05 July 2014

A Better Pawn Method

A short formality at the beginning of every chess960 game is to choose the start position for that game. If you're playing in a tournament, the position is always given, but if you're playing for fun, how do you select a random position? Here is a new method that was flagged to me by its creators: The Pawn Method: An easy way to set up Fischer Random Chess positions.
The Pawn Method is a simpler, faster way of setting up a starting position for Fischer Random Chess (aka Chess960). Unlike other methods, it doesn't require dice, coins, or a complicated process. It has been proven to select one of the 960 possible starting positions with perfect randomness. Created by Robby Walker and Tim Suzman.

Since web pages come and go, I'll quote the gist of the method here.

On the bottom of each Pawn, write the following: [White Pawns] Q B B N N... [Black Pawns] L1 L2 L3 L4 R1 R2 R3 R4

That only needs to be done once. For each game,

Begin by randomly shuffling the White Pawns in their starting row. Look under each Pawn. If the Pawn says Q, B, or N, place a Queen, Bishop, or Knight in the corresponding starting square.

Did the two Bishops land on the same color square? Randomly select one Black Pawn and look at the bottom. The number (1-4) tells you which number square of the opposite color to move the bishop to. Move the Bishop to that new square, swapping it with the piece already there if there is one.

Fill the three empty squares with Rook - King - Rook, in that order.

A few years ago I mentioned a similar, simpler method in Chess960 Waits for No One (see the last quote in the post). Ichabod, a professional statistician, commented on the post to point out that it didn't produce an even distribution over the 960 start positions. How does the Walker / Suzman method fare? I asked Ichabod for his opinion and he replied,

There are 1680 the letters on the bottom of the White Pawns can be arranged. 960 of those will be valid positions, leaving 720 invalid positions. The Black Pawns can rearrange each invalid position into 8 other positions, or 5,760 ways total. 5,760 / 960 = 6. That means that if the 5,760 ways are evenly distributed among the 960 legal positions, then the method works. My attempts to write a Python program to confirm that it does this have failed (they're giving invalid results meaning there is an error in the program), so I can't confirm it. My lunch hour is up, so I don't have time to fix the program.

A day later he confirmed that the method worked.

I redid the Python program. The system works. Each possible 960 position has the same probability of being chosen.

While that's a good start for the 'The Pawn Method', what about the statement that 'it doesn't require dice, coins, or a complicated process'. While it doesn't require dice or coins, it does require a special chess set. Lacking that set, how do you generate a new start position? You will still need dice, coins, or yarrow sticks. As for whether or not it is a 'complicated process', that requires a practical field test where real people judge its simplicity.

Chess960 certainly needs a simple, effective method to generate a new start position. Should chess sets be distributed with a mechanism to do that? Perhaps the time has already arrived.

No comments: