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Complexity of Large Variants. Some comments about the complexity of large chess/checkers variants.[All Comments] [Add Comment or Rating]
Sam Trenholme wrote on 2007-02-23 UTC
As it turns out, the number of possible opening positions increases even faster than EXPTIME when Chess Variant boards get bigger and bigger.

From a message I posted to the old Yahoo group:

There are 1,440 setups in 8x10 chess where the queen is to the left of the queen.

If you add a single faerie piece, there are 12,600 setups for 9x8 chess (with the queen to the left of the queen).

If you add two of a single colorbound faerie piece, there are 36,000 possible 10x8 setups (with the queen to the left and all that). If you add two of the same piece which isn't colorbound, there are 63,000 possible 10x8 setups. If you add two non-colorbound pieces, such as the archbishop (bishop + knight) and the marshall (rook + knight), there are 126,000 possible setups.

126,000 setups vs. 1,440 setups. No wonder why so many more are playable.

We can go even further: If you add three unique non-colorbound pieces to FIDE chess on an 11x8 board, 1,360,800 possible setups (680,400 if we add two of one kind of piece and one of anothe kind of piece, such as two archbishops and a marshall). If we add four unique non-colorbound pieces to the FIDE mix on a 12x8 board, we have 16,329,600 starting positions with the queen to the left of the king. If we insist on making it two pairs of colorbound pieces to a 12x8 board (such as two camels and two camels + bishops), this restricts us: We have only 1,296,000 possible starting positions.

And, even further: If we have a 'Grand Chess'/Shogi setup on a 10x10 board, with the pawns on the third row and two sets of Capablanca Chess pieces (we discard the second king) behind the pawns, we have some 92,201,259,150,000 total possible setups (with the king on the right hand side).

It might take a while for the chess variant community to come with a full opening theory for each and evey one of the above setups. :)

- Sam