# Variants using the same equipment

## Contents

## Hex with slightly different winning goal

The rules of Hex can be changed in this way. Red can win by connecting top, bottom and at least one among left and right. In alternative, it can win by having two groups, one connecting left, right and top and the other connecting left, right and bottom. Blue wins are symmetrical, with left and right swapping roles with top and bottom. Like Hex, there are no draws in this game, and there are also no "races" (unlike Chameleon). This game is nearly identical to Hex. If I recall correctly, it has been invented by Mark Steere.

## Reverse Hex

(Main article: Reverse Hex)

Reverse Hex is Hex played under the misère condition, that is, the first player to build a chain between their edges *loses*. Like Hex, the game cannot end in a tie. It has been proved with a non-constructive proof that the first player has a winning strategy on any empty NxN board if and only if N is even. The game seems quite interesting when played on small boards (like 8x8) and with the swap rule.

Little is know with certainty about Reverse Hex strategy. Apparently what is good Hex (centre, corners, "defensive" play and virtual connections) is really bad Reverse Hex. It is usually possible to see some moves ahead because good moves are often good for both players. Despite the appearances the opening and middle-game phases are fairly important.

## 1-2-2 Hex

1-2-2 Hex is a variant of Hex where the first player places one stone on the first turn, and subsequently, each player places two stones on each turn. The idea is to mitigate the first-player advantage by making the very first move only half as useful as all subsequent moves. Consequently, no swap rule is used in 1-2-2 Hex.

According to a letter from Martin Garner to Piet Hein on April 6, 1957, 1-2-2 Hex was suggested by John Tate at Princeton. See also chapter 6.2 of Hayward and Toft, "Hex, the full story". Since then, 1-2-2 Hex has been rediscovered many times, and is occasionally referred to by the names of the people who rediscovered it, such as Halsør's Hex and Esa Hex.

## Cooper's_Hex

Main article: Cooper's Hex.

Cooper's Hex is similar to 1-2-2 Hex, but uses the Thue-Morse sequence BWWBWBBWWBBWBWWB... to determine the move order.

## Random-turn Hex

Random-turn Hex is a variant of Hex where before each move, it is randomly decided whether it is Red's or Blue's turn. The strategy of random-turn Hex is entirely different from that of ordinary Hex, and the game has been completely analyzed in a paper by Peres, Schramm, Sheffield, and Wilson. The main result of that paper is that in each position, each player's winning probability is exactly the same as if the rest of the board were randomly filled with black and white stones. A consequence is that in each position, Black's optimal move is the same as White's optimal move, and is the cell most likely to change the outcome when the rest of the board is randomly filled. Because of these facts, it is easy to program a computer to play random-turn Hex as close to optimally as desired. The game holds very little interest to human players, because even under optimal play, the outcome seems to depend far more on the luck of the draw than on the particular skill of the players.

## Bidding Hex

Bidding Hex is a variant of Hex where instead of alternating, the players bid for the right to make the next move. The players start with an equal amount of money or tokens, and before each turn, each player announces how much they are willing to pay for the right to make the next move. The player who names the higher amount gets to make the move, and must pay the other player the agreed amount. A tie-breaking rule is usually added for the case when both players name the same amount. Bidding Hex has been analyzed in a paper by Payne and Robeva. Assuming money is continuous (can be divided into arbitrarily small amounts), bidding Hex is mathematically equivalent to random-turn Hex, but without the element of randomness. Therefore, computers can be programmed to play extremely close to optimally.

## Chameleon

*Adapted with permission from Cameron Browne's PBeM Help files.*

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.

### Rules

Two players, Vert and Horz, take turns placing either a red piece or a blue piece on the board.

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

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

### Examples

A win by Horz:

A win by Vert:

A win by the last mover:

### 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

- Ryan Hayward and Bjarne Toft. Hex: The Full Story. CRC Press, 2019. ISBN 978-0367144227.

- Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson. Random-Turn Hex and other selection games. American Mathematical Monthly, 114:373--387, May 2007. Available from arXiv:math/0508580.

- Sam Payne and Elina Robeva. Artificial intelligence for Bidding Hex. Games of No Chance 4, MSRI Publications 63 (2015), 207-214. Available from arXiv:0812.3677.