Question 173
Let $(a, b)$ be a pair of positive integers. Henning and Paul are playing a game: Initially, there are two piles of $a$ and $b$ stones, respectively, on a table. The pair $(a, b)$ is called the initial configuration of the game. The players proceed as follows:
- The players alternate and Henning begins.
- In each turn, a player either removes a positive number of stones from one of the two piles or the same positive number of stones from both piles.
- The player who removes the last stone from the table wins the game.
Let $A$ be the set of all positive integers $a$ for which there exists a positive integer $b\lt a$ such that Paul has a winning strategy for the initial configuration $(a, b)$. Order the elements of $A$ increasingly as $a_{1}\lt a_{2}\lt \ldots$
a. Prove that the set $A$ is infinite.
b. Prove that the sequence $\left(m_{k}\right)_{k \geq 1}$ defined by $m_{k}:=a_{k+1}-a_{k}$ for all $k \geq 1$ is not eventually periodic.
Remark: A sequence $\left(x_{k}\right)_{k \geq 1}$ is eventually periodic if there exists an integer $k_{0} \geq 0$ such that the sequence $\left(x_{k+k_{0}}\right)_{k \geq 1}$ is periodic.
- The players alternate and Henning begins.
- In each turn, a player either removes a positive number of stones from one of the two piles or the same positive number of stones from both piles.
- The player who removes the last stone from the table wins the game.
Let $A$ be the set of all positive integers $a$ for which there exists a positive integer $b\lt a$ such that Paul has a winning strategy for the initial configuration $(a, b)$. Order the elements of $A$ increasingly as $a_{1}\lt a_{2}\lt \ldots$
a. Prove that the set $A$ is infinite.
b. Prove that the sequence $\left(m_{k}\right)_{k \geq 1}$ defined by $m_{k}:=a_{k+1}-a_{k}$ for all $k \geq 1$ is not eventually periodic.
Remark: A sequence $\left(x_{k}\right)_{k \geq 1}$ is eventually periodic if there exists an integer $k_{0} \geq 0$ such that the sequence $\left(x_{k+k_{0}}\right)_{k \geq 1}$ is periodic.