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Billy and Matthew decide to play a game of Chess. They live far away, so they decide to do it online.

Matthew wants to play first, but the game randomly gave him the color black (who goes second.)

Is it possible for them to cooperatively move their knights in such a way that when put back them in their original places, it will be Matthew's turn to move? If not prove it's impossible.

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    $\begingroup$ Lateral thinking answer: They just shuffle their knight back and forth three times, causing a draw. Repeat until the game assigns Matthew white. $\endgroup$ – Milo Brandt May 23 '15 at 0:51
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    $\begingroup$ @StevenStadnicki But moving pieces other than knights require moving pawns and pawns can't be moved back $\endgroup$ – Richard Tingle May 23 '15 at 18:28
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    $\begingroup$ @Richard Tingle I think he's talking about the rooks (which are available after moving the knights) $\endgroup$ – warspyking May 23 '15 at 20:29
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    $\begingroup$ @warspyking True, but after moving the rooks, the game isn't in its original state anymore, because you cannot do castling anymore. Also, you can only move them back and forth (an even number of moves) so it won't help much, since you could do that with the knight just as well. $\endgroup$ – GolezTrol May 24 '15 at 10:31
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    $\begingroup$ White's king is on the right, and black's is on the left, so, even if this COULD be done, it wouldn't give either side a true game of chess. $\endgroup$ – JLee May 24 '15 at 19:03
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It is impossible.

Proof:

Initially, there are an even number of knights on white squares (namely, there are two of them, at b1 and g8).

Every time a knight moves, the number of knights on white squares either increases by one (if a knight on a black square moves) or decreases by one (if a knight on a white square moves).

Either way, the parity changes each turn. Thus, if the board ever returns to the initial condition, it must have done so after an even number of moves, meaning it will still be Billy going first.

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    $\begingroup$ Also works if you're allowed to move the rooks as well $\endgroup$ – Mike Earnest May 23 '15 at 1:54
  • $\begingroup$ Great answer, good explaination, couldn't have made a better answer myself :) $\endgroup$ – warspyking May 23 '15 at 3:48
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    $\begingroup$ @MikeEarnest Even if you could have gotten around the parity problem by moving the rooks, you can never get back to the starting position again after having moved a rook. That is because castling is only permitted with a rook which has never moved. $\endgroup$ – kasperd May 24 '15 at 15:39
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While (as other answers proved it) it is impossible to solve this just by using knights, the problem can still be solved.

Matthew says "As I want to play white, my first move isn't really a secret. So I tell you what my first move would be, and you open with the move what you would have moved as an answer. After that I make the move I said, and then we can move with our queens or bishops so that it will be my turn to move".

As all normal openings involve a pawn move which would enable the queen or one of the bishops to move, and both of these pieces can easily move 3 times to arrive back to the same position, the goals of the players can easilt be fulfilled.

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  • $\begingroup$ +1 for a perfect solution to the practical problem...Beautiful answer, I just can't express my astonishment at such a great answer. $\endgroup$ – User Not Found May 23 '15 at 12:51
  • $\begingroup$ Although I now realized that there is a problem in case that Billy wants to give f3 or f4( I know that both are dubious moves but still...) as return move $\endgroup$ – User Not Found May 23 '15 at 13:25
  • $\begingroup$ Not an answer to the problem, but still a brilliant solution to the task. Truly amazing. +1 $\endgroup$ – warspyking May 23 '15 at 15:26
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    $\begingroup$ White could also take two turns (instead of one) to move a pawn two spaces $\endgroup$ – BlueRaja - Danny Pflughoeft May 23 '15 at 16:57
  • $\begingroup$ Once we use @BlueRaja's method to deal with two-space pawn moves, there's still a problem if White opens by moving a knight or by playing f3, a3 or h3. Of the 20 possible opening moves, the only ones listed at chessgames.com.as having an average negative score for White are Nc3, f4 and a3. I was surprised that the list excludes h4. On average, Nf3 wins slightly more points for white than either e4 or d4, as does c4. $\endgroup$ – h34 May 23 '15 at 17:38
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Knights, no. Most chess games start with moving a pawn to the 4th rank, though. Assuming this how white intends to start they should agree and white moves his pawn to the third rank instead, then on the next move takes it to the 4th rank.

Cooperation is only needed to the extent that black does not play something that will interfere with this.

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Agreed with Mike. My explanation would be:

Knights always move to the opposite colour square to that which they're currently on. Therefore, for a knight to move somewhere else and then move back to where it started must take an even number of moves (they're moving to the same colour they started on). An even number of moves from both players will always end with white to move.

Edit (as Édouard rightly points out):

If a player swaps their knights that requires two lots of odd moves: still an even number of moves in total.

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    $\begingroup$ This argument is not sufficient, as the white or the black knights could very well be switching sides. This, of course, would require each of them an odd number of mouvement, and the sum of two odd numbers is even, but that’s not what you say. $\endgroup$ – Édouard May 23 '15 at 2:47
  • $\begingroup$ Good point Edouard. It is questionable whether that agrees with the requirements of "put back them in their original places" and I think that's why I didn't consider it. $\endgroup$ – Avon May 23 '15 at 9:31

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