Peg solitaire

Peg solitaire is a solo board game involving the movement of pegs or marbles across a board with holes.

Players aim to clear the board through strategic jumps. In the standard version, the board starts full except for one central hole. The ultimate objective is to make valid moves until only one solitary peg remains in that center hole. This game is known as solitaire in Britain, while Americans often call it peg solitaire to distinguish it from patience.

To play, you must follow a specific jumping mechanism. A valid move involves jumping a peg orthogonally, which means in a straight line, over an adjacent peg. The moving peg must land in an empty hole located two positions away. Once the jump is complete, you remove the jumped peg from the board. This process reduces the number of pegs with every turn. Some advanced players use a tactic called a catalyst. This is a single extra peg used to purge entire groups of three pegs at once.

There are two traditional board shapes used in the game: the English board and the European board. The English board is a common variant where the central hole is the starting point. The European board offers different challenges and different sets of alternative games. On a European board, players can start with a hole at one position and end with a peg in its mirrored position. Interestingly, if only orthogonal moves are permitted, there is no solution for the European board if the initial hole is centrally located.

History shows that the game has been cherished for centuries. The earliest evidence traces back to the court of Louis XIV in France. An engraving from 1697 by Claude Auguste Berey depicts Anne de Rohan-Chabot, the Princess of Soubise, with a puzzle by her side.

The August 1697 edition of the French magazine *Mercure galant* provided the first known written description of the rules and sample problems. This shows the game was a recognized pastime in high society long ago.

Mathematical analysis has revealed deep complexity within these simple moves. Researchers use a tool called a pagoda function to prove if certain problems are impossible to solve. In 1990, studies addressed generalized Hi-Q problems, which are equivalent to peg solitaire, showing they are NP-complete. By 1999, computers completely solved the game using exhaustive searches and symmetry. These digital studies show that only about 2.2% of all possible board positions can actually be reached starting from a center vacancy.

Specific records exist for the shortest possible solutions. For the standard English game, the shortest solution requires exactly 18 moves. Ernest Bergholt discovered this sequence in 1912. Later, in 1964, John Beasley provided the mathematical proof that it was the shortest possible way to finish.

On an English board, the final peg can only end up in specific spots that multiples of three permit. This mathematical constraint limits where a player can successfully finish a game.

Peg solitaire connects to several advanced fields of study. It is used to explore combinatorial optimization, which is the study of finding the best solution from many possibilities. In 1996, researchers discussed the properties of a "solitaire cone," which is a type of feasible region in math. The game also relates to abstract algebra, which helps prove there are only five fixed board positions where a game can successfully end with one peg. Whether played for fun or studied by scientists, the game remains a powerful model for logic and patterns.