TSTST 2015 Problem 6

A Nim-style game is defined as follows. Two positive integers $k$ and $n$ are specified, along with a finite set $S$ of $k$-tuples of integers (not necessarily positive). At the start of the game, the $k$-tuple $(~ n, 0,0, \ldots, 0)$ is written on the blackboard. A legal move consists of erasing the tuple $\left(a_{1}, a_{2}, \ldots, a_{k}\right)$ which is written on the blackboard and replacing it with $\left(a_{1}+b_{1}, a_{2}+b_{2}, \ldots, a_{k}+b_{k}\right)$, where $\left(b_{1}, b_{2}, \ldots, b_{k}\right)$ is an element of the set $S$. Two players take turns making legal moves, and the first to write a negative integer loses. In the event that neither player is ever forced to write a negative integer, the game is a draw. Prove that there is a choice of $k$ and $S$ with the following property: the first player has a winning strategy if $n$ is a power of 2 , and otherwise the second player has a winning strategy.