Peg Solitaire: How to Solve It

How the game works

Tap a peg and then the empty hole two along from it. On the cross, jumps go along rows and columns, never diagonally; on the triangle they also go along both slopes. Most puzzles ask for the last peg to finish in the centre hole, which is marked. If no jump is left, you are stuck and can undo.

Every solution to a board has the same number of jumps, one fewer than the pegs, because each jump takes off exactly one peg. So the game counts every jump you make, even the ones you undo, and three stars go to a clean solve.

Naming the holes

On the cross, columns are a to g from the left and rows 1 to 7 from the bottom, like a chessboard, so the centre hole is d4. The triangle’s holes are numbered 1 to 15, row by row from the top point, left to right. A jump is written as the peg’s hole and the hole it lands in: d5 to f5 means the peg on d5 jumps over e5 into f5.

Habits that help

• Work from the outside in. Pegs far out on the arms are the hardest to reach later, so clear them while there are still pegs nearby to jump over.
• Watch the corner holes. A peg in one of the corner holes at the end of an arm, such as c7 or g5, can never be jumped over. It can only leave by jumping itself, so keep a peg beside it until it does.
• Keep the pegs together. A peg left far from the others is hard to reach again: another peg has to come right next to it before either can jump.
• Think about the end. The last jump lands in the centre, so the last two pegs must sit in a line with it, one next to the centre and one two away.

The Little Cross

Six pegs: d6, c5, d5, e5, d4 and d3. Five jumps:

1. d5 to f5.
2. d3 to d5.
3. c5 to e5.
4. f5 to d5.
5. d6 to d4.

The Plus Sign

Nine pegs in a plus around the centre. Eight jumps:

1. d5 to d7. 2. d3 to d5. 3. b4 to d4. 4. d4 to d6. 5. d7 to d5. 6. f4 to d4. 7. d5 to d3. 8. d2 to d4.

The Fireplace

Eleven pegs: the top arm, the three holes below it, and c4 and e4 either side of the empty centre. Ten jumps:

1. d5 to f5. 2. e7 to e5. 3. c7 to e7. 4. e4 to e6. 5. e7 to e5. 6. f5 to d5. 7. d6 to d4. 8. d4 to b4. 9. c6 to c4. 10. b4 to d4.

The Triangle

Fifteen holes with only the top one, hole 1, empty. The last peg can finish anywhere. Thirteen jumps:

1. 4 to 1. 2. 6 to 4. 3. 1 to 6. 4. 7 to 2. 5. 10 to 3. 6. 12 to 5. 7. 13 to 6. 8. 2 to 9. 9. 3 to 10. 10. 15 to 6. 11. 6 to 13. 12. 14 to 12. 13. 11 to 13.

The Arrow and the Diamond

The arrow has seventeen pegs and the diamond twenty-four. There is no short list of moves to learn for these, but the habits above carry you through: clear the tip of the arrow and the far points of the diamond first, then work the pegs in towards the centre in pairs, and keep the last few pegs in a tight group around d4.

The Full Board

Thirty-two pegs and an empty centre: the classic. Thirty-one jumps:

1. d6 to d4. 2. b5 to d5. 3. c7 to c5. 4. e7 to c7. 5. d5 to b5. 6. a5 to c5. 7. e5 to e7. 8. g5 to e5. 9. c4 to c6. 10. c7 to c5. 11. a4 to c4. 12. c4 to c6. 13. e4 to c4. 14. g4 to e4. 15. e4 to e6. 16. e7 to e5. 17. c3 to c5. 18. c6 to c4. 19. a3 to c3. 20. d3 to b3. 21. e2 to e4. 22. g3 to e3. 23. c1 to c3. 24. c4 to c2. 25. e1 to c1. 26. c1 to c3. 27. b3 to d3. 28. d3 to f3. 29. e5 to e3. 30. f3 to d3. 31. d2 to d4.

A little history

A French engraving from 1697 shows a princess at the court of Louis XIV playing peg solitaire, so the game is well over three hundred years old. The board with 33 holes is often called the English board; a French board adds four more holes. When a chain of jumps by the same peg is counted as one move, the full board can be solved in 18 moves, found by Ernest Bergholt in 1912 and shown to be the fewest possible by John Beasley in 1964.