About 7 years ago I was asked this question. I remembered it right now, and I can't solve it.
Prove that no matter how you play, the game Hex will have a winner at the end.
2026-04-01 16:42:13.1775061733
Prove game Hex must have a winner
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There is a really nice proof given here.
Summary: After a game is completed, define the graph $G=(V,E)$, where $V$ is the set of vertices of the hexagons, and $E$ consists of only the edges which are between adjacent hexagons claimed by opposite players. Each internal vertex will have degree $0$ or $2$, according to whether it is surrounded by all hexagons belonging to the same player, or a mix of two of one player and one of the other. Therefore, the graph decomposes into a disjoint union of paths and cycles. You can then show that there must be two paths connecting the four corners of the board into tow pairs; each of these paths will trace out a winning chain of hexagons. Please refer to the linked document for the details.