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, opening in the acute corner wins). 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.