Mathematics

A legal jump preserves a position class, a parity invariant. On a square lattice, colour two sets of diagonals with repeating three-colour patterns and track the parity of total pegs minus each colour count. The six parities stay unchanged by a jump. If the initial and target positions have different classes, no solution exists. Equal classes are necessary, not sufficient: geometry can still prevent a solution. In particular, the European centre vacancy cannot finish with one peg anywhere, and the diamond centre complement is impossible.

Learn

1. Counting: start with N pegs and finish with one. Explain why exactly N−1 jumps are required. A full English game starts with 32 pegs and needs 31 jumps.
2. Planning: solve the 1-, 3- and 5-jump exercises. Before each jump, predict which pegs will remain. Compare two paths and discuss why an isolated peg cannot jump.
3. Parity: label diagonals with a repeating three-colour pattern. Record the odd/even colour counts before and after a jump. Explain what stays invariant and why that does not prove solvability.

The 37-hole European board adds four holes to the English shape. Our solvable opening empties c1 and finishes at e1 in 35 jumps. A d4 vacancy cannot finish with a single peg anywhere.

The 15-hole triangle has five rows, from one to five holes. Empty a1 and return the survivor to a1 in 13 jumps. Use all six lattice directions. The b3 complement and a1-to-b3 target are impossible.

The 41-hole diamond has rows of 1, 3, 5, 7, 9, 7, 5, 3, 1. Empty d2 and finish at f2 in 39 jumps. Do not confuse it with the 32-hole diamond. The centre complement is impossible.

Wiegleb’s 45-hole cross extends each arm by one row. Empty e5 and finish at e5 in 43 jumps. Unlike the central game, an e1-to-e1 complement is impossible.