Fun and Games on a Honeycombed Rhomboard.

Hex is an amusing game for two players, using a board or sheet of paper divided into hexagonal cells like a honeycomb. The playing board is rhomboidal in shape with an equal number of hexagons along each edge. Players take turns placing a counter or stone on a single cell of the board. One uses white stones, the other black. Or red and blue markers can be used on a paper board.

11 x 11 Hex Board. Image from  http://www.iggamecenter.com/

11 x 11 Hex Board. Image from iggamecenter.com

The goal for each player is to form a connected path of stones linking the opposite sides of the board marked by their colours, blocking their opponent from making a similar connection. The first player to make a complete connection wins the game. The four corner hexagons each belong to both adjacent sides.

The game was invented in 1942 by the Danish scientist, poet and inventor Piet Hein. He called it Con-Tac-Tix, but it became known in Denmark as Polygon. The mathematician John Nash thought up the game independently of Hein, during his graduate years at Princeton University. Parker Brothers marketed a version around 1952, calling it “Hex”, and that was the name that caught on. Martin Gardner (1959) brought Hex to the attention of a wide range of readers through his Scientific American column.

Left: Piet Hein (1905-1996). Right: John Nash (1928-2015).

Left: Piet Hein (1905-1996). Right: John Nash (1928-2015).

Strategy

John Nash proved that, assuming the board is symmetric, there is a winning strategy for the player who moves first. However, his proof is non-constructive: no general algorithm for winning is known. Winning strategies have been worked out for boards up to 10×10 cells but for larger boards Hex remains a game of skill.

One curious result of Brouwer’s fixed point theorem is that, in the game of Hex, there is always a winner. Both players try to build a continuous path from one side to the opposite one. If one player achieves this, the other one cannot break through: there cannot be a draw.

Variations

Hex is in a category called connection games. There is a long tradition of games of this sort, going back to the ancient oriental board-game Go. Many variations, embellishments and elaborations of Hex have been devised. For example, Pex is a similar game, played on a board of (irregular) pentagons:

11 x 11 Pex board. Image from  http://www.iggamecenter.com/

11 x 11 Pex board. Image from iggamecenter.com

It seems possible to devise a game based on any tiling of the plane. There are innumerable ways to divide the plane into polygons in patterns that are either periodic or aperiodic. The choice of game-boards is vast. Have fun exploring the possibilities.

Hex Boards can be printed using a PostScript file written by Larry Doolittle. The ASCII file is just fifty lines long, and can be downloaded and printed from this URL: http://doolittle.icarus.com/~larry/hex/. Alternatively, the picture below can be printed, or the game can be played online: iggamecenter.com

Click for larger version.

JPEG sheet of Hex boards generated from Larry Doolittle’s PostScript file. Click image for larger version.

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