How to play Reverse Hex

We present new results on how to play Reverse Hex, also known as Rex, or Misère Hex, on n×n boards. We give new proofs-and strengthened versions-of Lagarias and Sleator's theorem (for n×n boards, each player can prolong the game until the board is full, so the first/second player can always win if n is even/odd) and Evans's theorem (for even n, the acute corner is a winning opening move for the first player). Also, for even n≥4, we find another first-player winning opening (adjacent to the acute corner, on the first player's side), and for odd n≥3, and for each first-player opening, we find second-player winning replies. Finally, in response to comments by Martin Gardner, for each n≤5, we give a simple winning strategy for the n×n board.

► We extend a classic result of Lagarias and Sleator, and also one of Evans. ► For n-by-n boards, we give new strengthened proofs that each player can prolong the game until the board is full. ► For 2k-by-2k boards with k at least 2, we find a new first-player winning opening. ► In response to a comment of Martin Gardner, we give simple winning strategies on small boards.