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Another backgammon puzzle: more serious effort

Posted By: David Levy
Date: Friday, 1 March 2013, at 4:12 p.m.

In Response To: Another backgammon puzzle (Jason Lee)

Let G bet the set of all possible backgammon games. Assume it is finite--this will lead to a contradiction, negating the assumption.

G contains games G1...Gn where n is the (finite) number of games in the set. For each Gx, define a function f(Gx) = the number of times the opening position occurs in game Gx. [Aside: we know that the opening position can recur, see Kershaw's Backgammon Funfair]. Over the set G, f(Gx) can take on at most n different values, so there is some number, say N, between 1 and n+1 that did not occur.

It is easy to construct a backgammon game where the opening position recurs N times. That game could not have been in set G, the set of all backgammon games, a contradiction. Therefore the assumption that G is finite fails.

If Jason disagrees, I'm cheating with my set theory somehow...

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