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Another backgammon puzzle

Posted By: Timothy Chow
Date: Saturday, 2 March 2013, at 1:06 a.m.

In Response To: Another backgammon puzzle (Chuck Bower)

I'm struggling to find a variant of Chuck's question that isn't trivial.

Here's one candidate. Say that two backgammon games A and B are "the same" if there exists some number n such that the board position and dice in Game A on the nth roll are the same as the board position and dice in Game B on the nth roll. Are there infinitely many distinct backgammon games in this sense?

This rules out simple constructions like iterating a closed loop. And because n has to be the same in both Game A and Game B, it's fallacious to argue that since there are only finitely many distinct board/dice positions, there can only be finitely many distinct games.

This modified question isn't that difficult to answer either, but it's less trivial, and may be closer to what Chuck was thinking when he first posed the question.

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