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Subject: Re: turtle peg game
From: mwforum@xxxxxxxxxxxxxxxxxx (Mike Sandy)
Date: Sat, 19 Apr 2003 19:50:34 +0100

J notes in his peg-game that with starting position
10 it is more difficult to find a solution. This seemed
contradictory as 10 can be reached in 4 ways,
from 1, 3, 12 and 15, (unless he was being devious!).

The game-tree is short enough to be searched, even by
MW Pro in a "reasonable" time (20 mins).

There are only 4 different starting positions 1, 2, 3, and 7.
All others being equivalent by symmetry to one of these.
Results:

sp=starting position, g=number of games with one peg left.

peg.triang
(sp 1)
(g 29760 total 568630  5.23%)
(sp 2)
(g 14880 total 294543    5.05%)
(sp 3)
(g 85258 total 1149568  7.42%)
(sp 7)
(g 1550 total 137846  1.12%)

Thus 10(= 3) is the easiest sp.

A more difficult game is to end up in the sp, as in the
standard 33-hole solitaire.

Last peg in sp
(sp 1)
(g 6816 total 568630   1.20%)
(sp 2)
(g 720 total 294543   0.24%)
(sp 3)
(g 51452 total 1149568   4.48%)
(sp 7)
(g 0 total 137846  0.0%)

If you want to cheat, these figures are confirmed at
www.durangobill.com/PegSolitaire.html
and he gives some solutions, though not to games ending at sp.

I attach a 15-peg game based on two pentagrams,
but of less elegant design than J's.
The labelling (starting with 1) is from top to bottom, 
left to right, so the outer pentagram is (1 5 15 14 12)
the inner one (3 4 12 13 9).
Analysis:
peg.pent
(sp 1)
(g 895804 total 7254656   12.35%)
(sp 3)
(g 987872 total 8102512   12.19%)
(sp 6)
(g 188366 total 6023988   31.27%)

Mike

Attachment: peg_game(pent).mw2
Description: Binary data


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