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 12-13-2010, 03:14 AM #1 masque de Z Carpal \'Tunnel     Join Date: Aug 2009 Location: Stanford, CA USA Posts: 9,294 Total number of possible distinct backgammon boards What is the total number of possible legal backgammon board positions (any configurations of checkers including any borne off ones or checkers on the bar etc) that could ever exist ? Has this been calculated for fun before using combinatorics and checked for plausibility individually?
 12-13-2010, 07:14 AM #2 masque de Z Carpal \'Tunnel     Join Date: Aug 2009 Location: Stanford, CA USA Posts: 9,294 Re: Total number of possible distinct backgammon boards The combinatorics problem initially seems as equal to; Each of the white checkers can occupy anyone of the 24+2 bins (24 board +2 the having being hit bin for whites and bearing off bin for whites) leaving n empty bins in the board which is ranging from 24-15=9 to a full 24. Then the black ones have to be placed in the n empty bins plus the 2 outside ones (the black bearing off bin and the black having been hit bin). Clearly a product of ways for white and of ways for black for each n then sum over all possible n has to be performed. But then we must subtract a few cases that are simply impossible to appear such as both whites and blacks in bearing off bin or both whites and blacks in the bar bins. Agree? May get to it later or someone else could. It will be a huge number for sure.
12-13-2010, 04:36 PM   #3
plm
newbie

Join Date: Sep 2009
Posts: 39
Re: Total number of possible distinct backgammon boards

Quote:
 Originally Posted by masque de Z What is the total number of possible legal backgammon board positions ?
18,528,584,051,601,162,496

 12-13-2010, 05:14 PM #4 Taper_Mike adept     Join Date: Sep 2009 Posts: 1,030 Re: Total number of possible distinct backgammon boards When you do decide to "do the cominatorics," here is an academic paper I both learned from and enjoyed: Enumerating Backgammon Positions: The Perfect Hash By Arthur Benjamin and Andrew M. Ross http://ccdl.libraries.claremont.edu/...OPTR=312&REC=3

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