*Fischer Random Chess (or Chess960) is a chess variant created by former world champion Bobby Fischer. It randomizes the back rank pieces with two restrictions: the bishops must be on different colors and the king must be between the two rooks. This results in 960 possible positions, hence the alternate name of Chess960.*

Recently the St. Louis Chess Club hosted the Kings vs. Queens event, with a team of high rated women against a team of high rated men. It was an interesting event with non-standard Chess matches, including faster time controls and Fischer Random Chess. Mark Weeks over on Chess960 (FRC) posted a link to a podcast by the Full English Breakfast commentary on the event. There were apparently different methods used to choose the starting positions from the 960 possible positions. One of the methods used was to spin a roulette wheel with all of the pieces on it, and some “choice” results where the spinner can choose a piece. Each piece is chosen with a random spin and placed on the board in order from the a file to the h file, ignoring spins that would result in illegal positions. I’m not totally sure of the distribution of the pieces on the wheel, but assuming each piece is on the board the same number of times, and that the choices made by the spinner were also evenly distributed, the method doesn’t work.

Take the first spin, which is used to place the piece on a1. If I am right in my assumptions, each piece has the same chance of being on a1. The only exception is the king, which can’t be on a1 because it has to be between the two rooks. But if a king was the result of a spin, they would spin again until they got a non-king, making all the non-king pieces equally likely. The problem is that if you look at the legal FRC positions, there is not an even distribution of pieces on a1. The knight and the bishop appear on a1 in 240 position each (25% chance), but the queen is on a1 in only 120 positions (12.5% chance) and the rook is on a1 in 360 positions (37.5% chance). Therefore the method used at Kings vs. Queens was biased against rooks in the corner and biased for queens in the corner. As it turns out, the queen was on a1 twice at the event, in rounds one and three. I even remember some of the commentary saying what a problem this was, because it made developing the queen difficult. Looking up the binomial probability formula (dammit, I used to have these things memorized), we can see that a correct starting position generator would have led to that situation (queens on a1 two out of five games) 12.0% of the time, while the method used by Kings and Queens would have led to it 35.2% of the time.