What is Rex (Reverse Hex)?
Rex, also called Reverse Hex or Misère Hex, uses the board, the stones and the turn order of standard Hex, but turns the goal around. Red still owns the top and bottom edges and Blue the left and right edges, yet linking your own two edges is now the way to lose. Whoever completes such a chain hands the game to the opponent.
The game cannot end in a draw. A full Hex board always holds exactly one chain between a pair of opposite edges, so as the board fills up, someone is eventually forced to make one. Only the player who moves can complete a chain of their own colour, so the loser is always the player who placed the last stone. There is no passing: when every empty cell left would finish your chain, you must play one of them and lose.
Good Rex play looks very different from Hex. Instead of racing across the board, each player tries to spread stones that stay apart from each other and to leave the opponent with nothing but bad moves at the end. The swap rule and every board size from 7×7 to 19×19 work in Rex exactly as they do in standard Hex.
Rule changes
| Rule | Standard Hex | Rex (Reverse Hex) |
|---|---|---|
| How to win | Standard HexBe the first to link your own two edges with an unbroken chain of your stones. | Rex (Reverse Hex)Make your opponent link their own two edges. The player who completes a chain between their own edges loses. |
| End of the game | Standard HexThe game ends as soon as the mover completes a chain, and that player wins. | Rex (Reverse Hex)The game ends as soon as the mover completes a chain, and the opponent wins. You may not pass, so with only chain-completing cells left you must play one. |
| Draws | Standard HexImpossible: a full board always contains exactly one winning chain. | Rex (Reverse Hex)Also impossible, for the same reason: the board cannot fill up without one player linking their edges. |
| Swap rule | Standard HexOn by default: after Red's first stone, Blue may take it over instead of placing a stone. | Rex (Reverse Hex)The same, and it can be switched off in the table setup. In Rex an early stone can be a burden as well as a help, so deciding whether to swap is a different question. |
| Board | Standard HexAn 11×11 rhombus of hexagonal cells, with sizes from 7×7 to 19×19 in the table setup. | Rex (Reverse Hex)The same choice of board, from 7×7 to 19×19. |
History
Reverse Hex has been studied almost as long as Hex itself. Martin Gardner wrote about it in the late 1950s, after introducing Hex to the readers of his magazine column, reporting a discovery by Robert Winder: on a square n×n board without the swap rule, the first player can force a win when n is even and the second player when n is odd.
Jeffrey Lagarias and Daniel Sleator published a proof of that result in 1999, and showed more: the losing player can always hold out until the board is completely full, which makes Reverse Hex a long and tense game even when the outcome is settled. Because of this, games of Rex usually go much deeper into the board than games of Hex.
Ryan Hayward and colleagues at the University of Alberta, whose programs solved standard Hex on small boards, have since published studies of how to play Rex well, using the same computer methods.