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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Polyomino</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Ein <b>Polyomino</b> (Kunstwort, abgeleitet von <a href="Domino" title="Domino">Domino</a>) ist eine Fläche, die aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> zusammenhängenden <a href="Quadrat_(Geometrie)" class="mw-redirect" title="Quadrat (Geometrie)">Quadraten</a> besteht.
</p><p>Für kleine <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> sind auch die Bezeichnungen
<b>Monomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"></span>),
<b>Domino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy"></span>),
<b>Tromino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=3}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy"></span>),
<b>Tetromino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=4}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d928ec15aeef83aade867992ee473933adb6139d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=4}" loading="lazy"></span>),
<b>Pentomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=5}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=5}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdb41e9a10a8fd7179b9170149a8d70949ba5d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}" loading="lazy"></span>),
<b>Hexomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>6</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0365f0b9f2721ed3ebb488a96d7348d978acf8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}" loading="lazy"></span>),
<b>Heptomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=7}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>7</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=7}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a264db40fcdf1645e6dabe1885e21595de502623.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=7}" loading="lazy"></span>),
<b>Oktomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=8}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>8</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=8}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1344e2c194f016e12bb978493013955fe2f445f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}" loading="lazy"></span>),
<b>Nonomino</b> oder <b>Enneomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=9}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>9</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=9}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1029ce8384fde9f4da54009c5a79f17a9758085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=9}" loading="lazy"></span>),
<b>Dekomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=10}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>10</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=10}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fecadd7726f4d4c6c390c2f2e73533d7a3729ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=10}" loading="lazy"></span>),
<b>Undekomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=11}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>11</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=11}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e7f254d84dfad562e1c9ecd6680b456a5cae01b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=11}" loading="lazy"></span>),
<b>Dodekomino</b> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=12}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>12</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=12}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38fb7dfd38b5db65d78bb6f2c1bf9e6e6cd07c14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=12}" loading="lazy"></span>)
usw. üblich.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Ein Polyomino oder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Mino ist eine Figur <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>, die aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy"></span> <a href="Kongruenz_(Geometrie)" title="Kongruenz (Geometrie)">kongruenten</a> Quadraten besteht, für die gilt
</p>
<ol><li>je zwei Quadrate haben entweder keinen Punkt oder eine Ecke oder eine Kante gemeinsam,</li>
<li>zu je zwei verschiedenen Quadraten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8ea6463cb36d8278ff71214fb4d13127039ae53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.509ex;" alt="{\displaystyle Q_{1}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{*}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q^{*}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1785c0a77ab5a06684e8a7ac4f5e59d59ec0319.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.893ex; height:2.676ex;" alt="{\displaystyle Q^{*}}" loading="lazy"></span> aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> gibt es eine Folge <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{1}Q_{2}\ldots Q_{k-1}Q^{*}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>…<!-- … --></mo>
<msub>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msub>
<msup>
<mi>Q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q_{1}Q_{2}\ldots Q_{k-1}Q^{*}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c74c271de9a34f8f8d36e51697eb2ffac236d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.203ex; height:2.676ex;" alt="{\displaystyle Q_{1}Q_{2}\ldots Q_{k-1}Q^{*}}" loading="lazy"></span> von benachbarten Quadraten aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>.</li></ol>
<p>Dabei heißen zwei Quadrate benachbart, wenn die Menge ihrer gemeinsamen Punkte eine Seite ist. Folgende Beispiele stellen demnach keine Polyominos dar.
</p>
<ul class="gallery mw-gallery-traditional">
<li class="gallerybox" style="width: 115px">
<div class="thumb" style="width: 110px; height: 80px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">1. ungültig</div>
</li>
<li class="gallerybox" style="width: 115px">
<div class="thumb" style="width: 110px; height: 80px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">2. ungültig</div>
</li>
<li class="gallerybox" style="width: 115px">
<div class="thumb" style="width: 110px; height: 80px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">3. ungültig</div>
</li>
<li class="gallerybox" style="width: 115px">
<div class="thumb" style="width: 110px; height: 80px;"><span typeof="mw:File"></span></div>
<div class="gallerytext">4. ungültig</div>
</li>
</ul>
<p>Für besondere Anwendungen wird zusätzlich gefordert:
</p>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> bildet eine <a href="Einfach_zusammenh%C3%A4ngend" class="mw-redirect" title="Einfach zusammenhängend">einfach zusammenhängende</a> Punktmenge.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Darstellung_über_Zusammenhangsgraphen"><span id="Darstellung_.C3.BCber_Zusammenhangsgraphen"></span>Darstellung über Zusammenhangsgraphen</h3></div>
<p>Jedem Polyomino <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> lässt sich ein <a href="Zusammenhang_(Graphentheorie)" title="Zusammenhang (Graphentheorie)">Zusammenhangsgraph</a> zuordnen, indem man jedes Quadrat aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> als Knoten und das Benachbartsein zweier Quadrate durch eine Kante wiedergibt. Nachfolgend wird dies anhand der 5 Tetrominos dargestellt.
</p>
<ul class="gallery mw-gallery-nolines">
<li class="gallerybox" style="width: 125px">
<div class="thumb" style="width: 120px;"><span typeof="mw:File"></span></div>
<div class="gallerytext"></div>
</li>
<li class="gallerybox" style="width: 125px">
<div class="thumb" style="width: 120px;"><span typeof="mw:File"></span></div>
<div class="gallerytext"></div>
</li>
<li class="gallerybox" style="width: 125px">
<div class="thumb" style="width: 120px;"><span typeof="mw:File"></span></div>
<div class="gallerytext"></div>
</li>
<li class="gallerybox" style="width: 125px">
<div class="thumb" style="width: 120px;"><span typeof="mw:File"></span></div>
<div class="gallerytext"></div>
</li>
<li class="gallerybox" style="width: 125px">
<div class="thumb" style="width: 120px;"><span typeof="mw:File"></span></div>
<div class="gallerytext"></div>
</li>
</ul>
<div class="mw-heading mw-heading2"><h2 id="Konstruktion">Konstruktion</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Bestimmung_der_Anzahlen">Bestimmung der Anzahlen</h3></div>
<p>Es gibt verschiedene Ansätze, die Anzahl der Polyominos zu bestimmen. Am häufigsten wird nur bis auf <a href="Kongruenz_(Geometrie)" title="Kongruenz (Geometrie)">Kongruenz</a> unterschieden. In praktischen Sachverhalten sind jedoch häufig nur <a href="Orientierung_(Mathematik)" title="Orientierung (Mathematik)">orientierungserhaltende</a> Bewegungen für das <i>Zur-Deckung-Bringen</i> zugelassen, also nur Drehungen und Verschiebungen und keine <a href="Spiegelung_(Geometrie)" title="Spiegelung (Geometrie)">Achsenspiegelungen</a>. Auch bei dem Spiel <a href="Tetris" title="Tetris">Tetris</a> ist dies der Fall. Kongruente Polyominos, die eine unterschiedliche Orientierung besitzen, werden dabei als verschieden angesehen
</p>
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</style><div class="thumb tleft vl-mehrere-bilder" style="width:462px;"><div class="thumbinner"><div class="vl-mehrere-bilder-horizontal" style="width:224px;"><div class="thumbimage"><span typeof="mw:File"></span></div></div><div class="vl-mehrere-bilder-horizontal" style="width:224px;"><div class="thumbimage"><span typeof="mw:File"></span></div></div><div style="clear:both;"></div>
<div style="clear:both;"></div>
<div class="thumbcaption" style="clear:both;text-align:left;">nicht-orientierungserhaltende Bewegung</div></div></div>
<div style="clear:left;"></div>
<p><span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa563369b75d1fdb0b3eb905a0b698f27b7dda15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle A(n)}" loading="lazy"></span> bezeichnet die Anzahl Polyominos, die sich bis auf Kongruenz aus <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> Quadraten bilden lassen. <span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A'(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aca8b61f24bc777dfabe8551e2025ac5918ae81b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.632ex; height:3.009ex;" alt="{\displaystyle A'(n)}" loading="lazy"></span> ist die Anzahl unter Beachtung der Orientierung, d. h. zueinander spiegelbildliche (wie oben illustriert) werden als verschieden betrachtet. <span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A''(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b960039204a8d73349b0ccdfc67c640c9c03544d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.084ex; height:3.009ex;" alt="{\displaystyle A''(n)}" loading="lazy"></span> bezeichnet die Anzahl unter Beachtung der Orientierung und aller möglichen Lagen, dabei werden sogar zwei zueinander gedrehte, aber sonst gleiche Polyominos als verschieden angesehen. Vor allem <span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa563369b75d1fdb0b3eb905a0b698f27b7dda15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle A(n)}" loading="lazy"></span> ist von Interesse.
</p>
<table class="wikitable" style="text-align:right;">
<tbody><tr>
<th><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span></th>
<th><span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa563369b75d1fdb0b3eb905a0b698f27b7dda15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle A(n)}" loading="lazy"></span><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></th>
<th><span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A'(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aca8b61f24bc777dfabe8551e2025ac5918ae81b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.632ex; height:3.009ex;" alt="{\displaystyle A'(n)}" loading="lazy"></span><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></th>
<th><span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>A</mi>
<mo>″</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A''(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b960039204a8d73349b0ccdfc67c640c9c03544d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.084ex; height:3.009ex;" alt="{\displaystyle A''(n)}" loading="lazy"></span><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>
</th></tr>
<tr>
<td>1</td>
<td>1</td>
<td>1</td>
<td>1
</td></tr>
<tr>
<td>2</td>
<td>1</td>
<td>1</td>
<td>2
</td></tr>
<tr>
<td>3</td>
<td>2</td>
<td>2</td>
<td>6
</td></tr>
<tr>
<td>4</td>
<td>5</td>
<td>7</td>
<td>19
</td></tr>
<tr>
<td>5</td>
<td>12</td>
<td>18</td>
<td>63
</td></tr>
<tr>
<td>6</td>
<td>35</td>
<td>60</td>
<td>216
</td></tr>
<tr>
<td>7</td>
<td>108</td>
<td>196</td>
<td>760
</td></tr>
<tr>
<td>8</td>
<td>369</td>
<td>704</td>
<td>2.725
</td></tr>
<tr>
<td>9</td>
<td>1.285</td>
<td>2.500</td>
<td>9.910
</td></tr>
<tr>
<td>10</td>
<td>4.655</td>
<td>9.189</td>
<td>36.446
</td></tr>
<tr>
<td>11</td>
<td>17.073</td>
<td>33.896</td>
<td>135.268
</td></tr>
<tr>
<td>12</td>
<td>63.600</td>
<td>126.759</td>
<td>505.861
</td></tr>
<tr>
<td>13</td>
<td>238.591</td>
<td>476.270</td>
<td>1.903 890
</td></tr>
<tr>
<td>14</td>
<td>901.971</td>
<td>1.802.312</td>
<td>7.204.874
</td></tr>
<tr>
<td>15</td>
<td>3.426.576</td>
<td>6.849.777</td>
<td>27.394.666
</td></tr></tbody></table>
<p>Werden ausschließlich einfach zusammenhängende Polyominos gezählt, so ergeben sich von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=7}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>7</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=7}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a264db40fcdf1645e6dabe1885e21595de502623.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=7}" loading="lazy"></span> an abweichende Zahlen.<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>
<div class="mw-heading mw-heading3"><h3 id="Rekursive_Fortsetzung">Rekursive Fortsetzung</h3></div>
<div class="mw-heading mw-heading4"><h4 id="Algorithmus">Algorithmus</h4></div>
<p>Man kann leicht ein rekursives Verfahren beschreiben, das es gestattet, aus der Kenntnis aller <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span>-Minos <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\geq 2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n\geq 2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70a38cff271dc2319eefb37fe6d266b391317336.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.465ex; height:2.843ex;" alt="{\displaystyle (n\geq 2)}" loading="lazy"></span> alle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Minos zu gewinnen. Es lässt sich zunächst zeigen, dass es zu einem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Mino <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87858df7457aa93caaef5a316db87a7240cc8c29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}" loading="lazy"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\geq 2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>≥<!-- ≥ --></mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n\geq 2)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70a38cff271dc2319eefb37fe6d266b391317336.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.465ex; height:2.843ex;" alt="{\displaystyle (n\geq 2)}" loading="lazy"></span> ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span>-Mino <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/398f438d75434e6fbf48dc232c1ad7228a738568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}" loading="lazy"></span> und ein Quadrat <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> gibt, so dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}=P_{1}\cup Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>∪<!-- ∪ --></mo>
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P_{2}=P_{1}\cup Q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a20a22e425fa0cf4ab84d63282f8d49fcba6aec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.612ex; height:2.509ex;" alt="{\displaystyle P_{2}=P_{1}\cup Q}" loading="lazy"></span> ist. Dadurch kann von der Kenntnis der Klassen der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span>-Minos ausgegangen werden. Durch Anfügen eines Quadrates entsteht je ein Repräsentant der Klassen der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Minos. Auf diese Weise kann auch die Anzahl <span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa563369b75d1fdb0b3eb905a0b698f27b7dda15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle A(n)}" loading="lazy"></span> der Klassen bestimmt werden. Wir verfahren wie folgt.
</p><p>Wir nummerieren die Klassen der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span>-Minos durch und beginnen mit einem Repräsentanten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> der ersten Klasse und betrachten systematisch alle Lagen eines Quadrates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span>, die überhaupt zu einem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Mino <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\cup Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo>∪<!-- ∪ --></mo>
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P\cup Q}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19216ef2fbf1d6cdf589e5c793014d0a59e1be0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\cup Q}" loading="lazy"></span> führen können. Diese Lagen werden mit <span typeof="mw:File"></span> oder <span typeof="mw:File"></span> markiert, je nachdem, ob das entsprechende <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Mino zu den bisherigen kongruent ist oder nicht. Nach gleicher Weise wird bei den nächsten Klassen der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span>-Minos verfahren. Natürlich kann dabei ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Mino entstehen, welches zu einem aus vorangegangenen Schritten kongruent ist. Solche werden mit einem Lagekästchen <span typeof="mw:File"></span> bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i=1,2,\ldots )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (i=1,2,\ldots )}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ae117a3864c12514ecb0a729e380ff1bd35b8fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.826ex; height:2.843ex;" alt="{\displaystyle (i=1,2,\ldots )}" loading="lazy"></span>.
</p><p>Nach endlich vielen Schritten bricht das Verfahren ab und es liefert einen Repräsentanten für jede Klasse der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Minos.
</p>
<div class="mw-heading mw-heading4"><h4 id="Beispiel">Beispiel</h4></div>
<p>Der rekursive <a href="Algorithmus" title="Algorithmus">Algorithmus</a> soll bei der Ermittlung der Pentominos aus Tetrominos eingesetzt werden.
</p>
<div class="mw-heading mw-heading4"><h4 id="Computergestützt"><span id="Computergest.C3.BCtzt"></span>Computergestützt</h4></div>
<p>Auf der Grundlage dieses Verfahrens lassen sich die <span class="mwe-math-element mwe-math-element-inline"><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)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(n)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa563369b75d1fdb0b3eb905a0b698f27b7dda15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle A(n)}" loading="lazy"></span> mit Computern bestimmen. Dabei lassen sich Polyominos durch eine <a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> mit 0 und 1 wie in folgendem Beispiel beschreiben.
</p><p><span class="mw-default-size" typeof="mw:File"></span> wird zu <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&1&1&0\\1&0&1&1\\1&0&0&0\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1&1&0\\1&0&1&1\\1&0&0&0\end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6a66550e627bd21ead0217ccb23dd65f33400ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:16.436ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}1&1&1&0\\1&0&1&1\\1&0&0&0\end{pmatrix}}}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading4"><h4 id="Hexominos">Hexominos</h4></div>
<p>Eine Untergruppe von 11 der 35 Hexominos stellen geometrisch gesehen das <a href="W%C3%BCrfelnetz" title="Würfelnetz">Netz eines Würfels</a> dar, da er durch 6 quadratische Flächen begrenzt wird.
</p>
<div class="mw-heading mw-heading2"><h2 id="Zusammengesetzte_ähnliche_Polyominos_(Reptiles)"><span id="Zusammengesetzte_.C3.A4hnliche_Polyominos_.28Reptiles.29"></span>Zusammengesetzte ähnliche Polyominos (Reptiles)</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Allgemeine_Reptiles-Definitionen">Allgemeine Reptiles-Definitionen</h3></div>
<ul><li>Ähnliche Figuren, die sich lückenlos zu einer größeren Figur, die zu den kleineren Figuren ähnlich ist, zusammensetzen lassen, werden im Englischen als <i>Reptiles</i> (Abkürzung für <i>replicating tiles</i>) bezeichnet.</li>
<li>Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> die Anzahl der ähnlichen Teilfiguren, so wird die zusammengesetzte Figur <i>rep-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Figur</i> genannt.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Anwendungsbeispiele_zu_Polyominos">Anwendungsbeispiele zu Polyominos</h3></div>
<p>Im Folgenden sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"></span>. Unter den verschiedenen Arten von Polyominos gibt es rep-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>4</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 4^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/563b380d4a8d0bb311093679b5696422a1c6f66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 4^{k}}" loading="lazy"></span>-Figuren und rep-<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16^{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>16</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 16^{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b58837da3d0afb94df5f1d5b6620ef85a492b966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.414ex; height:2.676ex;" alt="{\displaystyle 16^{k}}" loading="lazy"></span>-Figuren <i>(Figuren 1, 2 und 3)</i>.<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><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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwendung">Verwendung</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Packungen">Packungen</h3></div>
<p>Welche notwendigen und hinreichenden Bedingungen bestehen für die positiv ganzzahligen Seitenlängen eines Rechteckes, damit dieses mit bestimmten Sorten von Polyominos gepackt werden kann.
</p>
<div class="mw-heading mw-heading3"><h3 id="Spieleindustrie">Spieleindustrie</h3></div>
<p>Die Spiele <a href="Domino" title="Domino">Domino</a> und <a href="Pentomino" title="Pentomino">Pentomino</a> (Begriff stammt vom amerikanischen Mathematiker <a href="Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a>) sind weit verbreitet. Tetrominos kommen unter anderen in dem vom russischen Programmierer <a href="Alexei_Paschitnow" class="mw-redirect" title="Alexei Paschitnow">Alexei Paschitnow</a> 1985 entwickelten Computerspiel <i><a href="Tetris" title="Tetris">Tetris</a></i> zum Einsatz, wobei komplexere Versionen dieses Spiels auch auf andere Polyominos – ggf. <a href="Polyw%C3%BCrfel" title="Polywürfel">Polywürfel</a>, z. B. <a href="BlockOut" title="BlockOut">BlockOut</a> – zurückgreifen. Ein Brettspiel, das dem Computerspiel <i>Tetris</i> nahe kommt, ist <i><a href="Fits" title="Fits">FITS</a></i> (2009) von <a href="Reiner_Knizia" title="Reiner Knizia">Reiner Knizia</a>. Es nahm sich das Computerspiel ausdrücklich zum Vorbild. Weitere neuere Brettspiele sind das 2000 erschienene <i><a href="Blokus" title="Blokus">Blokus</a></i> sowie <i><a href="Ubongo" title="Ubongo">Ubongo</a></i> (2005), wo jeweils die verschiedenen großen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>-Minos für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,...,5}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
<mo>,</mo>
<mn>5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=1,...,5}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd6658417e6dc333c59d1a20c7de634eefaeff5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.988ex; height:2.509ex;" alt="{\displaystyle n=1,...,5}" loading="lazy"></span> als Spielmaterial verwendet werden. Auch die Spiele <i><a href="Patchwork_(Spiel)" title="Patchwork (Spiel)">Patchwork</a></i> (2014) und <i><a href="Cottage_Garden" title="Cottage Garden">Cottage Garden</a></i> (2016) von <a href="Uwe_Rosenberg" title="Uwe Rosenberg">Uwe Rosenberg</a> sowie <i><a href="Die_Baumeister_von_Arkadia" title="Die Baumeister von Arkadia">Die Baumeister von Arkadia</a></i> (2006) von <a href="R%C3%BCdiger_Dorn" title="Rüdiger Dorn">Rüdiger Dorn</a>, <i><a href="NMBR_9" title="NMBR 9">NMBR 9</a></i> (2017) von Peter Wichmann und <i><a href="B%C3%A4renpark_(Spiel)" title="Bärenpark (Spiel)">Bärenpark</a></i> (2017) von <a href="Phil_Walker-Harding" title="Phil Walker-Harding">Phil Walker-Harding</a> nutzen diese Formen als Legeteile. Bei <i>Ein Fest für Odin</i> (2016) von Uwe Rosenberg sind die Plättchen rechteckig angeordnet. Auch dieses Spiel wird als Polyomino-Spiel eingestuft.<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> 2001 erschien das Spiel <i><a href="Rumis" title="Rumis">Rumis</a></i>, welches dreidimensionale Steine (Polywürfel) verwendet.<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>
</p>
<div class="mw-heading mw-heading3"><h3 id="Pädagogik"><span id="P.C3.A4dagogik"></span>Pädagogik</h3></div>
<p>Die Bausteine finden in der Grundschule Verwendung und dienen der Verbesserung der räumlichen Vorstellung. Ziel ist es, zu einer vorgegebenen Menge von Bausteinen Figuren zu finden oder für vorgegebene Figuren eine Zerlegung in die entsprechenden Bausteine.
</p><p>Es ist nicht möglich, aus allen 5 möglichen Tetronimos ein 5×4 Rechteck zu erstellen. Es ist auch nicht möglich, ohne Mehrfachverwendung eines Winkelstücks, ein 4×4 Quadrat aus Tetrominos zu erstellen. Die verwendeten Figuren werden, wenn für sie Tetrominos verwendet werden, die den Buchstaben L, T und Z ähnlich sind, auch LTZ-<a href="Parkettierung" title="Parkettierung">Parkette</a> genannt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Verwandte_Themen">Verwandte Themen</h2></div>
<ul><li><a href="Polyw%C3%BCrfel" title="Polywürfel">Polywürfel</a> (auch Polykuben) – das dreidimensionale Pendant mit Würfeln</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Solomon_W._Golomb" title="Solomon W. Golomb">Solomon W. Golomb</a>: <i>Polyominoes. Puzzles, Patterns, Problems, and Packings</i>. 2. erweiterte Auflage. Princeton University Press, Princeton 1994, ISBN 0-691-08573-0</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Commons"></span></span></div><b><span class=""><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Polyominoes?uselang=de"><span lang="en">Commons</span>: Polyomino</a></span></b> – Sammlung von Bildern, Videos und Audiodateien</div>
<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Polyomino.html">Polyominos bei Wolfram Mathworld</a></li>
<li><a rel="nofollow" class="external text" href="http://www.xs4all.nl/~gp/PolyominoSolver/Polyomino.html">Gerard's Universal Polyomino Solver</a></li>
<li><a rel="nofollow" class="external text" href="http://www.mathematik.ch/anwendungenmath/wuerfelpuzzle/">Puzzlespiel mit Polyominos 3D</a></li>
<li><style data-mw-deduplicate="TemplateStyles:r261891140">
/* start https://de.wikipedia.org/ */
.mw-parser-output .webarchiv-memento a{color:inherit}
/* end https://de.wikipedia.org/ */
</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080420070009/http://home.quicknet.nl/mw/prive/wil.laan/puzzle/large.html">Spiel zur Füllung mit Polyominos 2D</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 20. April 2008 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>)</li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070928070946/http://www.uni-landau.de/rasch/Geometrie/Vorlesungen/V5_Ebene_Figuren_Polygone.pdf">Ebene Figuren – Vielecke (Anwendung für Kinder)</a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 28. September 2007 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>)</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Folge <a href="https://oeis.org/A000105" class="extiw external" title="oeis:A000105">A000105</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Folge <a href="https://oeis.org/A000988" class="extiw external" title="oeis:A000988">A000988</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Folge <a href="https://oeis.org/A001168" class="extiw external" title="oeis:A001168">A001168</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Beispielsweise Folge <a href="https://oeis.org/A000104" class="extiw external" title="oeis:A000104">A000104</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Claudi Alsina, Roger B. Nelsen: <i>Perlen der Mathematik - 20 geometrische Figuren als Ausgangspunkte für mathematische Erkundungsreisen</i>, <a href="Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag GmbH <a href="Berlin" title="Berlin">Berlin</a> <a href="Heidelberg" title="Heidelberg">Heidelberg</a> 2015, ISBN 978-3-662-45460-2, Seiten 51 bis 54</span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">George E. Martin: <i>Polyominoes: A Guide to Puzzles and Problems in Tiling</i>, <a href="American_Mathematical_Society" title="American Mathematical Society">AMS</a>/<a href="Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a>, <a href="Washington%2C_D.C." title="Washington, D.C.">Washington</a> 1991</span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://boardgamegeek.com/boardgamefamily/24571/polyominoes">Übersicht Polyomino-Spiele</a> bei Boardgamegeek</span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.hall9000.de/rezi.pl?spiel=rumis_neu">Rezension</a> von Rumis bei hall9000.de</span>
</li>
</ol></div><!--htdig_noindex--><div><div class="zim-footer">
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