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Third attempt

Posted By: Tom Keith
Date: Sunday, 3 March 2013, at 5:46 p.m.

In Response To: Third attempt (Timothy Chow)

Say that two backgammon games are the same if they eventually agree, i.e., if after some finite number of rolls, the game positions coincide, and everything stays exactly the same after that. Are there infinitely many distinct backgammon games in this sense?

One way to do it would be to show that it is possible to create an infinite number of different repeating patterns. For example:

1111111111...
1010101010101010...
100100100100100100100100...
10001000100010001000100010001000...
(etc., each time adding an extra 0 between the 1's)

Relating this to the backgammon problem, you need to find two mutually reachable loops. Call one them "Loop Zero" and the other "Loop One". Then you can create an infinite number of different games using the pattern above. The first game continually loops around in Loop One (11111...). The second game alternates between between Loop One and Loop Zero (1010101...). The third game alternates between one iteration of Loop One and two iterations of Loop Zero (100100100...). And so on.

Now it is easy to see there are an infinite number of different games possible.

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