09 February 2013

Of Knights and Start Squares

After finishing last week's post on win/draw percentages for the pieces on different start squares -- Pieces, Start Squares, and Scores -- I created more tables for pairs of similar pieces. For example, what is the win/draw percentage for Bishops starting on the a- and b- files versus starting on the a- and d- files? While I was doing this, I discovered another geometry that I had bever noticed before. I'll introduce it with a question.

How many chess960 positions are there where the Knights start on the a- and b- files? Of the six remaining squares, three are available for one Bishop and three are available for the other. Placing the Bishops leaves four squares for the Queen. Putting this together means there are 3 x 3 x 4 possible positions, for a total of 36 positions.

How many positions are there where the Knights start on the a- and c-files? If you answered, 'also 36 of course!', then you're wrong. When the Knights start on a- and c-, they are both starting on the same color square. This leaves four squares for one Bishop and two squares for the other. Since 4 x 2 x 4 = 32, there are fewer chess960 positions than for the Knights' a- and b- configuration. The same is true for all start patterns with the Knights on the same color.

I thought I had already seen all of the start position anomalies, but this was a new one for me. I wonder if there are any others that I've overlooked.


GeneM said...

This seems related to the fact that, in chess960 FRC, --- the "BNQ" sequence of placing pieces when randomly (with one dice) generating a start setup is correct; and the "NBQ" sequence is wrong. (Page 65 in 'Play Stronger Chess by Examining Chess960').

The NBQ sequence warps the proper percentage chance that should be even for each setup.

Mark Weeks said...

Yes, there's a connection. A similar observation applies to the RKR patterns. When those three pieces all start on the same color, there are fewer SPs than when one of them is on a different color. - Mark