Author: Marc Bourzutschky
Date: 13:34:25 05/16/04
Go up one level in this thread
On May 16, 2004 at 16:14:15, Telmo C. Escobar wrote: >On May 16, 2004 at 14:28:48, Marc Bourzutschky wrote: > >>In a book review at www.chesscafe.com there is a discussion of an older version >>of Fischer Random Chess, called pre-chess. >> >>The main difference between FRC and pre-chess is that in pre-chess black and >>white can have different piece arrangements on the first row. The only >>constraints are that each side has to have opposite colored bishops, and that >>castling is only allowed like in classical chess (i.e., white can only castle if >>the king is on e1, and there is a rook on a1 or h1, etc). However, it is not >>required that the starting position must allow castling. >> >>How many game theoretically different positions are there in pre-chess? I know >>of 4 different suggestions, 2 by famous and 2 by less famous chess enthusiasts: >> >>Max Euwe: 4,147,200 >>Noam Elkies: 8,294,400 >>Paul Epstein: 5,317,600 >>Marc Bourzutschky: 5,149,368 >> >>Which one is correct, or all they all wrong? >> >>-Marc > > > This is a trivial exercise in elementary combinatorics. The natural to solve >it, I think, is the following: > > as the pawns are in the second (and seventh) file, the problem is just how many >ways there are to place the eight White pieces. Let us name this number as "W". >As the Black pieces are to be placed independently, the final solution will be W >squared (by comparison, in Fischerandom the solution will be just W). > > Let us speculate about W: > > as both bishops, poor thinks, have to be on different colored squares, there >are 4x4=16 ways to place them. On the remaining 6 squares, there are 6x5 ways to >place king + queen. Finally, in the four remaining squares, we must compute how >many ways to place two rooks and two knights. Obviously it suffices to compute >the rooks, and this is how many ways to select two squares, and this is six. > > In short, W=16 x 30 x 6= 16 x 6 x 30= 96 x 30= (100 - 4) x 30= 3000- 120= 2880. > > The final solution, W squared, is (3000 - 120) squared, in other words 9000000 >+ 14400 - (6000 x 120) = 9000000 + 14400 - 720000 = 9000000 - 7000000 - 20000 + >14400 = 8300000 - 20000 + 14400 = 8280000 + 14400 = 8294400. I don't have a >calculating machine at hand, so I had to make the calculations the human way, as >easy as possible. > > So that guy Noam Elkies was right, the other three were wrong. > > Max Euwe was a world chess champion (not significant information) and also he >held a PH.D. in math (very significant). So it's astonishing he was wrong about >this trivial problem. The other three I didn't know about. > > The calculation took less than one minute and one just have to know a minimal >high school algebra to solve it. It will make a useful exercise for teenage >students. > > Telmo Your calculation ignores the fact that all chess moves except castling are mirror symmetric, so you will double count some game theoretically equivalent positions. For example, suppose both sides start with a first row of: KQRRBBNN This is game theoretically equivalent to both sides starting with NNBBRRQK The tricky part (missed by Euwe) is that starting position such as RNNKBBRQ are not equivalent with the reflected position QRBBKNNR because in the latter position there is the possibility of short castling.
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