Question 56
Given a piece on the origin in the coordinate plane. Two persons $A$, $B$ act as the following. First $A$ marks on a lattice point on which a piece isn't put. Then $B$ moves a piece put on the point $(x, y)$ to the point $(x + 1, y)$ or $(x, y + 1)$ $m$ times ($1 \leq m \leq k$). Note that we may not move the piece to unmarked point. If $A$ wins when $B$ can't move any pieces, then find all possible integers $k$ such that $A$ will win at finite action in regardless of how $B$ moves each piece.