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 is to form a unbroken chain of stones that connects the top to the bottom, while the second player attempts to form an unbroken chain of her stones connecting the left side and the right.
Cell  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 ). If White places a stone in one , Black should place in the other . This is called a 2-bridge.
We can check for more information at the FAQ or in here. There's also a wiki about Hex.
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.
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.