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A Brief History of the Game of Hex

Author: Garikai Campbell [ profile | email ]
In 1942, Piet Hein [HP1] invented a game he called Polygon while contemplating the (then unsolved) four-color theorem. The game was later independently re-discovered by John Nash [NJ1] while he was a graduate student at Princeton University and it quickly became popular there (and at other schools) under the names of John and Nash. A couple of years later, in 1950, Parker Brothers began producing the game as Game of Hex.

Figure 1. A picture of the original game marketed by Parker Brothers. [AS1].

With so many names given to the same game, one might wonder why Hex is the name that endured. The fact that Parker Brothers put its marketing muscle behind that name is clearly a large contributing factor, but Martin Gardner's [GM1] 1957 Scientific American article "Concerning the game of Hex, which may be played on the tiles of the bathroom floor," [GM2]is undoubtedly another. This article and future writings by Gardner on the game were quite popular and brought the game to the attention of a vast number of mathematicians and computer scientists. In the article, Gardner states that
Hex may well become one of the most widely played and thoughtfully analyzed new mathematical games of the century.

So what exactly is this game of Hex, how does one play and what makes the game so special? First, Hex is a two person game played on an n x n board of hexagonal tiles arranged in a rhombus. We will denote the game of hex played on an n x n board as Hex(n). The Parker Brothers board shown above is an 11 x 11 board and is considered the "standard" size board. Two smaller 4 x 4 boards are illustrated below.

Figure 2. An empty 4 x 4 Hex board (left) and a 4 x 4 board containing a winning path for blue (right).

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How to play

Each player is assigned a pair of opposite sides of the board and given a set of counters. The players take turns placing one of their counters on any unoccupied tile of the board until one player has a path consisting of their own counters connecting their two sides of the board. Throughout the discussion here, one player is designated blue, the other red and we color the sides of the board and the counters to indicate to whom they belong. We also make the arbitrary choice of always letting the blue player go first. Figure 2 above illustrates a winning game of Hex(4) for blue.

What makes Hex special

Hex is a type of two player, perfect information, symmetric game which cannot end in a tie. Therefore, the game is susceptible to a strategy stealing argument and one can prove that the first player has a winning strategy.
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Where to go from here?

References

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