Question 20
Two players play a game. There is a heap of $m$ tokens and a heap of $n \lt m$ tokens. Each player in turn takes one or more tokens from the heap which is larger. The number he takes must be a multiple of the number in the smaller heap. For example, if the heaps are $15$ and $4$, the first player may take $4$, $8$ or $12$ from the larger heap. The first player to clear a heap wins. Show that if $m \gt 2n$, then the first player can always win. Find all $k$ such that if $m \gt k n$, then the first player can always win.