MathDB

Let

n

\geq

2

and

k

\geq

2

be positive integers. A cat and a mouse are playing Wim, which is a stone removal game. The game starts with

n

stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove

1

2

\dotsb

, or

k

stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates Wim 2, which is Wim but with the following additional rule: You cannot remove the same number of stones that your opponent removed on the previous turn. \\\\Find all values of

k

such that for every

n

, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

Problem Statement