Published online by Cambridge University Press: 30 May 2025
We give an [n+1/6]-cell handicap strategy for the game of Hex on an n x n board: the first player is guaranteed victory if she is allowed to colour [n+16] cells on her first move. Our strategy exploits a new kind of inferior Hex cell.
Hex was invented independently by Piet Hein [1942] and John Nash [1952]. The game is played by two players, Black and White, on a board with hexagonal cells. The players alternate turns, colouring any single uncoloured cell with their colour. The winner is the player who creates a path of her colour connecting her two opposing board sides. See Figure 1.
Hein and Nash observed that Hex cannot end in a draw [Hein 1942; Nash 1952]: exactly one player has a winning path if all cells are coloured [Beck et al. 1969]. Also, an extra coloured cell is never disadvantageous for the player with that colour [Nash 1952]. For n x n boards, Nash showed the existence of a first-player winning strategy [1952]; however, his proof reveals nothing about the nature of such a strategy. For 8x8 and smaller boards, computer search can find all winning first moves [Hayward et al. 2004; Henderson et al. 2009]. For the 9x9 board, Yang found by human search that moving to the centre cell is a winning first move.
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