![]() Hence a final position with only one peg can not be reached: then one of these numbers is one (the position of the peg, one is odd), while the other two numbers are zero, hence even. ![]() Hence after an even number of moves all these three numbers are even, and after an odd number of moves all these three numbers are odd. After every move the number of covered A positions increases or decreases by one, and the same for the number of covered B positions and the number of covered C positions. Initially with only the central position free, the number of covered A positions is 12, the number of covered B positions is 12, and also the number of covered C positions is 12. Divide the positions of the board into A, B and C positions as follows: This is easily seen as follows, by an argument from Hans Zantema. ![]() There is no solution to the European board with the initial hole centrally located, if only orthogonal moves are permitted. On the English board the equivalent alternative games are to start with a hole and end with a peg at the same position. This mirror image notation is used, amongst other reasons, since on the European board, one set of alternative games is to start with a hole at some position and to end with a single peg in its mirrored position. There are many different solutions to the standard problem, and one notation used to describe them assigns letters to the holes: Many people never manage to solve the problem. It is very easy to go wrong and find you have two or three widely spaced lone pegs. On an English board, the first three moves might be: Thus valid moves in each of the four orthogonal directions are: * A blue ยค is the hole the current peg moved from a red * is the final position of that peg, a red o is the hole of the peg that was jumped and removed. indicates a peg in a hole, * emboldened indicates the peg to be moved, and o indicates an empty hole.A valid move is to jump a peg orthogonally over an adjacent peg into a hole two positions away and then to remove the jumped peg.
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