Conway's Soldiers
Conway's Soldiers or the checker-jumping problem is a one-person mathematical game or puzzle devised and analyzed by mathematician John Horton Conway in 1961. A variant of peg solitaire, it takes place on an infinite checkerboard. The board is divided by a horizontal line that extends indefinitely. Above the line are empty cells and below the line are an arbitrary number of game pieces, or "soldiers". As in peg solitaire, a move consists of one soldier jumping over an adjacent soldier into an empty cell, vertically or horizontally (but not diagonally), and removing the soldier which was jumped over. The goal of the puzzle is to place a soldier as far above the horizontal line as possible. Conway proved that, regardless of the strategy used, there is no finite sequence of moves that will allow a soldier to advance more than four rows above the horizontal line. His argument uses a carefully chosen weighting of cells (involving the golden ratio), and he proved that the total weight can only decrease or remain constant. This argument has been reproduced in a number of popular math books. Simon Tatham and Gareth Taylor have shown that the fifth row can be reached via an infinite series of moves. If diagonal jumps are allowed, the 8th row can be reached, but not the 9th row. It has also been shown that, in the n-dimensional version of the game, the highest row that can be reached is . Conway's weighting argument demonstrates that the row cannot be reached. It is considerably harder to show that row can be reached. (Wikipedia).
