Question 31
Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled $1$ to $n$. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marble with the smallest label, say $i$, to either any empty box or the box containing marble $i+1$. Cathy wins if at any point there is a box containing only marble $n$.
Determine all pairs of integers $(n, k)$ such that Cathy can win this game.
Determine all pairs of integers $(n, k)$ such that Cathy can win this game.