Peg Solitaire

Why you finish with three stranded pegs

Nearly every failed game of peg solitaire dies the same death. Somewhere in the middlegame a region gets emptied, a peg or two are left behind on an outer arm of the cross, and twenty moves later they are still sitting there: nothing beside them to jump, nothing left to jump them. The board starts with thirty-two pegs and one empty hole in the centre, and since every jump removes exactly one peg, a perfect game is exactly thirty-one jumps. Not one of them can be spent flinging a peg somewhere lonely.

There are only four possible first moves, all mirror images of each other: a peg two holes from the centre jumps into the middle. Games diverge from the second move on, and the guiding principle from there is to treat the pegs as one crowd. Clear the arms of the cross inward, keep survivors within reach of one another, and when two jumps are both legal, prefer the one that lands in traffic.

Purges: the patterns worth learning

Serious solvers don't improvise thirty-one jumps; they erase the board in rehearsed chunks. The classic books on the game, John Beasley's The Ins and Outs of Peg Solitaire and the solitaire chapter of Winning Ways by Berlekamp, Conway and Guy, call these chunks packages and purges. A three-purge wipes out a row of three pegs, a six-purge clears a 2x3 block, and the L-purge handles the awkward corner shapes. Each one relies on a catalyst: a neighbouring peg that does the eating and finishes back on the square where it started, so the block vanishes without disturbing anything else on the board.

You don't need the whole catalogue to improve. Just start reading the board as blocks to be erased rather than pegs to be nibbled at random, and pick out which neighbour can serve as the eater for each block. Chain jumps are the workhorse: one peg hopping three, four, five times in a single move is how a purge clears a region in one stroke, and it is also the most satisfying thing this quiet little game lets you do.

The mathematics of getting stuck

Some finishes are not just hard but impossible, and the proof is tidy. Label the columns with a repeating three-letter pattern, do the same to the rows, and every legal jump preserves a hidden balance among the labels. The class of your final peg is therefore fixed before you make a single move, which is why no cleverness will ever reach certain endings. Beasley's book proves sharper limits with resource counts, numerical weights assigned to the holes that no jump can ever increase. The good news: the centre-start game on the English cross genuinely can finish with a single centre peg. It can even be done fast. In 1912 Ernest Bergholt published a solution in eighteen moves, counting an unbroken chain of jumps as one move, and in 1964 Beasley proved nothing shorter exists.

Habits that finish games

• Think in short chains: a single peg making several jumps in a row clears a cluster efficiently.
• Never empty a whole region early; isolated pegs on the outer arms are how games die.
• Play roughly symmetrically while you can, since balanced positions keep options open on both sides.
• Slow down once six pegs remain; one careless jump leaves two survivors that will never meet.
• For the centre finish, plan the last two or three jumps backward from a lone central peg.

A puzzle fit for a prisoner

Peg solitaire is old enough that in Europe the word solitaire meant this game long before it meant cards. Legend credits a French nobleman with inventing it while locked in the Bastille; charming, and unproven. What is documented is a 1697 engraving by Claude Auguste Berey showing the Princess de Soubise posed beside a solitaire board, which places the game at the court of Louis XIV at the latest. Gottfried Wilhelm Leibniz wrote about it admiringly in 1710 and confessed he preferred playing it in reverse, starting with an empty board and jumping pegs in to fill it, which he considered the more instructive direction.

Two classic boards survive. The English cross used here has 33 holes. The French board adds four more for 37, and its centre-start game is provably unsolvable down to one peg, which is why French rule books traditionally begin with the vacancy somewhere off-centre. Same pegs, same jump, one extra ring of holes, and what is possible quietly changes: this puzzle has always been mathematics wearing wooden clothes.