A Polyomino Tiling Problem of Thurston and Its Configurational Entropy.

This paper proves a conjecture of Thurston on tiling certain triangular region T sub (3N+1) of the hexagonal lattice with three-in-line ("tribone") tiles. It asserts that for all packings of T sub (3N+1) with tribones leaving exactly one uncovered cell, the uncovered cell must be the central cell. Furthermore there are exactly 2 sup N such packings. This result is analogous to closed formulae counting configurations for certain exactly solved models in statistical mechanics, and implies that the configurational entropy (per site) of tiling T sub 3N+1 with tribones with one defect tends to 0 as N -> inf.