Question 31
Aino and Väinö start to play the game GCD($m, n$) where $m$ and $n$ are positive integers. In the beginning there are two piles of stones on the table, one with $m$ stones, another with $n$ stones. The one whose turn it is, takes away a number of stones from one of the piles. This number is a multiple of the number of stones in the other pile. Aino starts, and the players take turns until one of the piles is empty. The one who manages to empty a pile, wins. Prove that there is an $\alpha \gt 1$ such that if $m$ and $n$ are positive integers with $m \gt \alpha n$, then Aino has a winning strategy in the game GCD($m, n$), whereas if $\alpha n \gt m \gt n$ Väinö has.