Every polyomino yields at most two square tilings

The problem of designing an efficient algorithm for deciding whether a given polygon tiles the plane becomes more tractable when restricted to polyominoes, that is, subsets of the square lattice Z whose boundary is a non-crossing closed path (see [6] for more on tilings and [3] for related problems). Here, we consider tilings obtained by translation of a single polyomino, called exact in [9]. Paths are conveniently described by words on the alphabet {0,1,2,3}, representing the elementary grid steps {→, ↑,←, ↓}. Beauquier and Nivat [1] characterized exact polyominoes by showing that the boundary word b(P ) of such a polyomino satisfies the equation b(P ) = X ·Y ·Z ·X ·Ŷ ·Ẑ, where at most one of the variables is empty and where Ŵ is the path W traveled in the opposite direction. Frow now on, this condition is referred as the BN-factorization. An exact polyomino is said to be a hexagon if none of the variables X, Y , Z is empty and a square if one of them is so. Note that a single polyomino may lead to many tilings