# Hex (game)

Hex / Polygon / con-tac-tix

Hex - The Zig-Zag Game (Parker Brothers)
Game data
author Piet Hein , John Nash
publishing company Parker Brothers (1952),
3M (1974)
Piet Hein
et al. a.
Publishing year 1942, 1947, 1952, 1974
Art Board game
Teammates 2
Duration 20 minutes
Age from 8 years
Hex board with 11 by 11 fields

Hex is a strategic board game invented by Piet Hein , a Danish poet and mathematician.

## history

Hein presented the game in 1942 under the name "Polygon" at the Niels Bohr Institute . On December 26, 1942, the Danish newspaper Politiken published the game, making it popular in Denmark.

In 1947, John F. Nash at Princeton University independently of Piet Hein invented the same game on the basis of game theory considerations. It is still often played in Princeton today and referred to as "John" or "Nash".

In 1952, Parker Brothers published the game under the name "Hex", which is still the best-known name of the game to this day. Piet Hein sold the game as "con-tac-tix".

## description

It is placed on a rhombus-shaped board with z. B. 11 times 11 hexagonal fields played. In the Nash biography " A beautiful Mind " it is shown that John Nash thinks 14 times 14 is the best size. Two opposite side lines are red, the other two are blue. The corner fields count towards both colors. A player wins if he connects the sides of his color with his pieces. Hex is closely related to other games such as compound TwixT or Y .

## Rules of the game

• You play on a diamond-shaped board made up of n by n hexagonal fields (picture). Piet Hein recommended an 11 × 11 board for this, while John Nash recommended a 14 × 14 board. Other common sizes are 13 × 13, 17 × 17, and 19 × 19. One field is adjacent to the six surrounding fields (a field on the edge of the board has four and a corner field has two or three neighbors).
• The parties (red and blue) alternately place a stone of their respective color on an unoccupied space, red begins.
• Two opposite sides of the board are marked with the same color. Each player has to connect the sides of his color with a chain of his stones to win. A chain consists of stones of the same color, all of which are directly or indirectly adjacent to one another.
• To compensate for the advantage of the first move, an exchange rule is usually used, according to which blue can change colors after the first red move. This means that blue takes over the opening move from red and continues to play with the red pieces. Red, on the other hand, gets the blue pieces and has to play against his own opening move. Red should therefore place his first stone in such a way that neither red nor blue have a recognizable advantage, because blue can always choose which side has the advantage.

## properties

• A cycle (repetition of positions) cannot occur because the number of stones on the board increases with each move.
• A tie is out of the question as the only way to block your opponent is to build a winning chain yourself. If all fields are filled, there is always a winning chain for exactly one player.
• It's a finite game with perfect information . If both sides play optimally, the winner is certain from the start. On the board sizes used in practice (about 9 × 9), however, it has not yet been possible to fully analyze the winning strategy and play it optimally.
• Without an exchange rule, blue has no winning strategy, as can be shown as follows: If blue had a winning strategy, then red could “steal” it from him by first making any move and then following the winning strategy. Red would have an additional stone on the originally occupied space, which cannot be a disadvantage for red. The fact that blue cannot bet on this square is only an advantage for red. If the strategy calls for red to bet on this space, then red makes any other move instead, which, as it were, shifts the additionally occupied space. Since there is no tie, red has a winning strategy.
• With an exchange rule, blue has a winning strategy because it can always choose the side for which there is a winning position.
• Without the exchange rule, red also has a great advantage in practice when playing between human players. When using the exchange rule, however, the chances of winning are almost the same on a board that is not too small.
• If the board is not the same size in both directions (m × n fields with m ≠ n), then a winning strategy is known for the player who has to connect the sides that are closer together.

## Online play opportunities

The following servers all offer a few other games besides Hex.

• Little Golem is a turn-based server, i. H. the duration of the game is more of a day here (English).
• Board Game Arena offers both turn-based play and real-time play (i.e. you complete the game in one go).
• igGameCenter is a real-time server. It can take a long time to find an opponent.

## Individual evidence

1. This is usually proven using Brouwer's Fixed Point Theorem or Sperner's Lemma . Piet Hein originally attributed it to the four-color problem . Conversely, the statement that a generalization of Hex to any dimensions always has exactly one winner is equivalent to Brouwer's fixed point theorem. See David Gale, The Game of Hex and the Brouwer Fixed-Point Theorem (PDF file; 282 kB).