Black Path Game

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The Black Path Game (also known by various other names, such as Brick) is a two-player board game described and analysed in Winning Ways for your Mathematical Plays. It was invented by Larry Black in 1960.[1]

Rules

The Black Path Game is played on a board ruled into squares. Any square that is not empty is filled with one of the following configurations:

  • Truchet-tile-1.svg
  • Truchet-tile-2.svg
  • Truchet-tile-3.svg

These tiles are the three ways to join the sides of the square in pairs. The first two are the tiles of the Truchet tiling.[2] One edge on the boundary of the board is designated to be the start of the path. The players alternate filling the square just after the end of the current path with one of the three configrations above, extending the path. The path may return to a previously filled square and follow the yet-unused segment on that square. The player who first causes the path to run back into the edge of the board loses the game.[1]

Strategy

The first player has a winning strategy on any rectangular board with at least one side-length even. Imagine the board covered with dominoes. The first player should always play so that the end of the path falls on the middle of one of the dominoes. If both sides of the board are odd, the second player can instead win by using a domino tiling including every square but the one containing the first player's first move.[1]

See also

References

  1. ↑ 1.0 1.1 1.2 Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (1982), "The Black Path Game", Winning Ways for your Mathematical Plays, Vol. 2: Games in Particular, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], pp. 682–683, MR 654502 .
  2. ↑ Browne, Cameron (2008), "Truchet curves and surfaces", Computers & Graphics 32 (2): 268–281, doi:10.1016/j.cag.2007.10.001, "Truchet-style tiles are used as the basis for several strategy games including Trax, Meander and the Black Path Game, all of which predate Smith's seminal 1987 article that associates such tiles with the work of Sébastien Truchet." 
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