The continuum hypothesis is equivalent to the assertion that for any set $X\subseteq\mathbb R$ either White or Black has a winning strategy in the following infinitely long game $G(X)$.

At the $n^\text{th}$ move, first White chooses a set $W_n\subseteq X$, and then Black chooses a set $B_n\in\{W_n,X\setminus W_n\}$. White wins if $\bigcap_{n\in\mathbb N}B_n\ne\varnothing$; Black wins if $\bigcap_{n\in\mathbb N}B_n=\varnothing$.

The game $G(X)$ is a win for White iff $|X|=2^{\aleph_0}$, a win for Black iff $|X|\le\aleph_0$, undetermined iff $\aleph_0\lt|X|\lt2^{\aleph_0}$.

I don't know the reference but I think this is due to Mycielski and Solovay.