The connection game that proves a fixed-point theorem — Hein, Nash, and topology at play.
Play It
Characterization
Hex is a connection game played on a rhombus of hexagonal cells — typically 11×11 — in which two players alternately claim cells, each trying to connect their two opposite sides of the board with an unbroken chain. The game was invented by Piet Hein in 1942 at the Niels Bohr Institute in Copenhagen; the first public description appeared in the Danish newspaper Politiken on 26 December 1942, where Hein introduced a daily puzzle series that ran for four months. It was independently reinvented in 1948 by John Nash at Princeton, where students played it on bathroom-tile floors and called it "John." Hex carries three philosophical treasures rarely found together. First, it cannot end in a draw — a topological fact whose proof is equivalent to the Brouwer fixed-point theorem. Second, Nash's strategy-stealing argument proves the first player must have a winning strategy without ever exhibiting that strategy — a non-constructive existence proof of rare elegance. Third, the underlying topology connects the game to the four-colour theorem and the Jordan curve theorem. Martin Gardner popularised the game in his July 1957 Mathematical Games column in Scientific American. The Academy hosts Hex in the Mind School because it is the purest example of a game in which knowing that something exists is not the same as knowing what it is — and because the child's act of connecting one side to the other turns out, at its mathematical root, to be a theorem about the continuity of space.
Lineage
Invented by Piet Hein (December 1942, Copenhagen); independently reinvented by John Nash (1948, Princeton). First public description in Politiken, 26 December 1942. Gardner's popularisation in Scientific American, July 1957. The strategy-stealing argument and the equivalence to the Brouwer fixed-point theorem are treated in Cameron Browne, Hex Strategy: Making the Right Connections (A K Peters, 2000), and in Ryan Hayward and Bjarne Toft, Hex, Inside and Out: The Full Story (CRC Press, 2019). The connection to the four-colour theorem explored in Anatole Beck, Michael Bleicher, and Donald Crowe, Excursions into Mathematics (1969).
From the Library
Quests
The Magus
Attempt to discover or articulate a partial strategy for Hex on a board of at least 7×7. Document your reasoning: which opening cells you prefer, which patterns you look for, and where the strategy breaks down. Compare your findings to any published analysis or the known fact that first-player wins but no explicit strategy is known for large boards.
The Adventurer
Play Hex to its conclusion with at least one opponent on a board of at least 11×11. Record one game in which the winning chain was not obvious until the final moves, and note the moment when the topology of the board — the impossibility of a draw — became tangible.
The Sage
Explain Nash's strategy-stealing argument for Hex and the equivalence between the no-draw property and the Brouwer fixed-point theorem. Cite at least two sources — Hayward and Toft, Cameron Browne, or Gardner's Scientific American column — and explain to a non-mathematician why a proof of existence without exhibition is philosophically remarkable.