontally</i> or <i>row convex</i> if its intersection with any horizontal line is convex. A polyomino is said to be <i>convex</i> if it is row and column convex.<sup id="cite_ref-W151_26-0" class="reference"><a href="#cite_note-W151-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>A polyomino is said to be <i>directed</i> if it contains a square, known as the <i>root</i>, such that every other square can be reached by movements of up or right one square, without leaving the polyomino. </p><p>Directed polyominoes,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> column (or row) convex polyominoes,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and convex polyominoes<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> have been effectively enumerated by area <i>n</i>, as well as by some other parameters such as perimeter, using <a href="/wiki/Generating_function" title="Generating function">generating functions</a>. </p><p>A polyomino is <a href="/wiki/Equable_shape" title="Equable shape">equable</a> if its area equals its perimeter. An equable polyomino must be made from an <a href="/wiki/Even_number" class="mw-redirect" title="Even number">even number</a> of squares; every even number greater than 15 is possible. For instance, the 16-omino in the form of a 4 × 4 square and the 18-omino in the form of a 3 × 6 rectangle are both equable. For polyominoes with 15 squares or fewer, the perimeter always exceeds the area.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Tiling_with_polyominoes">Tiling with polyominoes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=11" title="Edit section: Tiling with polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Recreational_mathematics" title="Recreational mathematics">recreational mathematics</a>, challenges are often posed for <a href="/wiki/Tessellation" title="Tessellation">tiling</a> a prescribed region, or the entire plane, with polyominoes,<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> and related problems are investigated in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Tiling_regions_with_sets_of_polyominoes">Tiling regions with sets of polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=12" title="Edit section: Tiling regions with sets of polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Puzzles commonly ask for tiling a given region with a given set of polyominoes, such as the 12 pentominoes. Golomb's and Gardner's books have many examples. A typical puzzle is to tile a 6×10 rectangle with the twelve pentominoes; the 2339 solutions to this were found in 1960.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Where multiple copies of the polyominoes in the set are allowed, Golomb defines a hierarchy of different regions that a set may be able to tile, such as rectangles, strips, and the whole plane, and shows that whether polyominoes from a given set can tile the plane is <a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">undecidable</a>, by mapping sets of <a href="/wiki/Wang_tile" title="Wang tile">Wang tiles</a> to sets of polyominoes.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>Because the general problem of tiling regions of the plane with sets of polyominoes is <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> tiling with more than a few pieces rapidly becomes intractable and so the aid of a computer is required. The traditional approach to tiling finite regions of the plane uses a technique in computer science called <a href="/wiki/Backtracking" title="Backtracking">backtracking</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Sudoku#Variants" title="Sudoku">Jigsaw Sudokus</a> a square grid is tiled with polyomino-shaped regions (sequence <span class="nowrap external"><a href="//oeis.org/A172477" class="extiw" title="oeis:A172477">A172477</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Tiling_regions_with_copies_of_a_single_polyomino">Tiling regions with copies of a single polyomino</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=13" title="Edit section: Tiling regions with copies of a single polyomino"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another class of problems asks whether copies of a given polyomino can tile a <a href="/wiki/Rectangle" title="Rectangle">rectangle</a>, and if so, what rectangles they can tile.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> These problems have been extensively studied for particular polyominoes,<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> and tables of results for individual polyominoes are available.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> <a href="/wiki/David_A._Klarner" title="David A. Klarner">Klarner</a> and Göbel showed that for any polyomino there is a finite set of <i>prime</i> rectangles it tiles, such that all other rectangles it tiles can be tiled by those prime rectangles.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Kamenetsky and Cooke showed how various disjoint (called "holey") polyominoes can tile rectangles.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>Beyond rectangles, Golomb gave his hierarchy for single polyominoes: a polyomino may tile a rectangle, a half strip, a bent strip, an enlarged copy of itself, a quadrant, a strip, a <a href="/wiki/Half_plane" class="mw-redirect" title="Half plane">half plane</a>, the whole plane, certain combinations, or none of these. There are certain implications among these, both obvious (for example, if a polyomino tiles the half plane then it tiles the whole plane) and less so (for example, if a polyomino tiles an enlarged copy of itself, then it tiles the quadrant). Polyominoes of size up to 6 are characterized in this hierarchy (with the status of one hexomino, later found to tile a rectangle, unresolved at that time).<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>In 2001 <a href="/wiki/Cris_Moore" class="mw-redirect" title="Cris Moore">Cristopher Moore</a> and John Michael Robson showed that the problem of tiling one polyomino with copies of another is <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tiling_the_plane_with_copies_of_a_single_polyomino">Tiling the plane with copies of a single polyomino</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=14" title="Edit section: Tiling the plane with copies of a single polyomino"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Conway_criterion_false_negative_nonominoes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Conway_criterion_false_negative_nonominoes.svg/180px-Conway_criterion_false_negative_nonominoes.svg.png" decoding="async" width="180" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Conway_criterion_false_negative_nonominoes.svg/270px-Conway_criterion_false_negative_nonominoes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Conway_criterion_false_negative_nonominoes.svg/360px-Conway_criterion_false_negative_nonominoes.svg.png 2x" data-file-width="544" data-file-height="512" /></a><figcaption>The two tiling nonominoes not satisfying the Conway criterion.</figcaption></figure> <p>Tiling the plane with copies of a single polyomino has also been much discussed. It was noted in 1965 that all polyominoes up to hexominoes<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> and all but four heptominoes tile the plane.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> It was then established by David Bird that all but 26 octominoes tile the plane.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Rawsthorne found that all but 235 polyominoes of size 9 tile,<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> and such results have been extended to higher area by Rhoads (to size 14)<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> and others. Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p>The study of which polyominoes can tile the plane has been facilitated using the <a href="/wiki/Conway_criterion" title="Conway criterion">Conway criterion</a>: except for two nonominoes, all tiling polyominoes up to size 9 form a patch of at least one tile satisfying it, with higher-size exceptions more frequent.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a <a href="/wiki/Rep-tile" title="Rep-tile">rep-tile</a> tiling of the plane. For instance, for every positive integer <span class="texhtml mvar" style="font-style:italic;">n</span>, it is possible to combine <span class="texhtml"><i>n</i><sup>2</sup></span> copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Tiling_a_common_figure_with_various_polyominoes">Tiling a common figure with various polyominoes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=15" title="Edit section: Tiling a common figure with various polyominoes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:PentominoCompatibilityTW.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/PentominoCompatibilityTW.svg/220px-PentominoCompatibilityTW.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/PentominoCompatibilityTW.svg/330px-PentominoCompatibilityTW.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/PentominoCompatibilityTW.svg/440px-PentominoCompatibilityTW.svg.png 2x" data-file-width="179" data-file-height="107" /></a><figcaption>A minimal compatibility figure for the T and W <a href="/wiki/Pentomino" title="Pentomino">pentominoes</a>.</figcaption></figure> <p>The <i>compatibility problem</i> is to take two or more polyominoes and find a figure that can be tiled with each. Polyomino compatibility has been widely studied since the 1990s. Jorge Luis Mireles and Giovanni Resta have published websites of systematic results,<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> and Livio Zucca shows results for some complicated cases like three different pentominoes.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> The general problem can be hard. The first compatibility figure for the L and X pentominoes was published in 2005 and had 80 tiles of each kind.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible. </p> <div class="mw-heading mw-heading2"><h2 id="Polyominoes_in_puzzles_and_games">Polyominoes in puzzles and games</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=16" title="Edit section: Polyominoes in puzzles and games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to the tiling problems described above, there are recreational mathematics puzzles that require folding a polyomino to create other shapes. Gardner proposed several simple games with a set of free pentominoes and a chessboard. Some variants of the <a href="/wiki/Sudoku#Variants" title="Sudoku">Sudoku</a> puzzle use nonomino-shaped regions on the grid. The video game <i><a href="/wiki/Tetris" title="Tetris">Tetris</a></i> is based on the seven one-sided tetrominoes (spelled "Tetriminos" in the game), and the board game <i><a href="/wiki/Blokus" title="Blokus">Blokus</a></i> uses all of the free polyominoes up to pentominoes. </p> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=17" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>polyomino</i> and the names of the various sizes of polyomino are all back-formations from the word <i><a href="/wiki/Dominoes" title="Dominoes">domino</a></i>, a common game piece consisting of two squares. The name <i>domino</i> for the game piece is believed to come from the spotted masquerade garment <i>domino</i>, from Latin <i>dominus</i>.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> Despite this word origin, in naming polyominoes, the first letter <i>d-</i> of <i>domino</i> is fancifully interpreted as a version of the prefix <i>di-</i> meaning "two", and replaced by other <a href="/wiki/Numerical_prefix" class="mw-redirect" title="Numerical prefix">numerical prefixes</a>. </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=Polyomino&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a>, the mathematical study of random subsets of integer grids. The finite connected components of these subsets form polyominoes.</li> <li><a href="/wiki/Young_tableau" title="Young tableau">Young diagram</a>, a special kind of polyomino used in number theory to describe integer partitions and in group theory and applications in mathematical physics to describe representations of the symmetric group.</li> <li><a href="/wiki/Blokus" title="Blokus">Blokus</a>, a board game using polyominoes.</li> <li><a href="/wiki/Squaregraph" title="Squaregraph">Squaregraph</a>, a kind of undirected graph including as a special case the graphs of vertices and edges of polyominoes.</li> <li><a href="/wiki/Polycube" title="Polycube">Polycube</a>, its analogue in three dimensions.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polyomino&action=edit&section=19" title="Edit section: Notes"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Golomb (<i>Polyominoes</i>, Preface to the First Edition) writes "the observation that there are twelve distinctive patterns (the pentominoes) that can be formed by five connected stones on a <a href="/wiki/Go_(game)" title="Go (game)">Go</a> board ... is attributed to an ancient master of that game".</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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="CITEREFGolomb1994" class="citation book cs1"><a href="/wiki/Solomon_W._Golomb" title="Solomon W. Golomb">Golomb, Solomon W.</a> (1994). <a href="/wiki/Polyominoes:_Puzzles,_Patterns,_Problems,_and_Packings" title="Polyominoes: Puzzles, Patterns, Problems, and Packings"><i>Polyominoes</i></a> (2nd ed.). Princeton, New Jersey: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02444-8" title="Special:BookSources/978-0-691-02444-8"><bdi>978-0-691-02444-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyominoes&rft.place=Princeton%2C+New+Jersey&rft.edition=2nd&rft.pub=Princeton+University+Press&rft.date=1994&rft.isbn=978-0-691-02444-8&rft.aulast=Golomb&rft.aufirst=Solomon+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></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="CITEREFGardner1960" class="citation journal cs1">Gardner, M. (November 1960). "More about the shapes that can be made with complex dominoes (Mathematical Games)". <i>Scientific American</i>. <b>203</b> (5): <span class="nowrap">186–</span>201. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican1160-186">10.1038/scientificamerican1160-186</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24940703">24940703</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=More+about+the+shapes+that+can+be+made+with+complex+dominoes+%28Mathematical+Games%29&rft.volume=203&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E186-%3C%2Fspan%3E201&rft.date=1960-11&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1160-186&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24940703%23id-name%3DJSTOR&rft.aulast=Gardner&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" 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="CITEREFWhittingtonSoteros1990" class="citation book cs1">Whittington, S. G.; <a href="/wiki/Chris_Soteros" title="Chris Soteros">Soteros, C. E.</a> (1990). "Lattice Animals: Rigorous Results and Wild Guesses". In Grimmett, G.; Welsh, D. (eds.). <i>Disorder in Physical Systems</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lattice+Animals%3A+Rigorous+Results+and+Wild+Guesses&rft.btitle=Disorder+in+Physical+Systems&rft.pub=Oxford+University+Press&rft.date=1990&rft.aulast=Whittington&rft.aufirst=S.+G.&rft.au=Soteros%2C+C.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrünbaumShephard,_G.C.1987" class="citation book cs1"><a href="/wiki/Branko_Gr%C3%BCnbaum" title="Branko Grünbaum">Grünbaum, Branko</a>; Shephard, G.C. (1987). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_0716711931"><i>Tilings and Patterns</i></a></span>. New York: W.H. Freeman and Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-1193-3" title="Special:BookSources/978-0-7167-1193-3"><bdi>978-0-7167-1193-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tilings+and+Patterns&rft.place=New+York&rft.pub=W.H.+Freeman+and+Company&rft.date=1987&rft.isbn=978-0-7167-1193-3&rft.aulast=Gr%C3%BCnbaum&rft.aufirst=Branko&rft.au=Shephard%2C+G.C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_0716711931&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRedelmeier1981" class="citation journal cs1">Redelmeier, D. Hugh (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2881%2990237-5">"Counting polyominoes: yet another attack"</a>. <i>Discrete Mathematics</i>. <b>36</b> (2): <span class="nowrap">191–</span>203. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2881%2990237-5">10.1016/0012-365X(81)90237-5</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Counting+polyominoes%3A+yet+another+attack&rft.volume=36&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E191-%3C%2Fspan%3E203&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2881%2990237-5&rft.aulast=Redelmeier&rft.aufirst=D.+Hugh&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0012-365X%252881%252990237-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Golomb, chapter 6</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIwan_Jensen" class="citation web cs1">Iwan Jensen. <a rel="nofollow" class="external text" href="http://www.ms.unimelb.edu.au/~iwan/animals/Animals_ser.html">"Series for lattice animals or polyominoes"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070612141716/http://www.ms.unimelb.edu.au/~iwan/animals/Animals_ser.html">Archived</a> from the original on 2007-06-12<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-05-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Series+for+lattice+animals+or+polyominoes&rft.au=Iwan+Jensen&rft_id=http%3A%2F%2Fwww.ms.unimelb.edu.au%2F~iwan%2Fanimals%2FAnimals_ser.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarequetBen-Shachar2024" class="citation book cs1">Barequet, Gill; Ben-Shachar, Gil (January 2024). <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/10.1137/1.9781611977929.10">"Counting Polyominoes, Revisited"</a>. <i>2024 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX) - Counting Polyominoes, Revisited</i>. Society for Industrial and Applied Mathematics. pp. <span class="nowrap">133–</span>143. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611977929.10">10.1137/1.9781611977929.10</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61197-792-9" title="Special:BookSources/978-1-61197-792-9"><bdi>978-1-61197-792-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Counting+Polyominoes%2C+Revisited&rft.btitle=2024+Proceedings+of+the+Symposium+on+Algorithm+Engineering+and+Experiments+%28ALENEX%29+-+Counting+Polyominoes%2C+Revisited&rft.pages=%3Cspan+class%3D%22nowrap%22%3E133-%3C%2Fspan%3E143&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=2024-01&rft_id=info%3Adoi%2F10.1137%2F1.9781611977929.10&rft.isbn=978-1-61197-792-9&rft.aulast=Barequet&rft.aufirst=Gill&rft.au=Ben-Shachar%2C+Gil&rft_id=https%3A%2F%2Fepubs.siam.org%2Fdoi%2F10.1137%2F1.9781611977929.10&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTomás_Oliveira_e_Silva" class="citation web cs1">Tomás Oliveira e Silva. <a rel="nofollow" class="external text" href="http://www.ieeta.pt/%7Etos/animals/a44.html">"Animal enumerations on the {4,4} Euclidean tiling"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070423213531/http://www.ieeta.pt/%7Etos/animals/a44.html">Archived</a> from the original on 2007-04-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-05-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Animal+enumerations+on+the+%7B4%2C4%7D+Euclidean+tiling&rft.au=Tom%C3%A1s+Oliveira+e+Silva&rft_id=http%3A%2F%2Fwww.ieeta.pt%2F%257Etos%2Fanimals%2Fa44.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.gathering4gardner.org/g4g10gift/math/Shirakawa_Toshihiro-Harmonic_Magic_Square.pdf">"Harmonic Magic Square, Enumeration of Polyominoes considering the symmetry"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Harmonic+Magic+Square%2C+Enumeration+of+Polyominoes+considering+the+symmetry&rft_id=https%3A%2F%2Fwww.gathering4gardner.org%2Fg4g10gift%2Fmath%2FShirakawa_Toshihiro-Harmonic_Magic_Square.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://oeis.org/A000105/a000105_1.pdf">"Counting size 50 polyominoes"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Counting+size+50+polyominoes&rft_id=https%3A%2F%2Foeis.org%2FA000105%2Fa000105_1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-Redelmeier,_section_3-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Redelmeier,_section_3_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Redelmeier,_section_3_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Redelmeier, section 3</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRedelmeier1981" class="citation journal cs1">Redelmeier, D.Hugh (1981). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2881%2990237-5">"Counting polyominoes: Yet another attack"</a>. <i>Discrete Mathematics</i>. <b>36</b> (2): <span class="nowrap">191–</span>203. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2881%2990237-5">10.1016/0012-365X(81)90237-5</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Counting+polyominoes%3A+Yet+another+attack&rft.volume=36&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E191-%3C%2Fspan%3E203&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2881%2990237-5&rft.aulast=Redelmeier&rft.aufirst=D.Hugh&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0012-365X%252881%252990237-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Golomb, pp. 73–79</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Redelmeier, section 4</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Redelmeier, section 6</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConway1995" class="citation journal cs1">Conway, Andrew (1995). "Enumerating 2D percolation series by the finite-lattice method: theory". <i>Journal of Physics A: Mathematical and General</i>. <b>28</b> (2): <span class="nowrap">335–</span>349. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995JPhA...28..335C">1995JPhA...28..335C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F28%2F2%2F011">10.1088/0305-4470/28/2/011</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:0849.05003">0849.05003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+A%3A+Mathematical+and+General&rft.atitle=Enumerating+2D+percolation+series+by+the+finite-lattice+method%3A+theory&rft.volume=28&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E335-%3C%2Fspan%3E349&rft.date=1995&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0849.05003%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F28%2F2%2F011&rft_id=info%3Abibcode%2F1995JPhA...28..335C&rft.aulast=Conway&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJensen2001" class="citation journal cs1">Jensen, Iwan (2001). "Enumerations of Lattice Animals and Trees". <i>Journal of Statistical Physics</i>. <b>102</b> (1): <span class="nowrap">865–</span>881. <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/cond-mat/0007239">cond-mat/0007239</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001JSP...102..865J">2001JSP...102..865J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1004855020556">10.1023/A:1004855020556</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Statistical+Physics&rft.atitle=Enumerations+of+Lattice+Animals+and+Trees&rft.volume=102&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E865-%3C%2Fspan%3E881&rft.date=2001&rft_id=info%3Aarxiv%2Fcond-mat%2F0007239&rft_id=info%3Adoi%2F10.1023%2FA%3A1004855020556&rft_id=info%3Abibcode%2F2001JSP...102..865J&rft.aulast=Jensen&rft.aufirst=Iwan&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJensen2003" class="citation conference cs1">Jensen, Iwan (2003). <i>Counting Polyominoes: A Parallel Implementation for Cluster Computing</i>. International Conference on Computer Science (ICCS). pp. <span class="nowrap">203–</span>212. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44863-2_21">10.1007/3-540-44863-2_21</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Counting+Polyominoes%3A+A+Parallel+Implementation+for+Cluster+Computing&rft.pages=%3Cspan+class%3D%22nowrap%22%3E203-%3C%2Fspan%3E212&rft.date=2003&rft_id=info%3Adoi%2F10.1007%2F3-540-44863-2_21&rft.aulast=Jensen&rft.aufirst=Iwan&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarequetBen-Shachar2003" class="citation conference cs1">Barequet, Gill; Ben-Shachar, Gil (2003). <i>Counting Polyominoes, Revisited</i>. Symposium on Algorithm Engineering and Experiments (SIAM). pp. <span class="nowrap">133–</span>143. <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/2310.20632">2310.20632</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/10.1137%2F1.9781611977929.1">10.1137/1.9781611977929.1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Counting+Polyominoes%2C+Revisited&rft.pages=%3Cspan+class%3D%22nowrap%22%3E133-%3C%2Fspan%3E143&rft.date=2003&rft_id=info%3Aarxiv%2F2310.20632&rft_id=info%3Adoi%2F10.1137%2F1.9781611977929.1&rft.aulast=Barequet&rft.aufirst=Gill&rft.au=Ben-Shachar%2C+Gil&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJensenGuttmann,_Anthony_J.2000" class="citation journal cs1">Jensen, Iwan; Guttmann, Anthony J. (2000). "Statistics of lattice animals (polyominoes) and polygons". <i>Journal of Physics A: Mathematical and General</i>. <b>33</b> (29): <span class="nowrap">L257 –</span> <span class="nowrap">L263</span>. <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/cond-mat/0007238v1">cond-mat/0007238v1</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000JPhA...33L.257J">2000JPhA...33L.257J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F33%2F29%2F102">10.1088/0305-4470/33/29/102</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:6461687">6461687</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physics+A%3A+Mathematical+and+General&rft.atitle=Statistics+of+lattice+animals+%28polyominoes%29+and+polygons&rft.volume=33&rft.issue=29&rft.pages=%3Cspan+class%3D%22nowrap%22%3EL257+-%3C%2Fspan%3E+%3Cspan+class%3D%22nowrap%22%3EL263%3C%2Fspan%3E&rft.date=2000&rft_id=info%3Aarxiv%2Fcond-mat%2F0007238v1&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6461687%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F33%2F29%2F102&rft_id=info%3Abibcode%2F2000JPhA...33L.257J&rft.aulast=Jensen&rft.aufirst=Iwan&rft.au=Guttmann%2C+Anthony+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarequetRoteShalah" class="citation journal cs1">Barequet, Gill; Rote, Gunter; Shalah, Mira. "λ > 4: An Improved Lower Bound on the Growth Constant of Polyominoes". <i>Communications of the ACM</i>. <b>59</b> (7): <span class="nowrap">88–</span>95. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2851485">10.1145/2851485</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=%CE%BB+%3E+4%3A+An+Improved+Lower+Bound+on+the+Growth+Constant+of+Polyominoes&rft.volume=59&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E88-%3C%2Fspan%3E95&rft_id=info%3Adoi%2F10.1145%2F2851485&rft.aulast=Barequet&rft.aufirst=Gill&rft.au=Rote%2C+Gunter&rft.au=Shalah%2C+Mira&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-:0-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarequetShalah2022" class="citation journal cs1">Barequet, Gill; Shalah, Mira (2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00453-022-00948-6">"Improved upper bounds on the growth constants of polyominoes and polycubes"</a>. <i>Algorithmica</i>. <b>84</b> (12): <span class="nowrap">3559–</span>3586. <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/1906.11447">1906.11447</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00453-022-00948-6">10.1007/s00453-022-00948-6</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Algorithmica&rft.atitle=Improved+upper+bounds+on+the+growth+constants+of+polyominoes+and+polycubes&rft.volume=84&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3559-%3C%2Fspan%3E3586&rft.date=2022&rft_id=info%3Aarxiv%2F1906.11447&rft_id=info%3Adoi%2F10.1007%2Fs00453-022-00948-6&rft.aulast=Barequet&rft.aufirst=Gill&rft.au=Shalah%2C+Mira&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs00453-022-00948-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlarnerRivest,_R.L.1973" class="citation journal cs1">Klarner, D.A.; Rivest, R.L. (1973). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061126083002/http://historical.ncstrl.org/litesite-data/stan/CS-TR-72-263.pdf">"A procedure for improving the upper bound for the number of <i>n</i>-ominoes"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a></i>. <b>25</b> (3): <span class="nowrap">585–</span>602. <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=10.1.1.309.9151">10.1.1.309.9151</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/10.4153%2FCJM-1973-060-4">10.4153/CJM-1973-060-4</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:121448572">121448572</a>. Archived from <a rel="nofollow" class="external text" href="http://historical.ncstrl.org/litesite-data/stan/CS-TR-72-263.pdf">the original</a> <span class="cs1-format">(PDF of technical report version)</span> on 2006-11-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-05-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=A+procedure+for+improving+the+upper+bound+for+the+number+of+n-ominoes&rft.volume=25&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E585-%3C%2Fspan%3E602&rft.date=1973&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.309.9151%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121448572%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.4153%2FCJM-1973-060-4&rft.aulast=Klarner&rft.aufirst=D.A.&rft.au=Rivest%2C+R.L.&rft_id=http%3A%2F%2Fhistorical.ncstrl.org%2Flitesite-data%2Fstan%2FCS-TR-72-263.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-W151-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-W151_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilf1994" class="citation book cs1"><a href="/wiki/Herbert_Wilf" title="Herbert Wilf">Wilf, Herbert S.</a> (1994). <i>Generatingfunctionology</i> (2nd ed.). Boston, MA: Academic Press. p. 151. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-751956-2" title="Special:BookSources/978-0-12-751956-2"><bdi>978-0-12-751956-2</bdi></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:0831.05001">0831.05001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Generatingfunctionology&rft.place=Boston%2C+MA&rft.pages=151&rft.edition=2nd&rft.pub=Academic+Press&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0831.05001%23id-name%3DZbl&rft.isbn=978-0-12-751956-2&rft.aulast=Wilf&rft.aufirst=Herbert+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBousquet-Mélou1998" class="citation journal cs1"><a href="/wiki/Mireille_Bousquet-M%C3%A9lou" title="Mireille Bousquet-Mélou">Bousquet-Mélou, Mireille</a> (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0012-365X%2897%2900109-X">"New enumerative results on two-dimensional directed animals"</a>. <i>Discrete Mathematics</i>. <b>180</b> (<span class="nowrap">1–</span>3): <span class="nowrap">73–</span>106. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0012-365X%2897%2900109-X">10.1016/S0012-365X(97)00109-X</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=New+enumerative+results+on+two-dimensional+directed+animals&rft.volume=180&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E73-%3C%2Fspan%3E106&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0012-365X%2897%2900109-X&rft.aulast=Bousquet-M%C3%A9lou&rft.aufirst=Mireille&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0012-365X%252897%252900109-X&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDelest1988" class="citation journal cs1">Delest, M.-P. (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0097-3165%2888%2990071-4">"Generating functions for column-convex polyominoes"</a>. <i>Journal of Combinatorial Theory, Series A</i>. <b>48</b> (1): <span class="nowrap">12–</span>31. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0097-3165%2888%2990071-4">10.1016/0097-3165(88)90071-4</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory%2C+Series+A&rft.atitle=Generating+functions+for+column-convex+polyominoes&rft.volume=48&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E12-%3C%2Fspan%3E31&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0097-3165%2888%2990071-4&rft.aulast=Delest&rft.aufirst=M.-P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0097-3165%252888%252990071-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBousquet-MélouFédou1995" class="citation journal cs1"><a href="/wiki/Mireille_Bousquet-M%C3%A9lou" title="Mireille Bousquet-Mélou">Bousquet-Mélou, Mireille</a>; Fédou, Jean-Marc (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2893%29E0161-V">"The generating function of convex polyominoes: The resolution of a <i>q</i>-differential system"</a>. <i><a href="/wiki/Discrete_Mathematics_(journal)" title="Discrete Mathematics (journal)">Discrete Mathematics</a></i>. <b>137</b> (<span class="nowrap">1–</span>3): <span class="nowrap">53–</span>75. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2893%29E0161-V">10.1016/0012-365X(93)E0161-V</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=The+generating+function+of+convex+polyominoes%3A+The+resolution+of+a+q-differential+system&rft.volume=137&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E53-%3C%2Fspan%3E75&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2893%29E0161-V&rft.aulast=Bousquet-M%C3%A9lou&rft.aufirst=Mireille&rft.au=F%C3%A9dou%2C+Jean-Marc&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0012-365X%252893%2529E0161-V&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPicciotto1999" class="citation cs2">Picciotto, Henri (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7gTMKr7TT6gC&pg=PA208"><i>Geometry Labs</i></a>, MathEducationPage.org, p. 208</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+Labs&rft.pages=208&rft.pub=MathEducationPage.org&rft.date=1999&rft.aulast=Picciotto&rft.aufirst=Henri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7gTMKr7TT6gC%26pg%3DPA208&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span>.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin1996" class="citation book cs1">Martin, George E. (1996). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/polyominoesguide00mart_0"><i>Polyominoes: A guide to puzzles and problems in tiling</i></a></span> (2nd ed.). <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-501-0" title="Special:BookSources/978-0-88385-501-0"><bdi>978-0-88385-501-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polyominoes%3A+A+guide+to+puzzles+and+problems+in+tiling&rft.edition=2nd&rft.pub=Mathematical+Association+of+America&rft.date=1996&rft.isbn=978-0-88385-501-0&rft.aulast=Martin&rft.aufirst=George+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpolyominoesguide00mart_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFC.B._HaselgroveJenifer_Haselgrove1960" class="citation journal cs1">C.B. Haselgrove; Jenifer Haselgrove (October 1960). <a rel="nofollow" class="external text" href="https://www.archim.org.uk/eureka/archive/Eureka-23.pdf">"A Computer Program for Pentominoes"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Eureka_(University_of_Cambridge_magazine)" title="Eureka (University of Cambridge magazine)">Eureka</a></i>. <b>23</b>: <span class="nowrap">16–</span>18.</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=A+Computer+Program+for+Pentominoes&rft.volume=23&rft.pages=%3Cspan+class%3D%22nowrap%22%3E16-%3C%2Fspan%3E18&rft.date=1960-10&rft.au=C.B.+Haselgrove&rft.au=Jenifer+Haselgrove&rft_id=https%3A%2F%2Fwww.archim.org.uk%2Feureka%2Farchive%2FEureka-23.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolomb1970" class="citation journal cs1">Golomb, Solomon W. (1970). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9800%2870%2980055-2">"Tiling with Sets of Polyominoes"</a>. <i>Journal of Combinatorial Theory</i>. <b>9</b>: <span class="nowrap">60–</span>71. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9800%2870%2980055-2">10.1016/S0021-9800(70)80055-2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory&rft.atitle=Tiling+with+Sets+of+Polyominoes&rft.volume=9&rft.pages=%3Cspan+class%3D%22nowrap%22%3E60-%3C%2Fspan%3E71&rft.date=1970&rft_id=info%3Adoi%2F10.1016%2FS0021-9800%2870%2980055-2&rft.aulast=Golomb&rft.aufirst=Solomon+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0021-9800%252870%252980055-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE.D._DemaineM.L._Demaine2007" class="citation journal cs1">E.D. Demaine; M.L. Demaine (June 2007). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s00373-007-0713-4">"Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity"</a>. <i>Graphs and Combinatorics</i>. <b>23</b>: <span class="nowrap">195–</span>208. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00373-007-0713-4">10.1007/s00373-007-0713-4</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:17190810">17190810</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Graphs+and+Combinatorics&rft.atitle=Jigsaw+Puzzles%2C+Edge+Matching%2C+and+Polyomino+Packing%3A+Connections+and+Complexity&rft.volume=23&rft.pages=%3Cspan+class%3D%22nowrap%22%3E195-%3C%2Fspan%3E208&rft.date=2007-06&rft_id=info%3Adoi%2F10.1007%2Fs00373-007-0713-4&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17190810%23id-name%3DS2CID&rft.au=E.D.+Demaine&rft.au=M.L.+Demaine&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs00373-007-0713-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFS.W._GolombL.D._Baumert1965" class="citation journal cs1">S.W. Golomb; L.D. Baumert (1965). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321296.321300">"Backtrack Programming"</a>. <i>Journal of the ACM</i>. <b>12</b> (4): <span class="nowrap">516–</span>524. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321296.321300">10.1145/321296.321300</a></span>.</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+ACM&rft.atitle=Backtrack+Programming&rft.volume=12&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E516-%3C%2Fspan%3E524&rft.date=1965&rft_id=info%3Adoi%2F10.1145%2F321296.321300&rft.au=S.W.+Golomb&rft.au=L.D.+Baumert&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F321296.321300&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">Golomb, <i>Polyominoes</i>, chapter 8</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReid" class="citation web cs1">Reid, Michael. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040116204311/http://www.math.ucf.edu/~reid/Polyomino/rectifiable_bib.html">"References for Rectifiable Polyominoes"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.ucf.edu/~reid/Polyomino/rectifiable_bib.html">the original</a> on 2004-01-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-05-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=References+for+Rectifiable+Polyominoes&rft.aulast=Reid&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.math.ucf.edu%2F~reid%2FPolyomino%2Frectifiable_bib.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReid" class="citation web cs1">Reid, Michael. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070416210221/http://www.math.ucf.edu/~reid/Polyomino/rectifiable_data.html">"List of known prime rectangles for various polyominoes"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.ucf.edu/~reid/Polyomino/rectifiable_data.html">the original</a> on 2007-04-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-05-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=List+of+known+prime+rectangles+for+various+polyominoes&rft.aulast=Reid&rft.aufirst=Michael&rft_id=http%3A%2F%2Fwww.math.ucf.edu%2F~reid%2FPolyomino%2Frectifiable_data.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlarnerGöbel,_F.1969" class="citation journal cs1">Klarner, D.A.; Göbel, F. (1969). "Packing boxes with congruent figures". <i>Indagationes Mathematicae</i>. <b>31</b>: <span class="nowrap">465–</span>472.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indagationes+Mathematicae&rft.atitle=Packing+boxes+with+congruent+figures&rft.volume=31&rft.pages=%3Cspan+class%3D%22nowrap%22%3E465-%3C%2Fspan%3E472&rft.date=1969&rft.aulast=Klarner&rft.aufirst=D.A.&rft.au=G%C3%B6bel%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlarner1973" class="citation web cs1">Klarner, David A. (February 1973). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071023221204/http://historical.ncstrl.org/litesite-data/stan/CS-TR-73-338.pdf">"A Finite Basis Theorem Revisited"</a> <span class="cs1-format">(PDF)</span>. Stanford University Technical Report STAN-CS-73–338. Archived from <a rel="nofollow" class="external text" href="http://historical.ncstrl.org/litesite-data/stan/CS-TR-73-338.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2007-10-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-05-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Finite+Basis+Theorem+Revisited&rft.pub=Stanford+University+Technical+Report+STAN-CS-73%E2%80%93338&rft.date=1973-02&rft.aulast=Klarner&rft.aufirst=David+A.&rft_id=http%3A%2F%2Fhistorical.ncstrl.org%2Flitesite-data%2Fstan%2FCS-TR-73-338.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKamenetskyCooke,_Tristrom2015" class="citation arxiv cs1">Kamenetsky, Dmitry; Cooke, Tristrom (2015). "Tiling rectangles with holey polyominoes". <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/1411.2699">1411.2699</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.CG">cs.CG</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Tiling+rectangles+with+holey+polyominoes&rft.date=2015&rft_id=info%3Aarxiv%2F1411.2699&rft.aulast=Kamenetsky&rft.aufirst=Dmitry&rft.au=Cooke%2C+Tristrom&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolomb1966" class="citation journal cs1">Golomb, Solomon W. (1966). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9800%2866%2980033-9">"Tiling with Polyominoes"</a>. <i>Journal of Combinatorial Theory</i>. <b>1</b> (2): <span class="nowrap">280–</span>296. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9800%2866%2980033-9">10.1016/S0021-9800(66)80033-9</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Combinatorial+Theory&rft.atitle=Tiling+with+Polyominoes&rft.volume=1&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E280-%3C%2Fspan%3E296&rft.date=1966&rft_id=info%3Adoi%2F10.1016%2FS0021-9800%2866%2980033-9&rft.aulast=Golomb&rft.aufirst=Solomon+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0021-9800%252866%252980033-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMooreRobson2001" class="citation web cs1"><a href="/wiki/Cris_Moore" class="mw-redirect" title="Cris Moore">Moore, Cristopher</a>; Robson, John Michael (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130617140921/http://www.santafe.edu/media/workingpapers/00-03-019.pdf">"Hard Tiling Problems with Simple Tiles"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.santafe.edu/media/workingpapers/00-03-019.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2013-06-17.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hard+Tiling+Problems+with+Simple+Tiles&rft.date=2001&rft.aulast=Moore&rft.aufirst=Cristopher&rft.au=Robson%2C+John+Michael&rft_id=http%3A%2F%2Fwww.santafe.edu%2Fmedia%2Fworkingpapers%2F00-03-019.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetersen1999" class="citation cs2">Petersen, Ivars (September 25, 1999), <a rel="nofollow" class="external text" href="http://www.sciencenews.org/pages/sn_arc99/9_25_99/mathland.htm">"Math Trek: Tiling with Polyominoes"</a>, <i><a href="/wiki/Science_News" title="Science News">Science News</a></i>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080320031732/http://www.sciencenews.org/pages/sn_arc99/9_25_99/mathland.htm">archived</a> from the original on March 20, 2008<span class="reference-accessdate">, retrieved <span class="nowrap">March 11,</span> 2012</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+News&rft.atitle=Math+Trek%3A+Tiling+with+Polyominoes&rft.date=1999-09-25&rft.aulast=Petersen&rft.aufirst=Ivars&rft_id=http%3A%2F%2Fwww.sciencenews.org%2Fpages%2Fsn_arc99%2F9_25_99%2Fmathland.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span>.</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1965" class="citation journal cs1">Gardner, Martin (July 1965). "On the relation between mathematics and the ordered patterns of Op art". <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. <b>213</b> (1): <span class="nowrap">100–</span>104. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican1265-100">10.1038/scientificamerican1265-100</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=On+the+relation+between+mathematics+and+the+ordered+patterns+of+Op+art&rft.volume=213&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E100-%3C%2Fspan%3E104&rft.date=1965-07&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1265-100&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1965" class="citation journal cs1">Gardner, Martin (August 1965). "Thoughts on the task of communication with intelligent organisms on other worlds". <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. <b>213</b> (2): <span class="nowrap">96–</span>100. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0865-96">10.1038/scientificamerican0865-96</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=Thoughts+on+the+task+of+communication+with+intelligent+organisms+on+other+worlds&rft.volume=213&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E96-%3C%2Fspan%3E100&rft.date=1965-08&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0865-96&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1975" class="citation journal cs1">Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes". <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. <b>233</b> (2): <span class="nowrap">112–</span>115. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican0875-112">10.1038/scientificamerican0875-112</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=More+about+tiling+the+plane%3A+the+possibilities+of+polyominoes%2C+polyiamonds+and+polyhexes&rft.volume=233&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E112-%3C%2Fspan%3E115&rft.date=1975-08&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican0875-112&rft.aulast=Gardner&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRawsthorne1988" class="citation journal cs1">Rawsthorne, Daniel A. (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2888%2990081-7">"Tiling complexity of small <i>n</i>-ominoes<br />(<i>n</i><10)"</a>. <i>Discrete Mathematics</i>. <b>70</b>: <span class="nowrap">71–</span>75. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2888%2990081-7">10.1016/0012-365X(88)90081-7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Mathematics&rft.atitle=Tiling+complexity+of+small+n-ominoes%3Cbr%2F%3E%28n%3C10%29&rft.volume=70&rft.pages=%3Cspan+class%3D%22nowrap%22%3E71-%3C%2Fspan%3E75&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0012-365X%2888%2990081-7&rft.aulast=Rawsthorne&rft.aufirst=Daniel+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0012-365X%252888%252990081-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRhoads2003" class="citation book cs1">Rhoads, Glenn C. (2003). <i>Planar Tilings and the Search for an Aperiodic Prototile</i>. PhD dissertation, Rutgers University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Planar+Tilings+and+the+Search+for+an+Aperiodic+Prototile&rft.pub=PhD+dissertation%2C+Rutgers+University&rft.date=2003&rft.aulast=Rhoads&rft.aufirst=Glenn+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text">Grünbaum and Shephard, section 9.4</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeatingVince,_A.1999" class="citation journal cs1">Keating, K.; Vince, A. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FPL00009442">"Isohedral Polyomino Tiling of the Plane"</a>. <i><a href="/wiki/Discrete_%26_Computational_Geometry" title="Discrete & Computational Geometry">Discrete & Computational Geometry</a></i>. <b>21</b> (4): <span class="nowrap">615–</span>630. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FPL00009442">10.1007/PL00009442</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+%26+Computational+Geometry&rft.atitle=Isohedral+Polyomino+Tiling+of+the+Plane&rft.volume=21&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E615-%3C%2Fspan%3E630&rft.date=1999&rft_id=info%3Adoi%2F10.1007%2FPL00009442&rft.aulast=Keating&rft.aufirst=K.&rft.au=Vince%2C+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FPL00009442&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRhoads2005" class="citation journal cs1">Rhoads, Glenn C. (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2004.05.002">"Planar tilings by polyominoes, polyhexes, and polyiamonds"</a>. <i>Journal of Computational and Applied Mathematics</i>. <b>174</b> (2): <span class="nowrap">329–</span>353. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005JCoAM.174..329R">2005JCoAM.174..329R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2004.05.002">10.1016/j.cam.2004.05.002</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft.atitle=Planar+tilings+by+polyominoes%2C+polyhexes%2C+and+polyiamonds&rft.volume=174&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E329-%3C%2Fspan%3E353&rft.date=2005&rft_id=info%3Adoi%2F10.1016%2Fj.cam.2004.05.002&rft_id=info%3Abibcode%2F2005JCoAM.174..329R&rft.aulast=Rhoads&rft.aufirst=Glenn+C.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.cam.2004.05.002&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNiţică2003" class="citation cs1">Niţică, Viorel (2003). "Rep-tiles revisited". <i>MASS selecta</i>. Providence, RI: American Mathematical Society. pp. <span class="nowrap">205–</span>217. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2027179">2027179</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Rep-tiles+revisited&rft.btitle=MASS+selecta&rft.place=Providence%2C+RI&rft.pages=%3Cspan+class%3D%22nowrap%22%3E205-%3C%2Fspan%3E217&rft.pub=American+Mathematical+Society&rft.date=2003&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2027179%23id-name%3DMR&rft.aulast=Ni%C5%A3ic%C4%83&rft.aufirst=Viorel&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091027093922/http://geocities.com/jorgeluismireles/polypolyominoes/">Mireles, J.L., "Poly<sup>2</sup>ominoes"</a></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.iread.it/Poly/">"Resta, G., "Polypolyominoes"<span class="cs1-kern-right"></span>"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110222051130/http://www.iread.it/Poly/">Archived</a> from the original on 2011-02-22<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-07-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Resta%2C+G.%2C+%22Polypolyominoes%22&rft_id=http%3A%2F%2Fwww.iread.it%2FPoly%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://sicherman.net/rosp/triplep.html">"Zucca, L., "Triple Pentominoes"<span class="cs1-kern-right"></span>"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-04-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Zucca%2C+L.%2C+%22Triple+Pentominoes%22&rft_id=http%3A%2F%2Fsicherman.net%2Frosp%2Ftriplep.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarbansCibulisLeeLiu2005" class="citation book cs1">Barbans, Uldis; Cibulis, Andris; Lee, Gilbert; Liu, Andy; Wainwright, Robert (2005). "Polyomino Number Theory (III)". In <a href="/wiki/Barry_Arthur_Cipra" title="Barry Arthur Cipra">Cipra, Barry Arthur</a>; Demaine, Erik D.; Demaine, Martin L.; Rodgers, Tom (eds.). <i>Tribute to a Mathemagician</i>. Wellesley, MA: A.K. Peters. pp. <span class="nowrap">131–</span>136. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-204-5" title="Special:BookSources/978-1-56881-204-5"><bdi>978-1-56881-204-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Polyomino+Number+Theory+%28III%29&rft.btitle=Tribute+to+a+Mathemagician&rft.place=Wellesley%2C+MA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E131-%3C%2Fspan%3E136&rft.pub=A.K.+Peters&rft.date=2005&rft.isbn=978-1-56881-204-5&rft.aulast=Barbans&rft.aufirst=Uldis&rft.au=Cibulis%2C+Andris&rft.au=Lee%2C+Gilbert&rft.au=Liu%2C+Andy&rft.au=Wainwright%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a>, 2nd edition, entry <i>domino</i></span> </li> </ol></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=Polyomino&action=edit&section=20" 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.eklhad.net/polyomino/">Karl Dahlke's polyomino finite-rectangle tilings</a></li> <li><a rel="nofollow" class="external text" href="http://www-cs-faculty.stanford.edu/~knuth/programs.html#polyominoes">An implementation and description of Jensen's method</a></li> <li><a rel="nofollow" class="external text" href="http://www.statslab.cam.ac.uk/~grg/books/hammfest/19-sgw.ps">A paper describing modern estimates (PS)</a></li> <li><span class="citation mathworld" id="Reference-Mathworld-Polyomino"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Polyomino.html">"Polyomino"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Polyomino&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPolyomino.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APolyomino" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath039.htm">MathPages – Notes on enumeration of polyominoes with various symmetries</a></li> <li><a rel="nofollow" class="external text" href="http://www.mayhematics.com/d/db.htm">List of dissection problems in Fairy Chess Review</a></li> <li><i><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/Tetrads/">Tetrads</a></i> by Karl Scherer, <a href="/wiki/Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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