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BGonline.org Forums
Are there more 4th roll positions than 3rd roll positions?
Posted By: Paul Weaver In Response To: Announcing a new software "3rd roll master" (Mochy)
Date: Sunday, 12 June 2016, at 5:25 a.m.
Are there more 4th roll positions than 3rd roll positions? It is trivial to see that the answer is yes. Is this true in general, that for all positive integers n, there are more (n+1)th roll positions than nth roll positions? The answer is NO!
Let’s back up and define what is meant by an nth roll position. If a position comes up on the nth roll but not any earlier than the nth roll, then it is called an nth roll position.
We know that the number of backgammon positions is finite. Suppose that for all n, there are more (n+1)th roll positions than nth roll positions. Then there would be no end to it all, ie, there would be an infinite number of positions. But we know the number of positions is finite, so this tells us that the supposition is false, and therefore there must be an integer n for which there are more nth roll positions than (n+1)th roll positions.
Here is the Weaver unsolved problem: Find the smallest integer N for which there are more Nth roll positions than (N+1)th roll positions. (Incidentally, here is a problem for Nack or anyone else: Find the backgammon position(s) that take(s) the greatest number of rolls to reach.)
I am offering a taxable cash prize of US $5 (please do not forget to report this to the IRS) to the first person who determines this value of N. If you are European, I offer a cash prize of five euros, provided you travel to the USA to collect. A Brit would get 5 quid. Hope no one changes citizenship over this.
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