Question 4
Let $n,k$ be positive integers. There are $n$ marbles labelled $1,\ldots,n$ and $k$ boxes. Initially all marbles are in one box. On each turn Cathy chooses a box and moves the marble with smallest label in it, say $i$, either to an empty box or to the box containing marble $i+1$. Cathy wins if some box contains only marble $n$. Determine all pairs $(n,k)$ for which she can win.