---------- | From: Robert M. Solovay <netmail!solovay@math.berkeley.edu> | To: Mike Markley | Cc: <cypherpunks@toad.com> | Subject: How Many Games of Chess? | Date: Friday, April 01, 1994 11:06AM | | Received: from math.Berkeley.EDU by netmail.microsoft.com with SMTP (5.65/25-eef) | id AA02131; Fri, 1 Apr 94 11:04:58 -0800 | Received: by math.berkeley.edu (8.6.8/1.33(math)Ow) | id LAA28894; Fri, 1 Apr 1994 11:06:45 -0800 | Message-Id: <199404011906.LAA28894@math.berkeley.edu> | In-Reply-To: Mike Markley's message of Fri, 1 Apr 94 10:20:55 | TZ <9404011831.AA05066@netmail2.microsoft.com> | | | mmarkley@microsoft.com writes: | | I seem to remember from way back in high school that the number of | potential moves by the third set of moves is on the order of billions | of legal moves. | | The number of moves in a given chess position is less than 64 | (number of starting squares) times 64 (number of destination squares) | x 4 [number of ways a pawn can promote]. Thus we get the bound 16, 384 | [which can be easily improved] which is way less than "billions of | possible moves". The same computation shows that the number of | possible games of length n grows at worst expoentially pace mr | markley. | | The right way to think about this is to get sharp upper bounds rather | than attempt a precise calculation. A crude upper bound would be | longerst possible game is about 6000 moves [using the 50 move rule]. | At most 2**16 mves per position so at most 10**[192 * 10**6] games. | I'm sure that sharper estimates are readily available. | | I should have said billions of potential states for the board after each move. If you think of the number of initial moves for the starting player its only 16 potential positions for the pawns and 4 for the knights. If the first player takes one of these positions then the second player moves he has the same 20 potential moves giving an potential state of 400 different positions after the first move. After the second move there is on the order of greater than 160,000 potential states for the board. After 3 moves it is greater than 2.56 * 10^10 potential states. I was thinking in terms of states rather than valid moves. Mike. -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Mike Markley || The opinions here do not represent the mmarkley@microsoft.com || opinions of my employer. Attempts to || associate the two are pointless. "I want to look at life, In the available light" - Neil Peart -