Chess puzzle with king

Question

There are 2 players who share the same king on a chessboard. The starting point is A1 and they want to finish at H8.The King can move up (A1-A2) right (A1-B1) and diagonaly (A1-B2) but back tracking isn't allowed. Each player move the king one space and then it's the other player's turn. What's the tactic of one of them to always win (so arriving first to H8)?

Answer 1

The first player wins by moving to b2 and then repeating the moves of the second player, keeping the king on an even-numbered file (b, d, f, h) and an even-numbered rank. Indeed, if these $16$ squares are marked on the chessboard:

• from any of the marked squares, only unmarked squares can be reached in one move – except h8, where the player presented with it loses. In this case the first player always presents the second player the king on a marked square
• from any of the $33$ unmarked squares reachable from b2, a marked square can always be reached. If the second player moves in one direction, the only marked square reachable is the one reached by moving in the same direction

This easily extends to $m×n$ boards: the second player wins iff both $m,n$ are odd.