HEX  

Copyright (c) 1942 Piete Hein

"In 1942, Danish poet & mathematician Piet Hein invented a game now called Hex. Piet invented the game while he was contemplating the famous four-color theorem of topology. The theorem, then unproved, is that four colors are sufficient to make any map so that no two countries of the same color have a common boundary. The game was introduced as the Niels Bohr Institute. The game became popular in Denmark under the name Polygon. The game was printed on pads of paper; and sold as a paper & pencil game. For months, Danish newspapers ran a series of "Polygon" problems. Albert Einstein kept the game on a shelf in his study." [Full text]

Hex is a two player strategy game played on a NxN rhombus of hexagons.

 

TURN - On each turn, each player drop one friendly stone in an empty cell. GOAL - The first player (say, Black) is to form a unbroken chain of stones that connects the SE and  NW sides of the board, while the second player (say, White) attempts to form an unbroken chain of her stones connecting the SW and NE sides of the board.

 

An example Cell [1] is essential to both White and Black, since if one drops a stone there it will cut the other. The two other black stones are can always be connected since there are two possible ways (the cells marked with [2]). If White places a stone in one [2], Black should place in the other [2]. This is called a 2-bridge.

I include here an idea from Michael Leigh, to equalize the game (he called it Vex): We add an additional neutral stone, say red, which is given to the second player who has the option to place this stone on the board on any empty hex of his choice either before the first player makes his initial move or on any subsequent turn in addition to playing his black stone. Vex is a draw because white can always pretend the neutral stone is an extra white stone (which therefore cannot disadvantage him forcing him to lose) and black may place the neutral stone before the start of the game and then by assuming the stone to be black and playing the 'hex' winning strategy cannot be forced to lose either. By using a neutral stone and giving the second player the option when to play it we no longer need to 'know' which hexes are winning hexes for the first player in order to make the game fair.

In 1995, Michael Howe proposed, on Games Magazine, three variants of Hex:

• Link: in this game after placing a stone a player may move any number of their stones which form a connected line, in of the directions of that line, up to as many spaces as the size of the group. Opposing stones in smaller groups may be pushed, but cannot be pushed off the board. [so, something similar to Abalone without captures]
• Tack: in this game a group of two or three stones in an adjacent line are captured if the other player places a stone to bracket the line on both ends. The captured stones are replaced by the capturing player's stones. [custodian capture Othello-style]
• Fan: in this game after placement, an optional capturing move is made by moving a piece one space so as to end up adjacent to an opposing stone, or to end up one space away from an opposing stone it started adjacent to, or both.

Here's a low-res picture of those pages:

Chameleon is a Hex variant, where both players may use stones of either color to achieve a vertical/horizontal connection, i.e.,  one player wins if he makes a vertical connection, the other by a horizontal connection. If a move makes both connections, wins the mover.

Words from Cameron Browne: Playing Chameleon is a constant tightrope act. In most connection games, each player can concentrate fully on pushing their connection as hard as possible. However in Chameleon players must keep their connections strong only in their direction or risk having them stolen. Players must consider the implications of each move very carefully. Chameleon has a similar feel to Jade but with clearer goals.

The game was available at gamerz.com, but the link stopped working, so here is the original text:

Rules

Two players, Vert and Horz, take turns placing either an 'x' piece or an 'o' piece on the board.

Vert wins by completing either a chain of 'x' pieces or a chain of 'o' pieces between the top and bottom board edges. Horz wins by completing either a chain of 'x' pieces or a chain of 'o' pieces between the left and right board edges.

If a move results in a connecting chain for both players, then the mover wins. For example:

. . . . . . .     . . . o . . .      . . . . . . x
 . . . . . . .     . . . o . . .      . . . . . x .
  . . . . . . .     . . . o . . .      . . . . x . .
   o o o o o o o .   . . . o . . .      . . . x . . .
    . . . . . . .     . . . o . . .      . . x . . . .
     . . . . . . .     . . . o . . .      . x . . . . .
      . . . . . . .     . . . o . . .      x . . . . . .
 
          Horz wins      Vert wins        Last mover wins

Notes

Playing Chameleon is a constant tightrope act. In most connection games, each player can concentrate fully on pushing their connection as hard as possible. However in Chameleon players must keep their connections strong only in their direction or risk having them stolen. Players must consider the implications of each move very carefully.

Chameleon has a similar feel to Jade but with clearer goals.

One of the most interesting aspects of Chameleon is that it inherently solves the first move advantage problem which plagues most connection games. While opening in the centre is a winning move in Hex, it is a death sentence in Chameleon. The first player's best opening move is well away from the centre and any opponent's edge.

Chameleon should be played on larger boards. Games smaller than 10x10 tend to degenerate into a race after only a few moves.

References and History

Chameleon was discovered by Randy Cox in early November 2003, then independently rediscovered mid November 2003 by Bill Taylor after an idea by Cameron Browne. Interestingly, there is a good reason for the proximity of these independent discoveries, as both were motivated by the upcoming deadline for the 2003 Shared Pieces game design competition.

The game was originally called Goofy Hex then Funky Hex by Randy, but was first made public under the name Chameleon and that has stuck. This name refers to the fact that players tend to change colours based on their environment; the fact that Bill's eyes pop out when he sees a good move has nothing to do with it.

Implementation and help file by Cameron Browne, November 2003.

Cameron Browne, in 2003, designed the following connection game for the Shared Pieces design competition:

Imnyss

• Players alternate taking turns, where each turn involves:
  1. Placing a piece of the nominated colour on the board, then
  2. Nominating what colour the opponent must play next turn.
• The opening piece may be of either colour.
• Goal
  • H wins by completing an 'x' chain or an 'o' chain left to right.
  • V wins by completing an 'x' chain or an 'o' chain top to bottom.
  • If a move results in winning chains for both players then the mover loses.

A sample match:

Imnyss   (game 1)
======                                 CamV   BilH
            C                          ====   ====
  1  . . . . . . . . .             1.  o f3   x e6
   2  . . . . . . . . .            2.  o d6   o g4
    3  . . . o . o x . x           3.  x d8   o d4
     4  . . . o . o o o .          4.  x g3   x c7
      5  . . . . . . . x .         5.  o f6   x f7
   B   6  . . . o x o o x .  B     6.  o b7   o f4
        7  . o x x . x . . .       7.  o b8   x g8
         8  . o x x . x x . .      8.  x i3   x b9
          9  o x . . . . . . .     9.  x h5   o g6
                                  10.  o a9   x f8
              a b C d e f g h i   11.  x h6   o h4
                                  12.  o d3   x d7
                                  13.  x c8   o resigns