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”)

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




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