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I recently read the following article giving a novel proof of the uncountability of $\mathbb{R}$ by analysing a particular game, amongst other results. http://people.math.gatech.edu/~mbaker/pdf/realgame.pdf
I'm aware that given a proof (or disproof) of a statement of the kind $\forall P \, \exists Q$ we can easily rephrase it as a game played by players selecting $P$ and $Q$.
This proof particularly interests me, however, as it seems most naturally formulated as a game which isn't obviously just identical to the statement and the result drops out as a corollary. Upon looking in to the references I found more proofs that also seemed most natural when formulated as games.
These proofs all come from the areas of topology or real analysis. I would like to know if anyone was aware of proofs from other mathematical areas that they think are naturally formulated as or were historically presented as analysis of particular games? (ideally these would be at undergraduate level so that I can understand them but I have no problem doing some background reading)

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In analytic number theory, Schmidt's $(\alpha,\beta)$ game and variations of it are sometimes very useful (see here, here for examples)

In particular, one can use the $(\alpha,\beta)$ game to prove that the set of badly approximable numbers, usually denoted by $\mathbf{Bad}$ (i.e. numbers such that there exists $c>0$: $|x-\frac pq|>\frac c{q^2}$) is $\sigma-$porous but it has Hausdorff dimension $1$ and it is incompressible, i.e. given $f_1,\dots, f_n,\dots$ diffeomorphisms of $\mathbb{R}$, $\cap f_n(\mathbf{Bad})$ has full Hausdorff dimension. These are already quite strong results but imo the real strength of this approach is that it generalizes to $\mathbb{R}^n$ very easily and also to more general situations, see e.g. here. With the same game, one can prove that the set of anormal numbers (numbers which are not normal in any given base) has $\dim_{\mathcal{H}}=1$.

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