Since a single marble solution doesn't exist, reducingĭown to two marbles like this is the best you can do. (2) has an even parity as do the resulting diagonals: A,B,D,E (0)Īnd C,F (2). Now ifĪ jump is made to the right you are left with two marbles. The total (3) has odd parity as does every diagonal (1). For example, a horizontal line just above the center would It offen occurs that an endgame is reached with three marbles inĪ row. Since this is not possible, there is no single marble solution. Single marble solution, diagonals A-F would all have to have odd As moves are played the parities change, but the Since each diagonal has the same parity (even) in the number of Terminology the position is "in phase" for every diagonal, The squares that have men on them initially: Diagonal A 10 (even) Diagonal B 10 (even) Diagonal C 4 (even) Diagonal D 10 (even) Diagonal E 4 (even) Diagonal F 10 (even) - Total men: 24 (even) First we label the diagonals as in diagrams 4 and 5: Beasley'sĪnalysis ("The Ins and Outs of Peg Solitaire", chapterĤ) on it. It to a standard type of puzzle means we can use John D. All the positions of the "otherĬolor" have been removed. This problem is equivalent to the standard type (orthogonal jump)
Look at diagram 2 diagonally, tilting it clockwise, you see that Two instances of the problem of reducing all the marbles on a singleĬolor: O:O:O:O: :O:O:O:O O:::::O: :O:::::O O:::::O: :O:::::O O:O:O:O: :O:O:O:O
THE PROOFīoard were checkered, then marbles would never leave their own color Is a removal of 44 marbles, leaving 4 marbles remaining on the board. It turns out that the substantial/difficultįeat they talk about is an impossibility. (1942) that "To have one peg on board is a very difficult feat."
Writes in "Mathematical Puzzles and Other Brain Twisters" Ahl writes that "It isĮasy to remove 30 to 39 checkers, a challenge to remove 40 to 44,Īnd a substantial feat to remove 45 to 47." Anthony S. The goal is to reduce the position to as few marbles as possible.Ĭomputer Games" (1978) David H. Jump diagonally as in Checkers and the jumped-over marble is removed. Tile and then choose "8x8 Diagonal Solitaire" from theĭiagonal Solitaire" 48 marbles are initially arranged on anĨx8 grid as folows: OOOOOOOO OOOOOOOO OO::::OO OO::::OO OO::::OO OO::::OO OOOOOOOO OOOOOOOO To find this game in Zillions of Games click the "Square Solitaire" Optimal Solution for the 8x8 Diagonal Solitaire Puzzle Zillions of Games - An Optimal Solution for the 8x8 Diagonal Solitaire Puzzle
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