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Fun Fact Question

Posted By: Max Stockslager
Date: Thursday, 2 May 2024, at 1:55 p.m.

In Response To: Fun Fact Question (ah_clem)

Here is my solution. (Disclaimer: I did not check this). I found it useful to distinguish between “2-checker plays” (where we move two separate checkers) and “1-checker plays” (where we move the same checker twice with both halves of our roll).

From the starting points (24/13/8/6), there are 4 legal ways to play 4s, 3s, and 2s (i.e., these numbers can be played from any of the starting points) but only 3 legal ways to play 6s, 5s, and 1s (since these can’t be played from the 6/24/13 respectively). Therefore, there are 9, 12, or 16 legal 2-checker plays for each opening roll (depending whether each of the two rolled numbers can be played 3 or 4 ways).

Since any opening roll can be played at least 9 legal ways, we can immediately rule out any number with pip total less than 9: this leaves 54 63 64 and 65 as candidates.

We can also rule out 54 and 63, because in addition to the minimum 9 legal 2-checker plays, these have at least one legal 1-checker play (e.g., 24/15 for both), so there are more than 9 legal plays, exceeding the 9-pip total.

We can also rule out 64, because the 6 can be played from three points (24, 13, and 8) and the 4 can be played from any of the four starting points, immediately giving us 3x4 (>10) legal 2-checker plays.

This just leaves 65 to check, which has 3x3 = 9 2-checker plays and 2 1-checker plays (24/13 and 13/2) for a total of 11 legal plays.

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