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Complexity of Large Variants. Some comments about the complexity of large chess/checkers variants.
George Duke wrote on 2010-03-01 UTC
```This 1999 information page, Sam Trenholme commented 4 years ago:
http://www.chessvariants.org/index/displaycomment.php?commentid=16245.```

Sam Trenholme wrote on 2007-07-14 UTC
Looks like you're right. I will draw it out:

• First, we place the bishops. There are 2025 ways of arranging the bishops.
• Next, we place the knights. For each arrangement of bishops, there are 1820 ways to put the knights on the board.
• Next, we place the rooks. For each arrangement of bishops and knights, there are 495 ways to place the rooks.
• Next, we place the Archbishops. For each arrangement of the minor pieces, there are 28 arrangements for the archbishops.
• Next, we place the Marshalls. For each arrangements of all of the above pieces, there are 15 Marshall setups.
• Next, we place the Queens. For all of the above setups, there are 6 ways to have the queens on the board.
• Next we place the king. There are two places he can go. However, this is balanced by the fact we remove all mirror images.
2025 * 1820 * 495 * 28 * 15 * 6 = 4,597,292,700,000

Sorry it took so long to verify your number. This is the first time I have had a computer with an arbitrary precision calculator and net access at the same time in a while (Let's hear it for Ubuntu live CDs).

Edit: Yes, 64-bit computer owners, the above number can easily be calculated using 64-bit integers. I'm still in the 32-bit stone age. Do they even make 64-bit laptops that weigh less than 6 pounds?

Andy wrote on 2007-02-24 UTC
Actually, if required two bishops on each color, number decreases to 4,597,292,700,000.

Andy wrote on 2007-02-24 UTC
Not to be argumentive, but I believe two sets of capablanca pieces on twenty squares, minus one king, accounting left-right symmetry, gives 'only' 10,999,448,460,000 arrays. Calculated by 20!/(2*2*2*2*4!*4!*4!). One of the 2's in denominator is left/right symmetry factor.

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

Greg Strong wrote on 2007-02-22 UTC
```This is actually good information, but without further description, it, in
itself, doesn't mean anything to non-comp-sci people. It does provide
references, though, but I am willing to bet that those won't make a lot
of sense to non-comp-sci people either.  Maybe some day I will try to
write more description about these concepts.

At the very least, this page is mis-classified.  This should not be a game
page, but an information page.```

Jeremy Good wrote on 2007-02-21 UTC
This was written for non-computer science people? I don't understand it myself.

Roberto Lavieri wrote on 2003-08-30 UTCExcellent ★★★★★
Interesting!

Anonymous wrote on 2003-08-26 UTCGood ★★★★