Symmetric polyomino tilings, tribones, ideals, and Gröbner bases

We apply the theory of Grobner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions TN = T3k?1 and TN = T3k in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120? rotation of the triangle if and only if either N = 27r ? 1 or N = 27r for some integer r > 0. The method applied is quite general and can be adapted to a large class of symmetric tiling problems.