What is Peg Solitaire with a centre finish?
With a centre finish, Peg Solitaire keeps its moves and boards but makes the goal stricter. It is not enough to leave one peg: the last peg must stand in the hole that was empty at the start. On the English and German boards that is the centre hole, so the game is to go from a single empty centre to a single peg in the centre.
On the triangle the empty hole at the start is the top corner, and the last peg must end there. The option is not offered on the European and diamond boards, where the last peg can never end in the hole that started empty.
The game ends as soon as no jump is possible. One peg in the starting hole wins; one peg anywhere else, or several pegs with no jump left, loses. To succeed you need to plan for the final jumps from the start, keeping pegs ready to come back to the middle.
Rule changes
| Rule | Standard Peg Solitaire | Peg Solitaire with a centre finish |
|---|---|---|
| Goal | Standard Peg SolitaireLeave one peg, anywhere on the board. | Peg Solitaire with a centre finishLeave one peg, in the hole that was empty at the start. |
| End of the game | Standard Peg SolitaireOne peg left wins. More pegs with no jump left loses. | Peg Solitaire with a centre finishOne peg in the starting hole wins. One peg anywhere else, or more pegs with no jump left, loses. |
| Boards | Standard Peg SolitaireAll five boards: English, European, triangle, diamond and German. | Peg Solitaire with a centre finishEnglish and German (the centre) and the triangle (the top corner). Not the European or diamond boards, where it is impossible. |
| Jumps | Standard Peg SolitaireA peg jumps over a neighbouring peg into the empty hole beyond; the jumped peg is removed. | Peg Solitaire with a centre finishUnchanged. Only the winning condition is stricter. |
History
On the 33-hole English board, starting with the centre empty and ending with the last peg in the centre is the classic problem of the game, often called the central game. Ernest Bergholt published a solution in 18 moves in 1912, counting a chain of jumps by one peg as a single move, and John Beasley proved in 1964 that no shorter solution exists.
Why the finish in the centre works on some boards and not on others is explained by position classes: every jump leaves certain counts over the board's diagonals unchanged, so from a given start only a few holes can ever hold the last peg. N. G. de Bruijn set out the argument in 1972, and Elwyn Berlekamp, John Conway and Richard Guy used it in Winning Ways for your Mathematical Plays (1982). On the French 37-hole board it shows that the last peg can never return to the hole that started empty.