MathDB

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

n

. Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marbles 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.

4