Competitive Tiling

Competitive Tiling consists of two players, a tile set, a region, and a non-negative integer $d$. Alice and Bob, our two players, alternate placing tiles on the untiled squares of the region. They play until no more tiles can be placed. Alice wins if at most $d$ squares are untiled at the end of the game, and Bob wins if more than $d$ squares are untiled. For certain regions and tile sets, we are interested in the smallest value of $d$ such that Alice can win. We call this the game tiling number. In this talk we will focus on finding the game tiling number for the game played with dominoes on $2 \times n$ rectangles, modified $2 \times n$ rectangles, and rectangular annular regions.