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For a great in-depth scientific notation on peg solitaire puzzles -- and history, one can view my good friend, George Bell's website  here . . . . . with many links to other internet areas which help supplement his wealth of information .

The following quote is taken from the book Creative Puzzles Of The World . . . . .
by authors Pieter van Delft and Jack Botermans:

"According to one old story, the game of peg solitaire often called simply solitaire, was invented in the 18th century by a French nobleman imprisoned in the Bastille, the grim fortress-prison in Paris.   He was modifying an already existing game called Fox and Geese.   His new invention, like the earlier game, used a board in which an array of holes -- called cells -- were bored as resting places for pegs or marbles.
There are many games and puzzles that can be devised for the solitaire board, but the method of making moves is common to all of them.

A peg (assuming that this is the marker used) can be moved only by jumping it over a neighboring peg to a vacant space directly on the other side.   Following such a move, the peg over which the jump was made is removed from the board.   Jumps can be made only along the lattice lines (as shown in the diagram); a peg cannot jump diagonally."

There are two kinds of boards used for peg solitaire.
One has 37 cells (top left diagram).
Another eliminates the corner cells, forming a cross-shaped pattern of cells (top right diagram).

"In the most widely known solitaire puzzle the 33-cell board is used.   All cells are filled except the one in the center.  The player is required to finish the game with a single peg in the central cell.

The number of jumps made in a game of solitaire equals the number of pegs removed.   However, a series of consecutive jumps made at one time with a single piece can be regarded as a single move; hence a player can aim not merely at solving a given puzzle, but also at finding the solution that requires the smallest number of moves.

The basic solitaire puzzle on a 33-cell board -- to begin with a single vacancy in the center and to finish with a single peg in the center -- requires 32 jumps, which can be grouped into 18 moves."

~ thus leading us to the following solution ~