Hex (board game)

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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.)

Contents

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. In 1952 Parker Brothers marketed a version. They called their version "Hex" and the name stuck.

Hex is an abstract strategy game that belongs to the general category of "connection" games. Other connection games include Omni, Y and Havannah. All of these games are related to the ancient Asian game of Go; Nash's version of Hex, in particular, was done as a response to Go. left

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.

Cameron Browne wrote a book entitled , 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.

Chameleon

Utilizing the same board and pieces as Hex, Chameleon gives the players the option of placing a piece of either color on the board. One player is attempting to connect the north and south edges, and the other is attempting to connect the east and west edges. The game is won when a connection between a player's goal edges is formed using either color. If a piece is placed that creates a connection between both players' goal edges (i.e. all edges are connected), the winner is the player who placed the final piece.

Chameleon is described in Cameron Browne's book Connection Games: Variations on a Theme (2005) and was independently discovered by Randy Cox and Bill Taylor.

The Shannon Switching game

See Shannon switching game. Unlike Hex, this isn't PSPACE hard.

External links and references

General links

[Game of Hex] -- From MathWorld
[HexWiki]
[Jack van Rijswijck's Hex page] includes proof that a draw is impossible.
[Hex game information center] (has not been updated since 1999)
[basic strategy well explained]
[Cameron's Hex Page] Author Cameron Browne explains
[Thesis on Hex by Thomas Maarup] Excerpts from a thesis on Hex, history, classification and complexity
[The Game of Hex - Templates] A catalogue of edge connection templates
[Hex] Has some good information on strategy
[Hex Theory and Proofs Project]
[Hex Can't End in a Draw] at cut-the-knot

Computer hex

[Six] The strongest Hex program out there (for Linux/Un*x).
[Hexy] The second strongest program out there. It's for Microsoft Windows and there are some publications on Hex algorithms on this site. Some of the papers there might be of interest to mathematicians willing to make a mathematical analysis of Hex.
[Queenbee]
[International Computer Games Association - Hex Information]

Playing on the internet

[Kurnik], the most popular realtime server for Hex
[Little Golem], a popular turn-based server with 13x13 and 19x19 Hex
[PBMserv], an email game server, any size Hex board supported
[BoardSpace.net], another realtime server for Hex
[Extended Y Java applet], can implement Hex

Solutions for small board sizes

[7x7, 8x8 and 9x9 solutions] by Jing Yang of the University of Manitoba

Game database

[OHex] This is an online database of Hex games. But please take it with a grain of salt since the database contains quite a number of mediocre games. There's also a Hex mailing list you can join there.

Hex tools

[Jhex] Jhex is a game tree editor for hex with many features, written in Java. It's useful for analyzing hex strategy. It also contains a fairly large downloadable database of Hex games.
[Jon's Hex Board] How to make a Hex board in HTML

Physical hex sets

[Hex Boards] Photos of wooden boards
[Miguel Garcia's Hex Board Gallery] Photos of a Hex board made with steel nuts and ball-bearings
[Making A Hex Game With A White-Erase Board and Magnets]
[Hex image gallery on BoardGameGeek] Mostly pictures of Hex sets.

Book

Browne, Cameron. Hex Strategy: Making the Right Connections. ISBN 1-56881-117-9

References

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