3D Conway’s Solitaire

We give a concrete process to send a peg to the seventh level in Conway’s Solitaire game in a three dimensional space. 1 Conway’s Solitaire Conway’s Solitaire is a game played by one person on a board. Pegs are placed at lattice points of the plane. One can move a peg jumping over another peg horizontally or vertically into an empty position. After the move the second peg is removed from the plane: \copyright \copyright $O $ $\Rightarrow $ $O $ $O $ \copyright (1) An initial configuration consists of finite number of pegs placed at positions in the lower half plane (below the line $y=1$). The goal of the game is to send a peg as high as possible on the plane. The problem is to find a configuration and a sequence of moves which send a peg at the possibly highest position. For example, starting with the T-configuration, we can go three steps up: O $O $ $O $ \copyright (2