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# Game of Hex

Hex is a board game played on a hexagonal grid, usually in the shape of an 11 by 11 rhombus. (the book "A Beautiful Mind" says Nash et al settled on 14 by 14 as best.)

History

The game was invented by the Danish mathematician Piet Hein in 1942, and independently by the mathematician John Nash in the late 1940s. It became known in Denmark under the name Polygon; Nash's fellow players at first called the game Nash.

Rules

Players have two colors, say "Red" and "Blue". They alternate turns placing a piece of their color inside a hexagon, filling in that hexagon with their color. Red's goal is to form a red path connecting the top and bottom sides of the parallelogram, and Blue's goal is to form a path connecting the left and right sides.

Red moved first in this game, and won.

Strategy

The game can never end in a tie, a fact found by Nash: the only way to prevent your opponent from forming a connecting path is to form a path yourself.

When the sides of the grid are equal, the game favors the first player. A standard non-constructive[?] strategy-stealing argument[?] proves that the first player has a winning strategy. First, a more general theorem shows that in any game that never goes on for an infinite number of moves, never ends in a tie, and has no hidden or random information, either the first or second player has a winning strategy. If the second player has a winning strategy, the first player could steal it by making an irrelevant move and then follow the second player's strategy. The extra move can only help the first player, i.e. if the strategy ever called for moving on the square already chosen, the first player makes another random move.

There are two ways to make the game fairer. One way is to make the second player's sides closer together, playing on a parallelogram rather than a rhombus. However, this has been proven to result in a win for the second player, so it theoretically doesn't improve matters. A better way is to allow the second player to choose his color after the first player makes the first move, or first three moves, which encourages the first player to intentionally even out the game. See the pie rule for a more detailed discussion.

Cameron Browne[?] wrote a book entitled Hex Strategy: Making the Right Connections[?], which covers Hex strategy at a greater level of detail than any preceding work. It is currently considered by many as necessary reading for any serious Hex enthusiast; another book, to be written by Jack van Rijswijck[?] and Ryan Hayward[?], was put on hold soon after the publication of Hex Strategy; it was to have a more mathematical bent than the somewhat conversational tone of Browne's book.

Variants

Hex had an incarnation as the question board from the British television game show Blockbusters.

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