Durango Bill’s

Peg Solitaire

How many ways are there to win the 15-hole version of Peg Solitaire?

(Click here for 33-hole Peg Solitaire - “Hi-Q”)

Peg Solitaire

How many ways are there to win the 15-hole version of Peg Solitaire?

(Click here for 33-hole Peg Solitaire - “Hi-Q”)

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

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

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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

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