Peg Solitaire: 18 Moves and One Impossible Board

Quick answer: peg solitaire is a board of holes filled with pegs, where you jump one peg over another and remove the one jumped. On the 33-hole English board the classic game can be solved in as few as 18 moves. On the 37-hole European board the same game is impossible, and that is a proof rather than an opinion.

Best for: anyone who likes a puzzle where the interesting part is which positions you can never reach.

Peg solitaire is the puzzle sitting on pub tables and in gift shops, usually in wood with marbles, usually with nobody able to finish it.

It is also one of the most thoroughly analysed objects in recreational mathematics, and the analysis produces a result most players never hear: one of the two standard boards cannot be solved at all.

How Does Peg Solitaire Work?

Every hole is filled with a peg except one. A legal move jumps a peg over an adjacent peg into the empty hole directly beyond it, and the peg that was jumped is removed.

Each move therefore removes exactly one peg. Start with 32 and the best possible finish is one peg remaining, which takes 31 removals. The classic challenge, usually called the central game, is to begin with the centre hole empty and finish with a single peg standing in it.

Moves are orthogonal: up, down, left, right. Diagonal variants exist and change the answers completely.

Where Did Peg Solitaire Come From?

The French court, and the record is unusually precise for a puzzle this old.

The first known reference in print is in the August 1697 edition of the French literary magazine Mercure galant, which describes the board, the rules and some sample problems. Roughly a decade later Claude Auguste Berey engraved a portrait of Anne de Rohan-Chabot, Princess of Soubise, with the puzzle beside her.

So this was a fashionable amusement at the court of Louis XIV, documented in a society magazine and depicted in a formal portrait. That is a very different origin from the workshop and the lock, which is where most of the puzzles in this field came from.

What Is the Shortest Solution?

Eighteen moves on the English board.

That is fewer than it sounds, because a single move can be a chain: one peg jumping several times in succession counts as one move provided it keeps going. Finding a solution at all is the first problem; finding one that uses long sweeping chains instead of 31 separate hops is the real one.

Ernest Bergholt produced an 18-move solution in 1912, and it was proved in 1964 that no shorter solution exists. Half a century between the achievement and the proof that it could not be beaten.

Why Is the European Board Impossible?

This is the result worth knowing, and it is the same shape of argument as the one that makes half of all 15 puzzle positions unsolvable.

Colour the board's holes in a repeating pattern and you can assign every possible position to one of 16 classes. A legal move never changes which class you are in. It cannot: the arithmetic of jumping preserves the property.

On the 37-hole European board, the starting position of the central game and the desired finishing position fall into different classes. There is therefore no sequence of moves connecting them, however long, however clever. The game is unsolvable with orthogonal moves, and the proof requires no search at all.

It is a beautiful piece of reasoning and a slightly cruel one. Two boards that look nearly identical, four extra holes, and one of them quietly cannot be finished.

What Are the Two Boards?

Worth being able to tell apart on sight, since one of them cannot be finished.

The English board has 33 holes in a cross: a three-by-three centre with three-by-two arms on each side. This is the one sold almost everywhere in the English-speaking world, and the central game on it is solvable.

The European board, sometimes called the French board, has 37 holes. It is the same cross with the four inner corners filled in, which makes it look rounder and more generous. Those four extra holes are what break it.

If you own one and have never managed the central game, count the holes before assuming you are bad at it.

How Do You Solve Peg Solitaire?

By thinking about the end rather than the beginning.

Work backwards from one peg. The final move is a single jump into the centre, which means the second-to-last position is highly constrained. Several moves can be reasoned out from the finish rather than guessed from the start.

Avoid stranding pegs at the edges. The arms of the cross are where pegs get marooned, because a peg with no neighbour and no gap beyond it is dead weight for the rest of the game.

Clear in blocks, not greedily. Removing the most pegs you can on each turn feels like progress and usually scatters what remains. Keeping pegs adjacent to one another is what keeps moves available later.

Chain your jumps. If you are aiming for a short solution rather than just a finish, look for a peg that can jump repeatedly in one turn. That is the entire difference between 31 moves and 18.

What Else Works Like This?

We do not make peg solitaire, and it is widely available in every material.

What it shares with things we do make is the invariant idea: a property of the position that no legal move can change, which decides what is reachable before you start. The Tower of Hanoi has an exact move count for the same kind of reason, and we make that one. The 15 puzzle has the parity version of the same argument.

The wider range is in mechanical puzzles.

Frequently Asked Questions

What are the rules of peg solitaire?

Every hole starts filled except one. A peg jumps over an adjacent peg into the empty hole directly beyond, and the jumped peg is removed. Moves are orthogonal. The usual goal is to finish with a single peg in the centre.

What is the fewest moves to solve peg solitaire?

Eighteen on the 33-hole English board, because a peg jumping several times in one turn counts as a single move. Ernest Bergholt found an 18-move solution in 1912, and it was proved in 1964 that none shorter exists.

Why can I never finish with one peg in the middle?

If you are playing the 37-hole European board, you cannot, and it is not your fault. Positions divide into 16 classes that no legal move can cross, and the start and target of that game sit in different classes. On the 33-hole English board it is solvable.

How old is peg solitaire?

The first known printed reference is from August 1697, in the French magazine Mercure galant, and a portrait by Claude Auguste Berey from about a decade later shows the Princess of Soubise with the puzzle. It was a fashionable amusement at the court of Louis XIV.

What is the trick to peg solitaire?

Work backwards from the final peg, keep pegs clustered rather than taking every capture available, and avoid stranding pegs in the arms of the cross. For a short solution, look for jumps that chain several captures into one move.