Handicap Achievement for Squares

Abstract

Given a polyomino $P$ with $n$ cells, two players $A$ and $B$ alternately color the cells of the square tessellation of the plane. In the case of $A$-achievement, player $A$ tries to achieve a copy of $P$ in his color and player $B$ tries to prevent $A$ from achieving a copy of $P$. The handicap number $h(P)$ denotes the minimum number of cells such that a winning strategy exists for player $A$. For all polyominoes that form a square of $n=s^2$ square cells, the handicap number will be determined to be $s^2–1$.