What is Diamond Peg Solitaire?
Diamond Peg Solitaire is played on 41 holes in the shape of a diamond: the rows hold 1, 3, 5, 7, 9, 7, 5, 3 and 1 holes, so the board comes to a point at the top, bottom, left and right.
The moves are the same as on the English board: a peg jumps over a neighbouring peg, along a row or a column, into the empty hole beyond, and the jumped peg is removed. You win by leaving a single peg.
With the centre empty this board cannot be solved. Here the game starts with e3 empty (columns a to i from the left, rows 1 to 9 from the top), the hole two below the top point, and the board can be cleared from there. The last peg can only end on e1, b4, e4, h4 or e7. With 40 pegs to start, this is a long game of 39 jumps.
Rule changes
| Rule | Standard Peg Solitaire | Diamond Peg Solitaire |
|---|---|---|
| Board | Standard Peg Solitaire33 holes in a cross on a 7×7 grid. | Diamond Peg Solitaire41 holes in a diamond on a 9×9 grid: rows of 1, 3, 5, 7, 9, 7, 5, 3 and 1. |
| Empty hole at the start | Standard Peg SolitaireThe centre hole, with 32 pegs on the board. | Diamond Peg Solitairee3, two holes below the top point, with 40 pegs on the board. The centre start cannot be solved on this board. |
| Where the last peg can end | Standard Peg SolitaireOnly in the centre (d4) or in the middle hole of an arm's outer edge (d1, a4, g4, d7). | Diamond Peg SolitaireOnly on e1 (the top point), b4, e4, h4 or e7. |
| Jumps in a solution | Standard Peg Solitaire31. | Diamond Peg Solitaire39. |
| Finish in the starting hole | Standard Peg SolitaireOffered as an option: the last peg must end in the centre. | Diamond Peg SolitaireNot offered: on this board the last peg can never end in the hole that was empty at the start, from any start. |
History
The 41-hole diamond is one of the board shapes shown beside the English cross, the French 37-hole board and Wiegleb's 45-hole board in modern accounts of the game.
Its difficulty with the centre start comes from position classes, the counting argument set out by N. G. de Bruijn in 1972 and in Winning Ways for your Mathematical Plays by Elwyn Berlekamp, John Conway and Richard Guy (1982): every jump leaves certain counts over the board's diagonals unchanged, and on the diamond those counts rule out a single last peg from the centre start, and rule out the last peg ending where the empty hole began from any start.