We consider the Maker-Breaker positional game on the vertices of the $n$-dimensional hypercube ${0,1}^n$ with $k$-dimensional subcubes as winning sets. We describe a pairing strategy which allows Breaker to win when $k = n/4 +1$ in the case where $n$ is a power of 4. Our results also imply the general result that there is a Breakers win pairing strategy for any $n geq 3$ if $k = leftlfloorfrac{3}{7}nrightrfloor +1$.
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