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Hex is a Board Game played on a Hexagonal Grid , usually in the shape of a 10 by 10 or 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 , who introduced the game in 1942 at the Niels Bohr Institute, and independently by the mathematician John Nash in the late 1940s. It became known in Denmark under the name ''Polygon'' (though Hein called it CON-TAC-TIX); Nash 's fellow players at first called the game ''Nash''. According to Martin Gardner , some of the Princeton University students also referred to the game as ''John'', because it was often played on the hexagonal tiles of bathroom floors. In 1952 Parker Brothers marketed a version. They called their version "Hex" and the name stuck. RULES Players have two colors, say "Red" and "Blue" (the colors are conventional and may vary from version to version). They take turns placing a piece of their own color on a hexagon. Red's goal is to form a red path connecting two opposite sides of the parallelogram, and Blue aims to connect the other two sides. Each of the four hexagons in the corners belong to ''two'' sides. 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. In other words, hex is a determined game. 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 . It is obvious that since hex is a finite, Perfect Information game which cannot end in a tie, either the first or second player has a winning strategy. Note that an extra move for either player in any position can only improve that player's position. Therefore, 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. If the strategy ever called for moving on the square already chosen, the first player makes another random move. This ensures a first player win. 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, using a simple pairing strategy, this has been proven to result in a win for the second player. A fairer way is to use the Pie Rule , by which the second player has the option of swapping colors after the first player makes the first move, or first three moves, thus encouraging the first player to even out the game. Nowadays, in most online sites, the swap rule is the default, with the swap made after only one move. In theory the swap rule ensures that the second player has a winning strategy, but in practice the first player can choose a hex for which no winning strategy is known. '', which covers Hex strategy at a greater level of detail than any preceding work. However, some hex players feel that this book contains many factual errors and advocates questionable strategies. 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. In Computational Complexity Theory , Hex has been proven to be PSPACE-complete . (Note that a number of other abstract strategy games, such as Checkers , Chess and Go , are EXPTIME-complete .) An important consequence of the Determinacy of hex is the Brouwer Fixed-point Theorem which was shown by David Gale. TEMPLATES An important concept in the theory of hex is the template. A template is a subset of the hexes with an assignment of red, blue or empty to each hex with two red edges set apart such that if blue were to move first, red would still be able to connect both red edges no matter what blue does. VARIANTS Blockbusters Hex had an incarnation as the question board from the television game show '' Blockbusters ''. In order to play a "move", contestants had to answer a question correctly. The board had 5 alternating columns of 4 hexagons; the solo player could connect top-to-bottom in 4 moves, while the team of two could connect left-to-right in 5 moves. The games of Y and Havannah Y and Havannah are considered by some to be generalizations of Hex; they differ primarily in requiring players to connect three or more edges of a polygonal board, rather than two selected edges of a parallelogram. The Shannon Switching game See Shannon Switching Game . Unlike Hex, this isn't PSPACE hard. SEE ALSO EXTERNAL LINKS AND REFERENCES General links
Computer hex
Playing on the internet
Solutions for small board sizes
Game database
Hex tools
Physical hex sets
Book
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