How to play Reverse Hex

We present new results on how to play Reverse Hex, also known as Rex, or Misere Hex, on n i? n boards. We give new proofs-and strengthened versions-of Lagarias and Sleator's theorem (for n i? 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 i? n board. Highlights? 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 2 k -by- 2 k 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.