Editing Polyomino (section)

===Tiling a common figure with various polyominoes===
[[Image:PentominoCompatibilityTW.svg|thumb|A minimal compatibility figure for the T and W [[pentomino]]es.]]
The ''compatibility problem'' 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,<ref>[https://web.archive.org/web/20091027093922/http://geocities.com/jorgeluismireles/polypolyominoes/ Mireles, J.L., "Poly<sup>2</sup>ominoes"]</ref><ref>{{Cite web |url=http://www.iread.it/Poly/ |title=Resta, G., "Polypolyominoes" |access-date=2010-07-02 |archive-url=https://web.archive.org/web/20110222051130/http://www.iread.it/Poly/ |archive-date=2011-02-22 |url-status=live }}</ref> and Livio Zucca shows results for some complicated cases like three different pentominoes.<ref>{{Cite web |url=http://sicherman.net/rosp/triplep.html |title=Zucca, L., "Triple Pentominoes" |access-date=2023-04-20 }}</ref> 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.<ref>{{cite book  |last1=Barbans |first1=Uldis |last2=Cibulis |first2=Andris |last3=Lee |first3=Gilbert |last4=Liu |first4=Andy |last5=Wainwright |first5=Robert |contribution=Polyomino Number Theory (III) |editor-last=Cipra |editor-first=Barry Arthur|editor-link = Barry Arthur Cipra |editor2-last=Demaine |editor2-first=Erik D. |editor3-last=Demaine |editor3-first=Martin L. |editor4-last=Rodgers |editor4-first=Tom |title=Tribute to a Mathemagician |place=Wellesley, MA |publisher=A.K. Peters |year=2005 |pages=131–136 |isbn=978-1-56881-204-5}}</ref> Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.