The Game You Can't Win

John Conway’s work stretched from nearly-inaccessible math theory to fun children’s games, and it defined how we thought about recreational mathematics in the 20th century. Conway’s Soldiers is the perfect example of a Conway-esque math game: the rules are simple, the gameplay is easy, and it becomes very hard very quickly… and in this case of a basic checkers game, it packs a surprising punch of becoming literally impossible for a reason that isn’t even close to obvious.

But the mathematical importance doesn’t end with an unexpected reveal of impossibility. Conway found that the secret to why it’s impossible depends on the Golden Ratio of 1.618, which has fascinated mathematicians since Pythagoras and Euclid. We see the Golden Ratio in aesthetically-pleasing magazine layouts, flower patterns, and pretty faces, but John Conway showed us that it’s at the heart of why we can’t reach the fifth row of a checker game.

Play the game yourself with the online simulator and check out Zvezdelina Stankova’s proof on Numberphile. It’ll make sense. John Conway may be gone now, but we’ll be playing his games and poring over his ideas for centuries.

Conway Checkers (proof), Numberphile2 | Zvezdelina Stankova:    • Conway Checkers (proof) - Numberphile  

Conway’s Soldiers Interactive Game (Clever Learning): https://www.cleverlearning.co.uk/blog...

** SOURCES **

Berlekamp, E. R.; Conway, J. H; and Guy, R. K. "The Solitaire Army." In Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, pp. 715-717 and 729, 1982.

Honsberger, Ross. "A Problem in Checker Jumping." Ch. 3 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 23-28, 1976.

Tatham, Simon and Taylor, Gareth. “Reaching Row Five in Solitaire Army.” https://tartarus.org/gareth/maths/stu...

plus magazine, “Mathematical Mysteries: The Solitaire Advance.” https://plus.maths.org/content/os/iss...

Bell, George. “The Peg Solitaire Army.” http://www.gibell.net/pegsolitaire/ar...

Bell, Hirschberg, Guerrero-Garcia. “The Minimum Size Required of a Solitaire Army.” https://arxiv.org/pdf/math/0612612.pdf

Taylor, Jasper. “Conway’s Soldiers.” https://sites.math.washington.edu/~mo...

John Conway’s obituary, New York Times: https://www.nytimes.com/2020/04/15/te...

Roberts, Siobhan. “Genius at Play: The Curious Mind of John Horton Conway” (2015). https://www.amazon.com/Genius-At-Play...

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