Peg solitaire

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Peg solitaire is a board game for one player involving movement of pegs on a board with holes. Some sets use marbles in a board with indentations. The game is known simply as Solitaire in the United Kingdom where the card games are called Patience. It is also referred to as Brainvita (especially in India).

According to a popular story, the game was invented by a French aristocrat in the 17th century, when incarcerated in the Bastille, explaining the game's less common name Solo Noble. John Beasley (author of "The Ins and Outs of Peg Solitaire") has extensively searched for evidence to support this, and has found it lacking. The first reference to this story appeared in 1810, more than a hundred years after the alleged event. He believes that the colorful tale is fiction, yet it persists. In other sources, the invention of the game is attributed to the Native Americans—there is also no evidence to support this.

The first evidence of the game can be traced back to the court of Louis XIV, and the specific date of 1697, with an engraving made that year by Claude Auguste Berey of Anne de Rohan-Chabot, Princess of Soubise, with the puzzle by her side. Several works of art from that time show peg solitaire boards, demonstrating that the game was highly fashionable.

The standard game fills the entire board with pegs except for the central hole. The objective is, making valid moves, to empty the entire board except for a solitary peg in the central hole.



There are two traditional boards, graphically depicted as follows ('.' as an initial peg, 'o' as an initial hole):

   English          European
    · · ·             · · ·
    · · ·           · · · · ·
· · · · · · ·     · · · · · · ·
· · · o · · ·     · · · o · · ·
· · · · · · ·     · · · · · · ·
    · · ·           · · · · ·
    · · ·             · · ·

[edit] Play

A valid move is to jump a peg orthogonally over an adjacent peg into a hole two positions away and then to remove the jumped peg.

In the diagrams which follow, * indicates a peg in a hole, the peg to be moved * emboldened, and o indicates an empty hole. A green ¤ is the hole the current peg moved from; a red * is the final position of that peg, a red o is the hole of the peg that was jumped and removed.

Thus valid moves in each of the four orthogonal directions are:

* * o  →  ¤ o *)  Jump to right
o * ** o ¤  Jump to left 
*     ¤
*  →  o  Jump down
o     *
o     *
*  →  o  Jump up
*     ¤

On an English board, the first three moves might be:

    * * *             * * *             * * *             * * * 
    * * *             * ¤ *             * o *             * * * 
* * * * * * *     * * * o * * *     * ¤ o * * * *     * o o o * * *
* * * o * * *     * * * * * * *     * * * * * * *     * * * ¤ * * *
* * * * * * *     * * * * * * *     * * * * * * *     * * * * * * *
    * * *             * * *             * * *             * * *
    * * *             * * *             * * *             * * *

[edit] Strategy

It is very easy to go wrong and find you have two or three widely spaced lone pegs. Many people never manage to solve the problem.

There are many different solutions to the standard problem, and one notation used to describe them assigns letters to the holes:

   English          European
    a b c             a b c
    d e f           y d e f z
g h i j k l m     g h i j k l m
n o p x P O N     n o p x P O N
M L K J I H G     M L K J I H G
    F E D           Z F E D Y
    C B A             C B A

This mirror image notation is used, amongst other reasons, since on the European board, one set of alternative games is to start with a hole at some position and to end with a single peg in its mirrored position. On the English board the equivalent alternative games are to start with a hole and end with a peg at the same position.

There is no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted. This is easily seen as follows, by an argument from Hans Zantema. Divide the positions of the board into A, B and C positions as follows:

    A B C
  A B C A B
  B C A B C
    A B C

Initially with only the central position free, the number of covered A positions is 12, the number of covered B positions is 12, and also the number of covered C positions is 12. After every move the number of covered A positions increases or decreases by one, and the same for the number of covered B positions and the number of covered C positions. Hence after an even number of moves all these three numbers are even, and after an odd number of moves all these three numbers are odd. Hence a final position with only one peg can not be reached: then one of these numbers is one (the position of the peg, one is odd), while the other two numbers are zero, hence even.

There are, however, several other configurations where a single initial hole can be reduced to a single peg.

A tactic that can be used is to divide the board into packages of three and to purge (remove) them entirely using one extra peg, the catalyst, that jumps out and then jumps back again. In the example below, the * is the catalyst.:

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