Many years ago I was shown this puzzle. It's a type of solitaire or peg-jumping puzzle. One places some arbitrary arrangement of pieces on a rectangular grid, below the grid dividing line (the bar).
Then you perform a sequence of jumps.
A jump is allowed for any 3 cells in line horizontally or laterally, and follows the standard peg-jumping rule, ie: $X X . \implies . . X$ (the center piece is removed).
The objective is to project a piece as far above the bar as possible.
For example:
$$ \begin{bmatrix} \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\\hline \cdot & \cdot & X & X & X & \cdot & \cdot \\ \cdot & \cdot & X & X & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{bmatrix}$$
This layout allows one to project a piece up to the 2nd row above the bar.
It is known that, even with an infinite grid size and unlimited supply of pieces, it is impossible to project a piece beyond the $4$-th row above the bar.
I was shown an elegant proof of this limit back in 1975, and my question is simply this - does anybody recognise this puzzle? If so, can somebody point me to the proof?
The name of it is the Conway's Soldiers Puzzle.