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Most legal plays

Posted By: Ian Shaw
Date: Friday, 14 January 2011, at 4:18 p.m.

In Response To: Most legal plays (Timothy Chow)

As Timothy notes, you can find this with gnubg. To find this many moves using gnubg, the default move limit must be increased with the command "set analysis limit 3000" or the Settings, Analysis, Move Limit dialogue box. Make sure you've selected only 0-ply!

When I had a crack at calculating this a while ago, I got bogged down trying to calculate it for doubles.

For non-doubles, I got: There are three different types of move.

  1. one chequer twice
  2. two chequers from different points
  3. both dice from different chequers on the same point.

case 1:

If there are men on n points, then the number of possible moves is 2n. Each man may be moved with the sequence d1d2 or d2d1.

case 2:

There are nP2 permutations of moves, which is n(n-1).

Cases 1 and 2 represent all the cases that move at most one man from each point. Combining cases 1 and 2 we have the number of possible moves = 2n + n(n-1) = n2 + n = (n+1)n = n+1P2.

case 3:

For each point, p, with more than one chequer you can play both dice with different chequers.

The maximum number of moves with two dice from n occupied points including p made points is therefore n2 + n + p = (n+1)n + p.

With 15 chequers the theoretical limit is 240 moves, but because there are only 24 points available the practical limit seems to be 224.

Board Diagram

Gnubg Position ID for 11: REREpJBSSimlAA:cInkAAAAAAAA

Gnubg Position ID for 21: REREpJBSSimlAA:cAngAAAAAAAA

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