Puzzle #3
Whose turn is it to play?
Given a chess position, can you determine whose turn it is to play?
We have another chess puzzle!
The Problem
You are organising a chess tournament. The games had just started when the clock struck one, and all players promptly left their games and headed for lunch! When they returned, they found to their dismay, they had all forgotten whose turn it was to play. Can you help them out?
For each game, you know only the positions of the pieces and nothing else. From this information alone, you need to decide whether it’s White or Black to play next. You may assume that the players have been playing legal chess moves, but not necessarily optimal ones.
“Is this even possible?”, you ask. Does a chess position have the information required to deduce which player plays next? This is what you will try to reason out.
For each of the following games, find which one of the following statements holds:
- It must be White’s turn to play
- It must be Black’s turn to play
- It could be either White or Black to play
- It is an impossible position. (i.e. One cannot reach this position by playing legal moves only. Someone might have disturbed the board.)
Game 1
Here’s Game 1 of the tournament. This position looks like the initial position of a game.
It might be the case that the game hasn’t started, which would mean that it could be White to move. It remains to be answered if this position could be Black to move.
Could it be the case that the players played some moves and then arrived back at this initial position? If not, then we are done. Otherwise, we need to find out if there exists a sequence of moves that results in this position with Black to move.
Game 2
Something’s up with the knights… They seem to have switched places. Whose turn is it to play?
Game 3
One of Black’s knights has vanished from the board! Whose turn is it to play?
Game 4
All the knights are gone!
Whose turn is it to play?
Game 5
Uh-oh! Black is in trouble. Black is down a lot of material. Whose turn is it to play?
Game 6
In a standard game of chess, White goes first. So what happened here? Is this position even possible? If yes, then whose turn is it to play?
Game 7
White might be preparing for Fool’s mate here. Whose turn is it to play?
Game 8
Black has moved a pawn. All other pieces are at their initial position. Whose turn is it to play?
Game 9
White’s knight has forked Black’s rook and queen! Whose turn is it to play?
Game 10
And here’s the final game in this puzzle. All the pieces are on the board.
Whose turn is it to play?
Be careful: this is a tricky one!
Hints
If you have spent some time thinking about this problem and need some help to proceed, then these hints might be helpful. (Click on a hint to reveal)
Hint 1
Given a chess position, how do we show that it’s White’s turn to play next? We must show that for all sequences of legal moves that result in the given position, the last move of the sequence must have been played by Black. That would mean that it’s White to move in the given position. Moreover, we must also show that at least one such sequence exists.
Showing that a position is Black to play uses similar reasoning.
How do we show that a position could be either White to move or Black to move? This case is simpler. It is sufficient to show two sequences of legal moves that result in the given position: one ending with a move played by White, and the other ending with a move by Black. There’s no way of knowing from the position alone which player must play next, and therefore, the position could be either White or Black to play in the game.
Some positions are impossible in chess. That is, there does not exist any sequence of legal moves that result in the position. To show that a position is impossible, we must show exactly this: There does not exist any sequence of legal moves that results in the position.
Hint 2
At any point of time in a game, look at the parities of the number of moves played by White and the number of moves played by Black. What does it mean if both are odd? What happens when one is odd and one is even?
Generalisation
Try to come up with a fast algorithm that, given a position, decides whether the position is White to play, Black to play, either, or none.
You could begin by finding a simple characterization for positions that must be White to play. What are the necessary and sufficient conditions for a position to necessarily be White to play?
It might help if you try out more chess positions. If you have access to a physical chess board, that’s great! Otherwise, you can play with the board online on the Lichess board editor.
Good luck, have fun!
If you solved the problem, how did you solve it? Share your solutions, and the best ones will be featured with the next puzzle.
If you are stuck, share your progress. What have you tried so far? I will give you a hint.
Let me know if there’s an error or something’s not clear in the puzzle.
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