Conway’s Soldiers

The other day I learned about Conway’s Soldiers: Consider playing peg solitaire on an infinite grid of holes, where initially an infinite number of pegs fill the entire half-plane south of the line \(y=0\). The goal is to advance one peg as far north as possible, by repeatedly jumping and removing pegs in the usual peg-solitaire fashion.

John Horton Conway explored the problem and proved the surprising fact that no matter how many pegs you jump, it’s impossible to advance any peg beyond the line \(y=4\) in a finite number of moves!

However, Simon Tatham observes
that it is possible to advance a single peg to \(y=5\), if you are
permitted an *infinite* number of moves to do so. He provides this animation
of the winning strategy, which provably must use every single one of the pegs:

Incidentally, Conway died earlier this year, of COVID-19, causing me to update “Hello Muddah, Hello Faddah (Coronavirus Version)” (2020-04-08) with another verse:

Rest in peace, John / Horton Conway,

You have gone A- / nacreon way;

You’ve ascended / Gardner’s column,

And now Heaven has another angel problem!