Gale's 1979 proof that the Brouwer fixed-point theorem implies Hex's no-draw property is based on a suggestion from John Stallings, modified by Michael Todd

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The converse direction of David Gale's 1979 equivalence proof — that the Brouwer fixed-point theorem implies Hex's no-draw property — uses the fact that the Hex board $B_k$ triangulates the $k \times k$ square. Gale states the technique directly: "The proof that 'Brouwer' implies 'Hex,' based on a suggestion of John Stallings modified by one of Michael Todd, makes use of the fact that the Hex board Bk gives a triangulation of the k × k square I2 in R2."

Gale's introduction gives the fuller priority account: he had "over the years" heard the equivalence asserted informally as "cocktail conversation," and his Berkeley colleague John Stallings had already shown him "an argument which derives the Hex Theorem from familiar topological facts which are equivalent to the Brouwer Theorem" — this direction — before Gale wrote the paper. The other direction (Hex implies Brouwer, see claim-gale-1979-hex-implies-brouwer-via-covering-argument) "only occurred to [Gale] recently," and he explicitly hedges even that: "it may well be that others have been aware of it." He claims only the n-dimensional generalization as possibly new. Read against claim-gale-1979-hex-draw-impossibility-equivalent-to-brouwer-fixed-point, this means the 1979 paper's actual originality is narrower than "Gale proved the equivalence" suggests — the equivalence itself was folklore, one direction came from Stallings (via Todd's modification), and Gale's stated contribution is packaging both into a rigorous paper plus the higher-dimensional extension.