# Moving a knight to a new square each turn: who wins?

Alice and Bob are playing a game. Alice starts by placing a knight on the chessboard. Then they take turns moving the knight to a new square (standard chess rules apply: the knight moves in an "L" shape). The first player who cannot move the knight to a new square loses the game. Who wins if both players play optimally, and what is the winning strategy?

This is a puzzle from Presh Talwalkar's youtube channel (https://www.youtube.com/watch?v=ZGWZM8PcUlY&ab_channel=MindYourDecisions) and the solution Presh gives is very clever and neat based on colouring 4×2 regions of the chessboard which is a bit unintuitive in my opinion or at least the video doesn't give an intuitive explanation on why consider these grids.

Since this seems like a classical nim as the game will eventually terminate with a winner I'm wondering if this is solvable without the clever usage of the 4×2 grids and colouring?

• When you say 'new', does this mean a square not walked on by either player, or a square not walked on by the player making the move? Jul 20, 2023 at 22:19
• It doesn't matter. Knights change color every move, so if Alice moves onto a square, Bob can't move onto that square. Jul 20, 2023 at 22:31
• This is overkill, but the existence of knight's tours implies such a matching -- just take pairs of adjacent elements along the tour.
– xnor
Jul 21, 2023 at 4:20
• @xnor did you mean to post this comment to the answer?
– Bass
Jul 21, 2023 at 7:45