Hex is a board game played on a hexagonal grid, usually in the shape of a 10 by 10 or a 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". (Of course, the colors are merely a convention and the actual colors vary from
board to board and from version to version.) 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.
In the image on left, 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. 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 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.
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.)
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 game of Y
Y is considered by some to be superior to Hex, because Hex is a subset of
Y.
The Shannon Switching game
See Shannon switching game. Unlike Hex, this isn't
PSPACE hard.
Havannah
Havannah is a similar game to hex.
See also:
External links and references
General links
Computer hex
- Six (http://six.retes.hu/) The strongest Hex program out there (for Linux/Un*x).
- Hexy (http://home.earthlink.net/~vanshel/) 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 (http://www.cs.ualberta.ca/~queenbee/)
- International Computer Games Association - Hex Information (http://www.cs.unimaas.nl/icga/games/hex/)
Playing on the internet
- Kurnik (http://www.kurnik.org/), currently the best realtime server for Hex
- Little Golem (http://www.littlegolem.net/), a popular turn-based server with 13x13 and 19x19 Hex
- PBMserv (http://www.gamerz.net/pbmserv/), an email game server, any size Hex board supported
Solutions for small board sizes
Game database
- OHex (http://www.kosmanor.com/hex-bin/board) 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
- Jon's Hex Board (http://folk.uio.no/jkleiser/hex/hexboard.html) How to make a Hex board in HTML
Physical hex sets
Book
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