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Yeah. It would be possible for both players to give themselves the same column and give each other the same row. So they'd gain the same space on that turn. That's pretty unsatisfying.
I think this kind of pitfall is unavoidable if you want perfect symmetry, though. If both players are playing identical games and make identical moves, their outcomes will be identical.
You're right. Suits aren't necessary to introduce this kind of symmetry. I think we can also do away with the double meaning of the tiles by altering the selection rule.
Imagine if each player sees the same board (not transposed!) and they pick columns for themselves and rows for their opponent, with the restriction that the index of a chosen row can't match the index of a chosen column. This way we don't even need an extra condition on the board generation.
I was curious whether there would be any fair boards under that rule, so I came up with a classification. Every such board can be row/column permuted into a board that has (1,1), (2,2), (3,3), etc going down the diagonal and the tile in position (i,j) is the pair of the tile in position (j,i). There are on the order of 10^16 distinct boards that look like that.
There might be a way to make a variant of this game perfectly symmetrical if there are six suits and six ranks. You could make the values one player sees as suits the same ones the other sees as ranks, so a fair board would be one where the transpose of the rank-swapped board was a row and column permutation of the original.
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