| Another is Conway's Soliders: 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.
(https://en.wikipedia.org/wiki/Conway%27s_Soldiers) How many soldiers have to be put below the line before the game starts to enable at least one soldier to reach a given height over the line? For height 1, you need 2 soldiers. For height 2, you need 4. For height 3, you need 8. For height 4, you need ... 20. For height 5, it's impossible, by the golden ratio. |
Strictly speaking, it's Conway's proof uses the golden ratio. But it could be that there's an alternate proof that doesn't use the golden ratio.