Game · 2026-07-13

Peg solitaire, on the English board.

Thirty-three holes in a cross. Thirty-two pegs, one empty hole in the middle. A move jumps a peg over an adjacent peg into an empty hole, removing the jumped peg. The classic target is one peg left, standing in the centre.

Pegs 32

Moves 0

Status Ready

Where the last peg can land

Colour each hole by (row + column) mod 3. The board splits into three classes of eleven holes each. Every legal jump touches three consecutive cells in a row or column, so it touches one hole of each class. It removes two pegs from two different classes and adds one peg to the third.

Read that mod two. A jump flips the parity of the peg count in each of the three classes at the same time, so the pair (p_A ⊕ p_B, p_B ⊕ p_C) never changes. That is a game invariant, and it rules the last peg out of any hole whose class differs from the centre's.

Now do the same argument with a second colouring: (row − column) mod 3. Jumps still touch three distinct classes, and a second, independent invariant appears. Together the two rule the last peg out of every hole whose class differs from the centre's under either colouring.

What survives is the centre and the four arm-tips: five holes out of thirty-three, discovered by pure arithmetic before you place a peg. Reiss showed this in 1857. The full board is symmetric under a ninety-degree rotation, and so is the surviving set.

A second look at the counter

Each jump removes exactly one peg, so the trip from thirty-two pegs down to one takes thirty-one moves. The counter above ticks moves, not pegs, so a full solution ends at thirty-one.

Sources

The three-colour invariant is credited to Michel Reiss, Beitrag zur Theorie des Solitär-Spiels, Journal für die reine und angewandte Mathematik 54 (1857), 344-379. A modern treatment sits in John Beasley, The Ins and Outs of Peg Solitaire, Oxford University Press, 1985. The 33-hole English board is documented in court records from Louis XIV's reign; the earliest known illustration is a 1697 engraving by Claude-Auguste Berey of Anne de Rohan-Chabot playing at the game.