  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 if $n$ is a
power of 4 and $k \ge n/4 +1$. Our results also imply that for all $n \geq 3$
there is a Breaker's win pairing strategy if $k \ge
\left\lfloor\frac{3}{7}n\right\rfloor +1$.
