A variant on the "classic" peg solitaire layout described by
wangmu is a fifteen-peg game commonly found at the Cracker
Barrel with which the idea is to jump pegs and end up with one
left. The pegs are arranged in a triangular fashion:
*
* *
* * *
* * * *
* * * * *
To begin, the player removes any peg from the board. The game
proceeds in the same manner as the classic game by jumping
pegs and removing the jumped pegs. The goal is, of course, to
finish with one peg.
Here's the instructions on the Cracker Barrel game:
"Jump All But One Games"
Jump each tee and remove it
Leave only one -- you're genius
Leave two and you're purty smart
Leave three and you're just plain dumb
Leave four or mor'n you're just plain
"EG-NO-RA-MOOSE"
Cracker Barrel Old Country Store
Being the hacker type, I wrote a little Perl script to
analyze the possible ways to jump the pegs so that one peg
remains. To simplify my description, I'll label the points the
board as follows:
A
B C
D E F
G H I J
K L M N O
Now, accounting for symmetry and rotation of the board, there
are four logically different starting points — A, B, D, and
E. K and O are the same as A; C, G, J, L, and N are all the same
as B; F and M are equivalent to D; finally, H and I can be
translated to E.
According to my script (which, I should add, was slapped
together and not extensively checked, so the resulting numbers
might be inaccurate) the number of different winning combinations
and the possibilities for the last position are largely
dependent on what peg is taken out first.
Peg | Wins | Final position
----+-------+---------------
D | 85258 | C, D, I, L, O
A | 29760 | A, M, F, J
B | 14880 | B, K, N, F
E | 1550 | M
(Note: these are the "raw" numbers, not accounting for
symmetry/rotation.)
Therefore, the best bet for winning the game is to start by
removing peg D (or, if you like, F or M). I've noticed many
people start with E/H/I, which is perhaps why they always end up
with four or five pegs left.
As an interesting (?) side note, it does not
appear to be possible to jump the pegs in any way that results in
the mythical eight-peg endgame. (I read somewhere that it was
possible but that there were only a few ways to do it, and now
I'm quite sure that the person who wrote that was just making shit up.)
If you'd like to impress your friends with your amazing ability
to move little pieces of plastic, but don't want to spend
several hours figuring out how to get it right, here's a few
ways to end up with one peg, and even in the same position
the first peg was taken from:
-
Remove A, D-A, F-D, G-B, A-D,
O-F, L-E, N-L, K-M, C-J, M-F, J-C, D-F, F-A
-
Remove B, G-B, M-D, B-G, C-H,
J-C, A-F, K-M, N-L, F-M, L-N, O-M, M-D, G-B
-
Remove D, M-D, K-M, N-L, J-H,
C-J, O-F, D-M, A-D, G-B, B-I, L-N, N-E, F-D
Cracker Barrel distributes the game with a cheat sheet
describing another solution, though it doesn't leave the last peg
in the starting position:
Remove A, D-B, F-D, A-F, G-B,
L-D, I-H, B-G, G-I, O-L, K-N, F-L, N-K, J-L