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Alan Baljeu wrote on 2005-05-13 UTC
Here's a fair way of shuffling the pieces.  All 960 positions are equally
probable, and no extra equipment is needed.  It works by splitting the
placement into two phases for dark and light squares.

1. Randomly shuffle the 6 pieces not including the Bishops.  Divide these
pieces into two groups of 3 each.
2. Take the first 3 pieces and one bishop.  Shuffle these 4 pieces. 
Place
them on a1, c1, e1, and g1.
3. Take the remaining pieces and 1 bishop, and randomly shuffle them. 
Place these pieces on b1, d1, f1, and h1.
4. Correct the king position by swapping with a rook if needed.

Here's the math: 
The first shuffle yields 6-choose-3 or 6*5*4/3/2/1 ways to divide the
pieces.
The second shuffle yields 4*3*2 permutations of 4 pieces.
The third shuffle yields  4*3*2 permutations of 4 pieces.
Divide by 2 because the knights are the same.  
Divide by 6 because of the king/rook unshuffle.
End result = 5*4*4*4*3 = 960 positions.

To make the shuffle totally fair, label the bottom of your eight pawns
RRRBBNNQ and shuffle them instead of your pieces.

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Comment on the page Fischer Random Chess: Manual Procedure for Generating Piece Plac

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