# A game-theoretic chess puzzle — Proof verification

I recently came up with the following chess puzzle (which has almost nothing to do with one's Chess skills):

Puzzle: Consider a variant of chess where black has to start with $$1...e5$$ regardless of white's first move. All other rules remain unchanged. Prove that in this variant, white can at least force a draw.

I have told this it to several people, and some of them seem to think that my solution is wrong.

Solution: consider the position where white has a pawn on $$e4$$ and black has a pawn on $$e5$$, with all other pieces at their starting positions. Let's refer to this position as $$P$$, the player with the next move as player $$1$$, and the other player as player $$2$$. Then according to Zermelo's Theorem, starting from position $$P$$, either player $$1$$ can force a win, or player $$2$$ can force a win, or they can both force a draw. We consider them one by one:

• If they can both force a draw starting with position $$P$$, all white has to do is to reach this position by playing $$1.e4$$, as black will have to go $$1...e5$$ which creates position $$P$$.

• If player $$1$$ can force a win, white has to reach position $$P$$ and be the player with the next move. In this case, white will still play $$1.e4$$, which is followed by $$1...e5$$. At this point, position $$P$$ has been reached and white is the player with the next move.

• If player $$2$$ can force a win. White will play $$1.e3$$ and after black plays $$1...e5$$, white will play $$2.e4$$. Now we're in position $$P$$, and it's black to move. This means that black is player $$1$$, which makes white player $$2$$.

They argue that position $$P$$ is not the same position when player $$1$$ is white and when it is black. When player $$1$$ is black, their king is on their right side, whereas when player $$1$$ is white, their king is on their left (same goes for the queen).

My answer is that starting with any symmetric position (like $$P$$, or the starting position in standard chess or this position) if white had a winning (drawing) strategy $$\mathcal{A}$$ starting with position $$P$$, then if black were to start instead, it would have a winning (drawing) strategy $$\mathcal{A}'$$ where for every move (black's or white's) in $$\mathcal{A}$$, row $$i$$ is replaced with $$9-i$$, where $$1 \leq i \leq 8$$ (e.g. $$Nc6$$ would become $$Nc3$$). Thus position $$P$$ is the same regardless of who starts.

Is my proof correct?

• Strictly speaking, you should notice that there are no possible en passant captures (since that's the one potential difference between your 1. e4 and 2. e4 positions). – Micah Feb 10 at 6:08