Wikipedia:Reference desk/Archives/Mathematics/2015 April 2

= April 2 =

Symmetry in chess
I know that chess conventions of the starting board has the white queen to the right of her king, with the black pieces mirrored across from them. And it's clear that swapping one of those pairs would change the game to some extent. Now I am not a very experienced player (maybe a dozen full games in my short adult life), but my engineer's mind says that as long as the queens are in the same file as each other, it shouldn't actually matter whether it is the d or e file, as that would just create an identical, if just mirrored, version of the game board, where each move would have the same effect on the game as if it was not mirrored.

However, when talking with a friend who has actually spent a good amount of time learning chess strategies and tactics, I am met with the argument that I don't actually know what I'm talking about. I also can't see how the additional convention that white always goes first would change anything about this, since in my mind swapping the kings and queens is essentially the same as having black go first instead (player with queen on the right goes first). The only think I could think of as to why this would matter would have to do with players who have studied and memorized certain configurations, where different characteristics are given to black and white respectively, and also perhaps on the "queenside" and "kindside" halves of the board.

Am I missing anything here? I tried looking around on google for this, but all I keep getting are discussions on the effectiveness of mirroring your opponent's moves, but that's a completely different question. —Akrabbimtalk 17:17, 2 April 2015 (UTC)


 * You are correct that reflecting the board produces an isomorphic game, where every move in normal chess corresponds to its mirrored move in this alternate version of chess. Your friend is either misunderstanding what you're describing or simply wrong.--80.109.80.31 (talk) 19:13, 2 April 2015 (UTC)


 * A proper mathematical explanation would speak of isomorphisms, as 80.109.80.31 does, but your engineer's intuition sounds good to me, as you could set up the board your way but play with both you and your opponent looking at its reflection in a mirror. This works because the rules of motion, capture, check, and check-mate are symmetric under reflection, and depended only on the relative position of pieces.  You are also correct that playing a mirrored board might through off experienced players. -- ToE 19:47, 2 April 2015 (UTC)


 * To play the devil's advocate here, the "handedness" of the game is of psychological importance for players. Call the version of chess with the queen "on her color" right-handed chess, and the one with queen on opposite color left-handed chess.  Although it's true that the two games would be isomorphic, they would not "feel" isomorphic.  If you're willing to humor a slippery-slope argument for a moment, at a very crude level "the game of chess" is just some directed graph whose vertices are the positions of a game, and whose edges represent the valid moves, together with some nodes distinguished as "winning positions" and "stalemate".  If you are presented with such a graph, it is by no means clear that it should represent the game of chess.  Navigating to a "winning position" would be extremely complicated task, and would be almost impossible for a human to do.  The "game of chess", to actual human players, is more than just the isomorphism class of the game.   Sławomir Biały  (talk) 21:16, 2 April 2015 (UTC)
 * I agree with all replies and in particular anon's. Your friend has either misunderstood or is wrong.
 * And it's ok if he's wrong, even if he's a chess expert. Because it's not a question about chess, it's a question about mathematics.
 * Likewise, you could have presented to him the following argument: In chess, there are 3 possibilities:
 * White can guarantee a win
 * Black can guarantee a win
 * Both players can guarantee a draw
 * He could claim you're mistaken (saying, for example, that for every player, you can find a player that beats him consistently regardless of side). And he'd be wrong, because this, too, is not a question about chess.
 * PS. From White's perspective, his queen is to the left of the king.
 * PS2. Not a chess expert myself, but to build up on your comment "swapping one of those pairs would change the game to some extent" - it would pretty much throw away most of opening theory; it wouldn't change the principles of midgame, but change the situations you are likely to encounter in it; it would have no effect on endgame.
 * There are popular chess variants that have to do with shuffling the base rank pieces; the most notable is probably Chess960, but actually it doesn't allow this kind of asymmetrical shuffle. -- Meni Rosenfeld (talk) 12:15, 3 April 2015 (UTC)