  We apply the theory of Groebner 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 T_N=T_{3k-1} and T_N=T_{3k} in a
hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones)
symmetric with respect to the 120 degrees rotation of the triangle if and only
if either N=27r-1 or N=27r for some integer r.
