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Suppose you come across a chessboard where the pieces are colored red and green and there are no labels on the squares. You don't know which move number it is.
What are the positions where, by only knowing whose move it is, you can determine which of the red or green pieces are the 'White' pieces?
There are many positions where if you know whose move it is, you can determine who is Black and who is White for parity reasons, e.g. :
[FEN "nrb1kb1N/1ppppppr/p6p/8/nN6/P6P/1PPPPPP1/1RBK1B1R w - - 0 1"]
In this position, White has played an odd number of moves:
Meanwhile, Black has played an odd number of moves:
We can deduce that both sides have played the same number of moves, and thus it's White to play.
Conversely, if you meet this position with red and green pieces and you know which side is on move, you can deduce that this side is White.
Alter the position with Pa7 -> Pa8, and the conclusions are reversed : the side on move will be Black.
This will work for any legal position where no unit but queens has been captured, and all the pawns stand on their second rank, but the rook-pawns that may be on their second or third ranks.
For White, the king starts to the right of the queen; for Black, it's the opposite. Hence, if for only one side, the king, queen, bishops and b- up to g-pawn are still on their original square, you don't even need to know whose move it is to know who's White and who's Black.
[FEN "1nbqkb2/1pppppp1/8/8/8/8/1PPPPPP1/1NBQKB2 w - - 0 1"]
(the knights are there because some piece must have made the last move).
In some cases, like the diagram below, those pawns could actually be on the seventh row; if there are sufficient pieces from the other color, retrograde analysis can prove the position is not possible.
[FEN "2bqkb2/1pppppp1/8/8/8/8/8/4K3 w - - 0 1"]
I have a hard time thinking of a position where you can tell who's White and who's Black, unless you know who is to move. I suspect it might not be possible at all.
In the initial position, with red and green pieces, if you are told it's red turn then red must be white.
From the initial position, you can only reach the initial position by playing an even number of moves, thus both side must have played the same number of moves and thus it is white's turn.
Probably openings, checks and stalemate situations. Openings because you can for the first moves count who is to play, checks because it is always the turn of the player in check and stalemates because it is the move of the player that can make a move otherwise it already was a draw
