Question 63
Let $k$ be a positive integer. Players $A$ and $B$ play a game according to the following rule: Initially, a chess piece is placed at the origin $(0, 0)$ of the $xy$-plane. The player $A$ will start the game followed by $B$ and repeat, each choosing a strategy among those specified by the following rules:
* Possible strategies for $A$: Choose a lattice point, which is not occupied by the chess piece and mark the point by a ✓.
(Here, by a lattice point we mean a point in the $xy$-plane whose $x$-coordinate and $y$-coordinate are both integers).
* Possible strategies for $B$: Repeat the process of moving the chess piece located at $(x, y)$ to either $(x+1, y)$ or $(x, y+1)$ $j$ times where $1 \le j \le k$. However, in each move of the process he is not allowed to move the chess piece into a lattice point marked by a ✓.
$A$ wins the game if $B$ gets into the situation where he cannot move the chess piece. Determine all possible values for $k$ for which $A$ can win the game after a finite number of steps no matter how $B$ chooses his strategies.
* Possible strategies for $A$: Choose a lattice point, which is not occupied by the chess piece and mark the point by a ✓.
(Here, by a lattice point we mean a point in the $xy$-plane whose $x$-coordinate and $y$-coordinate are both integers).
* Possible strategies for $B$: Repeat the process of moving the chess piece located at $(x, y)$ to either $(x+1, y)$ or $(x, y+1)$ $j$ times where $1 \le j \le k$. However, in each move of the process he is not allowed to move the chess piece into a lattice point marked by a ✓.
$A$ wins the game if $B$ gets into the situation where he cannot move the chess piece. Determine all possible values for $k$ for which $A$ can win the game after a finite number of steps no matter how $B$ chooses his strategies.