Suppose you take the infinite square lattice and put solitaire pieces on all points (x,y) that lie on or below the x-axis. Using solitaire moves, can you reach a position where the point (0,5) is occupied?

Note: A solitaire move consists in a piece jumping over a neighbouring piece to a vacant square and removing (or “taking”) the neighbouring piece.

You can attempt this challenge interactively here.

Hint: if you can’t get there, don’t feel too bad about it.

A description of the original proof based on coming up with an invariant involving the golden ratio:

A description of the original proof based on coming up with an invariant involving the golden ratio, but in this case in real-time, while thinking out loud:

A description of a more recent proof based on a rather clever and beautiful use of Fibonacci numbers:

And by the way, if you are really keen on reaching row 5 here’s how you can do it with infinitely many moves!