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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?

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The name of it is the Conway's Soldiers Puzzle.

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  • $\begingroup$ And thank you too! $\endgroup$ – Jim White Jul 3 '18 at 16:46

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