Rules of the game:
The 15-hole version of the game consists of 15 holes (see diagram
below) and 14 pegs. To start the game, a player places 14 pegs in the
holes and leaves the 15th hole (player’s choice) empty. (The game
may also be played by simply drawing the diagram on a piece of paper
and then using any 14 markers as the pegs.) Then, moves are made by
taking any peg, jumping over another peg and landing in an empty hole.
The moves may be in any direction, must be in a straight line, and each
jumped over peg is removed from the board.
As an example of a legal move, assume the starting position has a hole
at position 1, and all other holes are filled. The 2 possible legal
moves are: 4 over 2 to 1 (and remove the peg at 2), or 6 over 3 to 1
(and remove the peg at 3).
If a player can make 13 moves (leaving 1 peg on the board), the player
wins.
1
/ \
2 - 3
/ \ / \
4 - 5 - 6
/ \ / \ / \
7 - 8 - 9 -10
/ \ / \ / \ / \
11 -12 -13
-14 -15
A few questions can be raised at this point:
How many ways are there of winning?
What differences are there for different starting holes?
Can you leave the final peg at the original hole position?
Can you leave the final peg in any of the center holes?
Can you win if the initial hole uses one of the center positions?
What is the shortest possible game (no more legal moves)?
The following table will answer most of these questions. Then we will
give a few observations.
Peg Solitaire Solutions
Computer Program by Bill Butler
Last
Peg Shortest Total
Initial
Total
Center At
Initial Game Possible
Hole At Solutions
Solutions Hole Sols. Nbr. Moves
Games
------------------------------------------------------------------------
1
29,760
0
6,816
6
598,390
2
14,880
0
720
6
309,423
3
14,880
0
720
6
309,423
4
85,258 1,550 51,452 7 1,234,826
5 1,550 0
0
4
139,396
6
85,258 1,550 51,452 7 1,234,826
7
14,880
0
720
6
309,423
8
1,550
0
0
4
139,396
9
1,550
0
0
4
139,396
10
14,880
0
720
6
309,423
11
29,760
0
6,816
6
598,390
12
14,880 0
720
6
309,423
13
85,258
1,550
51,452
7 1,234,826
14
14,880
0
720
6 309,423
15
29,760
0
6,816
6 598,390
------------------------------------------------------------------------
Totals
438,984
4,650
179,124
7,335,390
Observations with the initial hole at “1”.
(11 and 15 are similar.)
If the initial hole is at position “1”, it is impossible to
leave the final peg in one of the center holes. However, there are
still 29,760 ways to win, and 6,816 of these leave the final peg in the
initial hole position. An example (using the above board number system)
would be:
6->1, 13->6, 15->13, 12->14, 10->3, 4->13, 14->12,
11->13, 3->8, 1->4, 7->2, 13->4, 4->1
The shortest possible game hits a dead end after 6 moves. There are two
ways of doing this. (The other is a mirror image of the following)
6->1, 13->6, 7->9, 10->8, 4->13, 1->4
Observations with the initial hole at “2”.
(3, 7, 10, 12, and 14 are similar.)
If the initial hole is at position “2”, it is again
impossible to leave the final peg in one of the center holes. However,
there are still 14,880 ways to win, but only 720 of these leave the
final peg in the initial hole position. An example would be:
7->2, 13->4, 15->13, 12->14, 6->13, 14->12,
11->13, 2->7, 1->6, 10->3, 3->8, 13->4, 7->2
The shortest possible game hits a dead end after 6 moves. The only way
this can be done is:
7->2, 6->4, 14->5, 2->9, 13->6, 11->13
Observations with the initial hole at “4”.
(6 and 13 are similar.)
There are 85,258 different ways to win if the original hole is at
position “4”. 1,550 of these leave the peg in center hole
position “9”. (There are no solutions for the other center
holes.) An example would be:
11->4, 2->7, 6->4, 1->6, 7->2, 10->3, 13->4,
2->7, 15->13, 12->14, 14->5, 3->8, 7->9
There are also 51,452 different ways the final peg can be left in the
initial hole position. An example would be:
13->4 15->13, 12->14, 10->8, 7->9, 6->13, 14->12,
11->13, 3->8 2->7, 13->4, 7->2, 1->4
The shortest possible game uses 7 moves. There are many combinations.
Observations with the initial hole at “5”.
(8 and 9) are similar.)
There are 1,550 different solutions, but all of these end with the
final peg at position 13. A typical solution would be:
14->5, 12->14, 7->9, 10->8, 3->10, 15->6, 2->7,
6->4, 7->2, 1->4, 4->13, 14->12, 11->13
(Note that any game that starts with a hole in the center is a reverse
image of a game that leaves the final peg in the center.)
The shortest possible game hits a dead end in only 4 moves. This can be
done as follows (also the mirror image.):
14->5, 2->9, 12->5, 9->2
Return to Durango Bill's Home page
Web page generated via
KompoZer