Tiling a common figure with various polyominoes thumb|A minimal compatibility figure for the T and W pentominoes. 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,Mireles, J.L., "Poly2ominoes" and Livio Zucca shows results for some complicated cases like three different pentominoes. 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. Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.
Tiling a common figure with various polyominoes thumb|A minimal compatibility figure for the T and W pentominoes. 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,Mireles, J.L., "Poly2ominoes" and Livio Zucca shows results for some complicated cases like three different pentominoes. 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. Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.