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If you really want to go for the ultimate in symmetry, I would suggest we need to do away with the notion of a square board. A square has only eight symmetries: reflection NS or EW, 180 degree rotation, or any (or no) combination of these. Indeed, the ultimate in symmetry would be to do away with the board's edges: the board should be infinite, hence giving it translational as well as reflectional symmetry. And we should do away with the notion of cells within the board: the most symmetrical 2-dimensional object being the entire Euclidean plane, in which any point is equivalent to any other. Then we have complete rotational symmetry, about any point, as well as translations and reflections. But since we're pursuing symmetry as the ultimate goal here, we need to embolden ourselves to take the next vital step as well. To do away with the last vestiges of ugly asymmetry, we must also abolish the pieces: for once pieces are introduced into our pristine continuum, they render the game asymmetrical again, by causing some points and directions to have more importance than others: in particular, the points pieces occupy, and the directions they would need to move to attack other pieces, would have special importance. Our ultimate, perfectly symmetrical chess must therefore consist of an infinite plane with NO PIECES AT ALL. It might be objected that without pieces it will be difficult to state rules of movement, capture, initial setup, and object. But clearly, since we desire a perfectly symmetrical game, we must abolish these notions as well: because the perfectly symmetrical chess game must be symmetrical in time as well as in space, and therefore it must have no beginning, no end, and no change: the state of the game at any point must be the same as its state at any other point. And so, at last, we have our perfectly symmetrical game: no cells, no pieces, no goal, no players: is not its perfect, chaste serenity a thing of beauty? Have we not achieved true theoretical perfection? And can we not get back to discussing real chess games now?
My reflection on these games has led to the conclusion that Symposium Chess, named for a story told in Plato's Symposium, is the variant that most closely approximates Chess while having perfect bilateral symmetry. Almost any Game of Chess can be played as a game of Symposium Chess. So can almost any game of Sinister Queens Chess. A game of Symposium Chess could also play out as the mathematically equivalent mirror image of either Chess or Sinister Queens Chess. White's first move cannot determine which of these the game will be played as. After White's first move, all four possibilities remain. So each move White can make gives him about four times as many possible games as the first move in Chess makes possible. But half of these are mathematically equivalent to the other half, and it turns out that half of White's moves lead to games mathematically equivalant to what would follow from the other half. If we consider only the moves available to White on one side of the board, say the left side, each possible combination of King and Queen remains distinct, so that each move leads to four times the possible games as a move in Chess does. And if we consider the moves available on the other side, the same will be true, but each game in one set will be paired with a mathematically equivalent game in the other set. So White's first move let's him choose from only about twice as many possible games as he has to choose from in Chess. If White had 40 opening moves to choose from, he would have more control over the course of the game on his opening move, but with only ten mathematically distinct opening moves available, White's first move exerts less control over the course of the game than a first move in Chess does. Besides taking away some of the control White has on his opening move, Symposium Chess gives more control to Black. Both players have the power to differentiate their Monarchs into King and Queen. This gives players power not only over the course that the game will take from there but also power over the significance of previous moves. Now, although most games of Chess can be played as games of Symposium Chess, an actual game of Symposium Chess would probably play out differently than a game of Chess, because it does change the dynamics of the opening game. Until a player differentiates his King and Queen, his opponent doesn't have a clear target for his attack, and if he does pick one Monarch as the focus of his attack, that just increases the odds that this Monarch will be the Queen and not the King. Overall, it would seem that Symposium Chess puts White and Black on a more equal footing, so that White gains less of an advantage from moving first. But questions remain to be asked. How great a difference is it? What significance does it make? Will the difference be seen only by statistically comparing game scores of top players? Does this difference make Symposium Chess the better game? If so, how much better? My inclination is that it makes only a slight difference, and that if it is the better game, it is not better by that much. I would not presume that Chess is junk for being asymmetical while Symposium Chess, which is not that different from Chess, is so much greater for being symmetrical.
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