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Old 12-13-2010, 03:14 AM   #1
masque de Z
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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?
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Old 12-13-2010, 07:14 AM   #2
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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.
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Old 12-13-2010, 04:36 PM   #3
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Re: Total number of possible distinct backgammon boards

Quote:
Originally Posted by masque de Z View Post
What is the total number of possible legal backgammon board positions ?
18,528,584,051,601,162,496
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Old 12-13-2010, 05:14 PM   #4
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Talking 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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