# 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!

Posted 2020-10-17