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<!doctype html><html lang="en"><head><title data-rh="true">How Do You Mathematically Guard A Museum? | by Cole Frederick | Science Spectrum | Nov, 2024 | Medium</title><meta data-rh="true" charset="utf-8"/><meta data-rh="true" name="viewport" content="width=device-width,minimum-scale=1,initial-scale=1,maximum-scale=1"/><meta data-rh="true" name="theme-color" content="#000000"/><meta data-rh="true" name="twitter:app:name:iphone" content="Medium"/><meta data-rh="true" name="twitter:app:id:iphone" content="828256236"/><meta data-rh="true" property="al:ios:app_name" content="Medium"/><meta data-rh="true" property="al:ios:app_store_id" content="828256236"/><meta data-rh="true" property="al:android:package" content="com.medium.reader"/><meta data-rh="true" property="fb:app_id" content="542599432471018"/><meta data-rh="true" property="og:site_name" content="Medium"/><meta data-rh="true" property="og:type" content="article"/><meta data-rh="true" property="article:published_time" content="2024-11-19T07:02:04.932Z"/><meta data-rh="true" name="title" content="How Do You Mathematically Guard A Museum? | by Cole Frederick | Science Spectrum | Nov, 2024 | Medium"/><meta data-rh="true" property="og:title" content="How Do You Mathematically Guard A Museum?"/><meta data-rh="true" property="al:android:url" content="medium://p/6971a775e35a"/><meta data-rh="true" property="al:ios:url" content="medium://p/6971a775e35a"/><meta data-rh="true" property="al:android:app_name" content="Medium"/><meta data-rh="true" name="description" content="One of my favorite things about mathematics is how it can be applied to such a wide variety of situations. 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href="/@colefp?source=post_page---byline--6971a775e35a--------------------------------">Cole Frederick</a></p></div></div></div><span class="ji jj" aria-hidden="true"><span class="bf b bg z du">·</span></span><p class="bf b jf jg du"><span><a class="jk jl ah ai aj ak al am an ao ap aq ar ex jm jn" rel="noopener follow" href="/m/signin?actionUrl=https%3A%2F%2Fmedium.com%2F_%2Fsubscribe%2Fuser%2F24757aa678c6&operation=register&redirect=https%3A%2F%2Fmedium.com%2Fscience-spectrum%2Fhow-do-you-mathematically-guard-a-museum-6971a775e35a&user=Cole+Frederick&userId=24757aa678c6&source=post_page-24757aa678c6--byline--6971a775e35a---------------------post_header-----------">Follow</a></span></p></div></div></span></div></div><div class="l jo"><span class="bf b bg z du"><div class="ab cn jp jq jr"><div class="js jt ab"><div class="bf b bg z du ab ju"><span class="gt l jo">Published in</span><div><div class="l" aria-hidden="false"><a class="af ag ah ai aj ak al am an ao ap aq ar jh ab q" 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https://miro.medium.com/v2/resize:fit:2000/format:webp/0*u3EvBy1mfYrSj-sn 2000w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 1000px" type="image/webp"/><source data-testid="og" srcSet="https://miro.medium.com/v2/resize:fit:640/0*u3EvBy1mfYrSj-sn 640w, https://miro.medium.com/v2/resize:fit:720/0*u3EvBy1mfYrSj-sn 720w, https://miro.medium.com/v2/resize:fit:750/0*u3EvBy1mfYrSj-sn 750w, https://miro.medium.com/v2/resize:fit:786/0*u3EvBy1mfYrSj-sn 786w, https://miro.medium.com/v2/resize:fit:828/0*u3EvBy1mfYrSj-sn 828w, https://miro.medium.com/v2/resize:fit:1100/0*u3EvBy1mfYrSj-sn 1100w, https://miro.medium.com/v2/resize:fit:2000/0*u3EvBy1mfYrSj-sn 2000w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 1000px"/><img alt="" class="bh mt og c" width="1000" height="667" loading="eager" role="presentation"/></picture></div></div><figcaption class="oh ff oi ns nt oj ok bf b bg z du">Photo by <a class="af ol" href="https://unsplash.com/@julietfurst?utm_source=medium&utm_medium=referral" rel="noopener ugc nofollow" target="_blank">Juliet Furst</a> on <a class="af ol" href="https://unsplash.com/?utm_source=medium&utm_medium=referral" rel="noopener ugc nofollow" target="_blank">Unsplash</a></figcaption></figure></div></div></div><div class="ab cb"><div class="ci bh fz ga gb gc"><p id="c779" class="pw-post-body-paragraph om on hd oo b hx op oq or ia os ot ou ov ow ox oy oz pa pb pc pd pe pf pg ph gw bk">One of my favorite things about mathematics is how it can be applied to such a wide variety of situations. Unexpected problems can turn into a fun math puzzle that leads to a whole new series of theorems being discovered! I’ve written many articles about these situations before, and they are often my favorite topics to dive into.</p><figure class="nv nw nx ny nz nm ns nt paragraph-image"><div class="ns nt pi"><picture><source srcSet="https://miro.medium.com/v2/resize:fit:640/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 640w, https://miro.medium.com/v2/resize:fit:720/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 720w, https://miro.medium.com/v2/resize:fit:750/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 750w, https://miro.medium.com/v2/resize:fit:786/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 786w, https://miro.medium.com/v2/resize:fit:828/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 828w, https://miro.medium.com/v2/resize:fit:1100/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 1100w, https://miro.medium.com/v2/resize:fit:1034/format:webp/1*_rvk-CDYTJSVVSopPYr2xQ.png 1034w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 517px" type="image/webp"/><source data-testid="og" srcSet="https://miro.medium.com/v2/resize:fit:640/1*_rvk-CDYTJSVVSopPYr2xQ.png 640w, https://miro.medium.com/v2/resize:fit:720/1*_rvk-CDYTJSVVSopPYr2xQ.png 720w, https://miro.medium.com/v2/resize:fit:750/1*_rvk-CDYTJSVVSopPYr2xQ.png 750w, https://miro.medium.com/v2/resize:fit:786/1*_rvk-CDYTJSVVSopPYr2xQ.png 786w, https://miro.medium.com/v2/resize:fit:828/1*_rvk-CDYTJSVVSopPYr2xQ.png 828w, https://miro.medium.com/v2/resize:fit:1100/1*_rvk-CDYTJSVVSopPYr2xQ.png 1100w, https://miro.medium.com/v2/resize:fit:1034/1*_rvk-CDYTJSVVSopPYr2xQ.png 1034w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 517px"/><img alt="" class="bh mt og c" width="517" height="359" loading="eager" role="presentation"/></picture></div><figcaption class="oh ff oi ns nt oj ok bf b bg z du">Victor Klee (left) and Václav Chvátal (right) (<a class="af ol" href="https://en.wikipedia.org/wiki/Victor_Klee" rel="noopener ugc nofollow" target="_blank">Source</a>, <a class="af ol" href="https://en.wikipedia.org/wiki/V%C3%A1clav_Chv%C3%A1tal" rel="noopener ugc nofollow" target="_blank">Source</a>)</figcaption></figure><p id="dd76" class="pw-post-body-paragraph om on hd oo b hx op oq or ia os ot ou ov ow ox oy oz pa pb pc pd pe pf pg ph gw bk">A classic example of this is the museum problem. It was first posed by Victor Klee in 1973 and was solved later by Václav Chvátal, both pictured above. It is now considered a quintessential example of the usefulness and flexibility of graph theory. The museum problem has become famous because of how visually intuitive it is and how easy it is to understand. In this problem, we are given a museum as a series of rooms like in the examples below.</p><figure class="nv nw nx ny nz nm ns nt paragraph-image"><div role="button" tabindex="0" class="oc od fj oe bh of"><div class="ns nt pj"><picture><source srcSet="https://miro.medium.com/v2/resize:fit:640/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 640w, https://miro.medium.com/v2/resize:fit:720/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 720w, https://miro.medium.com/v2/resize:fit:750/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 750w, https://miro.medium.com/v2/resize:fit:786/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 786w, https://miro.medium.com/v2/resize:fit:828/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 828w, https://miro.medium.com/v2/resize:fit:1100/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 1100w, https://miro.medium.com/v2/resize:fit:1400/format:webp/1*o3j9VYM8p-N1jIGYSueQ9w.png 1400w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 700px" type="image/webp"/><source data-testid="og" srcSet="https://miro.medium.com/v2/resize:fit:640/1*o3j9VYM8p-N1jIGYSueQ9w.png 640w, https://miro.medium.com/v2/resize:fit:720/1*o3j9VYM8p-N1jIGYSueQ9w.png 720w, https://miro.medium.com/v2/resize:fit:750/1*o3j9VYM8p-N1jIGYSueQ9w.png 750w, https://miro.medium.com/v2/resize:fit:786/1*o3j9VYM8p-N1jIGYSueQ9w.png 786w, https://miro.medium.com/v2/resize:fit:828/1*o3j9VYM8p-N1jIGYSueQ9w.png 828w, https://miro.medium.com/v2/resize:fit:1100/1*o3j9VYM8p-N1jIGYSueQ9w.png 1100w, https://miro.medium.com/v2/resize:fit:1400/1*o3j9VYM8p-N1jIGYSueQ9w.png 1400w" sizes="(min-resolution: 4dppx) and (max-width: 700px) 50vw, (-webkit-min-device-pixel-ratio: 4) and (max-width: 700px) 50vw, (min-resolution: 3dppx) and (max-width: 700px) 67vw, (-webkit-min-device-pixel-ratio: 3) and (max-width: 700px) 65vw, (min-resolution: 2.5dppx) and (max-width: 700px) 80vw, (-webkit-min-device-pixel-ratio: 2.5) and (max-width: 700px) 80vw, (min-resolution: 2dppx) and (max-width: 700px) 100vw, (-webkit-min-device-pixel-ratio: 2) and (max-width: 700px) 100vw, 700px"/><img alt="" class="bh mt og c" width="700" height="218" loading="lazy" role="presentation"/></picture></div></div><figcaption class="oh ff oi ns nt oj ok bf b bg z du">Four example museums (<a class="af ol" href="https://math.mit.edu/~apost/courses/18.204_2018/Nicole_Chesnokov_paper.pdf" rel="noopener ugc nofollow" target="_blank">Source</a>, modified)</figcaption></figure><p id="baf4" class="pw-post-body-paragraph om on hd oo b hx op oq or ia os ot ou ov ow ox oy oz pa pb pc pd pe pf pg ph gw bk">We want to know how many guards are needed to watch over this museum. For the purposes of this problem, a guard is represented as a…</p></div></div></div></div></section></div></div></article></div><div class="l"></div><footer class="pk pl pm pn gv ab q po jb c"><div class="l ae"><div class="ab cb"><div class="ci bh fz ga gb gc"><div class="ab cp pp"><div class="ab q ll"><div class="pq l"><span class="l pr ps pt e d"><div class="ab q ll lm"><div class="pw-multi-vote-icon fj gt ln lo lp"><span><a class="af ag ah ai aj ak al am an ao ap aq ar as at" data-testid="footerClapButton" rel="noopener follow" href="/m/signin?actionUrl=https%3A%2F%2Fmedium.com%2F_%2Fvote%2Fscience-spectrum%2F6971a775e35a&operation=register&redirect=https%3A%2F%2Fmedium.com%2Fscience-spectrum%2Fhow-do-you-mathematically-guard-a-museum-6971a775e35a&user=Cole+Frederick&userId=24757aa678c6&source=---footer_actions--6971a775e35a---------------------clap_footer-----------"><div><div class="bm" aria-hidden="false"><div class="lq ao lr ls lt lu am lv lw lx lp"><svg 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