Hex (board game)
From Wikipedia, the free encyclopedia
| The Game of Hex | |
|---|---|
Hexy, a Hex computer program |
|
| Players | 2 |
| Age range | 4+ |
| Setup time | 1 minute |
| Playing time | 15 minutes (11x11 board) |
| Random chance | None |
| Skills required | Tactics, strategy, position |
Hex is a board game played on a hexagonal grid, theoretically of any size and several possible shapes, but traditionally as a 11x11 rhombus. Other popular dimensions are 13x13 and 19x19 as a result of the game's relationship to the older game of Go. According to the book A Beautiful Mind, John Nash (one of the game's inventors) advocated 14x14 as the optimal size.
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[edit] History
The game was invented by the Danish mathematician Piet Hein, who introduced the game in 1942 at the Niels Bohr Institute, and also independently invented by the mathematician John Nash in 1947 at Princeton University. 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 (according to some sources this was because they played the game using the mosaic of the bathroom floor). 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.
[edit] Rules
Each player has an allocated color, Red and Blue being conventional. Players take turns placing a stone of their color on a single cell within the overall playing board. The goal is to form a connected path of your stones linking the opposing sides of the board marked by your colors, before your opponent connects his sides in a similar fashion. The first player to complete his connection wins the game. The four corner hexagons each belong to two sides.
Since the first player to move in Hex has a distinct advantage, the pie rule is generally implemented for fairness. This rule allows the second player to choose whether to switch positions with the first player after the first player makes the first move.
[edit] Strategy
The game can never end in a tie, a fact found by John 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, aka the swap 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 Hex Strategy: Making the Right Connections, 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.
[edit] Theory and Proofs
John Nash proved that a game of Hex cannot end in a tie.
In computational complexity theory, Hex has been proven to be PSPACE-complete. A number of other abstract strategy games, such as checkers, chess and go, are EXPTIME-complete. (It is known that an EXPTIME-complete problem is at least as hard as a PSPACE-complete problem. Most experts believe that EXPTIME-complete problems are harder than PSPACE-complete problems, although this is not proved.)
Hex has been solved for all symmetrical playing grids up to and including 9x9: that is, a perfect strategy has been demonstrated for the first player to move, such that they are always able to win.
The determinacy of hex has other mathematical consequences: it can be used to prove the two-dimensional Brouwer fixed-point theorem, as David Gale showed in 1979, and the determinancy of higher-dimensional variants proves the fixed-point theorem in general.
[edit] 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.
[edit] Variants
[edit] 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.
[edit] 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.
[edit] 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.
[edit] The Shannon Switching game
See Shannon switching game. Unlike Hex, this isn't PSPACE hard.
[edit] See also
[edit] References
- Browne, Cameron. Hex Strategy: Making the Right Connections. ISBN 1-56881-117-9
- Gale, D. (1979). "The Game of Hex and Brouwer Fixed-Point Theorem". The American Mathematical Monthly 86: 818-827.
[edit] External links
- HexWiki, a wiki dedicated to Hex
- Thesis on Hex history, classification and complexity
- Game of Hex at MathWorld with links to related mathematical papers
- 1x1 to 6x6 opening strategy by Jack van Rijswijck of the University of Alberta
- 7x7, 8x8 and 9x9 solutions by Jing Yang of the University of Manitoba
