Why 7n+1 ?

by Walter Trice
15 January 2009

Walter Trice

A stack of 15 checkers on an ace point will be borne off in 6.9795917638 rolls, on average, which is about as useless a piece of numerical information as any I can imagine. But if you multiply that ungainly quantity by 8.1666666667, which is the average number of pips in one roll of the dice in backgammon, you get 56.99999939. For all practical purposes that is 57. Similarly the values for mean number of pips rolled while bearing off are close to simple integers for all ace-point stack positions. The general formula, with n pairs of checkers to bear off, is 7n + 1. Thus the sequence, increasing from a single pair of checkers, goes 8, 15, 22, 29, 36, 43, 50, 57.
 

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