Tribone Tilings of Triangular Regions that Cover All but Three Holes

This paper extends the literature of tiling n -row triangular arrays ( T n ) with copies of 1 × 3 trominos discussed by Thurston, Conway, and Lagarias, focusing on tilings which cover all but three holes. We find a set of 2 Ω ( n 2 ) such tilings, disproving the conjecture from 1993 that there are only 2 o ( n 2 ) such tilings. Furthermore, we show that if three cells are randomly removed from T n when n ≡ 0 , 2 ( mod 3 ) , then the probability that the remaining region can be tiled by tribones is nonzero.