# A Game of ChessTron

The rules of ChessTron are as follows.

1. Alice picks a chess piece (not the queen, as it is considered too powerful), and places it in the corner of a standard 8x8 chessboard.
2. Bob picks a chess piece (again not the queen, but could be the same as Alice's choice), and places it in the opposite corner.
3. Players alternate moving their pieces starting with Alice. Capturing is not allowed. Every time a piece leaves or moves through a square, the square is destroyed, and cannot be moved to or through again in the game.
4. If a player is "trapped" without a legal move, that person loses the game.

Who wins and why?

• If queens are allowed, I believe it is an Alice victory, but I wasn't every able to find a good argument as to why Alice could win with a queen if Bob chooses a knight. Jun 24, 2015 at 16:28
• If a knight is selected as one of the pieces, which squares are removed when it moves? Jun 26, 2015 at 6:57
• @Kevin: When a knight moves, only the square it leaves is removed from the board. Jun 26, 2015 at 20:49

Bob wins.

I'm assuming the eligible pieces are the knight, the bishop, the rook, and the king.

If Alice picks the knight, the rook, or the king, Bob can win by picking the same piece. On each of his turns, Bob can play the move that is the 180 degree rotation of the move Alice plays. This works because the starting position of the board is rotationally symmetric, so after Alice's first move, Bob can make the 180 degree rotated move. This puts the board back into a symmetric position (including forbidden squares), allowing Bob to mimic Alice's next move in the same way. Eventually, Alice will run out of moves.

This strategy will not work if Alice picks the bishop. If Bob also picks the bishop, then on Alice's first move she can move her bishop across the diagonal and place it next to Bob's bishop, preventing Bob from moving at all on his turn.

If Alice picks the bishop, Bob can win by picking the king. Assume that the players put their pieces on the black diagonal. Bob's strategy is to advance the king along the black diagonal toward the opposite corner until this is no longer possible. At this point, Alice's bishop becomes trapped one one side of the board, while Bob's king can take advantage of all the black squares on the other side, plus all of the white squares, which Alice cannot move to. Alice will run out of moves before Bob does.

A note about the rook and bishop cases: Alice is able to thwart the symmetry strategy in the bishop case because she can move multiple squares directly toward Bob, moving through the square Bob needs to occupy. For example, say Alice's bishop starts on a1, and Bob's on h8. If Alice moves to g7, Bob's symmetric move is to b2. However, Alice moved through b2 to get to g7, so Bob can't move there. This cannot happen in the rook case. In order for Alice to move her rook directly toward Bob's, the two rooks must occupy the same rank or same file on Alice's turn. However, Bob's strategy ensures that the board is always rotationally symmetric at the beginning of Alice's turn, and no position with the rooks in the same rank or same file is rotationally symmetric.

The symmetric king case does not fall into the same trap as the bishop case, either. Although Alice can move directly toward Bob along the diagonal, she can only move one square at a time. In any rotationally symmetric position with the two kings on the same diagonal, there are an even number of squares between them, so if Alice moves closer to Bob's king, Bob will be able to make the corresponding move toward Alice's king.

• Very good, this is essentially a full solution, with one exception. I'd like more explanation about when Bob can win by moving symmetrically. Why are some cases symmetric while others aren't? Especially since I think you mis-catagorized one of the cases. Jun 24, 2015 at 5:47
• Nevermind, I missed one of the cases! Jun 24, 2015 at 6:16
• Alice can thwart the symmetry strategy in the bishop vs. bishop case by moving directly toward Bob and moving through the square Bob would need to move to. This is not possible in the other symmetric cases. I'll add a note about the rook case, since your comment caused me to double check that that one couldn't get into the same situation as the bishops. :) Jun 24, 2015 at 6:19
• A pawn would also be a valid choice, wouldn't it? It's not a very good choice as Bob could win quickly with a bishop, rook, or knight by simply blocking her advancement. Jun 24, 2015 at 12:33
• @JohnStevens: Thanks, I like what you added about the rook. This was the case I really had to think about, but the case I messed up on was actually the king case: I thought it wasn't a symmetry case, but of course it is. Jun 24, 2015 at 16:27