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.

*** 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.

*** LINKS ***

Vsauce2:

Hosted and Produced by Kevin Lieber

Research and Writing by Matthew Tabor

Editing by John Swan

Huge Thanks To Paula Lieber

Get Vsauce's favorite science and math toys delivered to your door!

#education #vsauce2 #maths

[anythingg]

My guy really drew us a whole board just to tell us that he cant use it. The dedication.

[bunsenn]

I actually managed to win this game, I used a strategy called “cheating whenever I’m at a disadvantage”

[brighton]

"The game you cant win"

Me: pretty much every game i've ever played.

[thomasfieschi-rose]

Rest In Peace, John Conway. The man that gave us life.

[jmzlolo]

Thanks for making a tribute to John Conway, I'm sure he would have appreciated it

[ahuman]

Every time I hear about Conway the guy sounds cooler, his game of life is still one of my favorite things

[FrancisACNH]

Kevin: “This video is spo-“

Me: *furiously taps right side of the screen*

[teridactyl]

‘It’s like trying to invade Russia in winter’
*laughs in Mongolian*

[MrOvergryph]

Kevin: "You can solve it if you bend time"
Also Kevin: "Using 27-digit-dimensional space"
Still Kevin: "He called it the Monster"
*boldy explains nothing to audience*
More Kevin: "And as always, Thanks for watching!"

[jax]

“The goal of this game is to get these checkers to the other side of the li-“

Me: *shoves all the checkers to the other side of the board with my arm*

[trackeralias]

2n+1 for the moves seems complicated.. if the goal is reached when the final piece hits the target level, and each move removes one piece, then you'll always make total pieces - 1 moves? as you can't have 'pieces left over' as any piece used effectively deletes itself.

[crckem]

I can imagine Kevin bullying his future child with the whole "right? WRONG!" thing. Poor kid.

[huhneat]

This feels a lot like Minecraft piston extenders

[spoopy]

It's Like trying to invade russia in winter, its impossible
*_heh im just glad hitler didnt know that_*

[danieltreshner]

The main difficulty I ran into using the game in the link, is that as you progress, you slowly create a void behind your leading soldier. That void prevents other soldiers from progressing in that column purely because they can't reach it.

[DrakiniteOfficial]

This is the first I heard about Conway's death. Rest in peace, my friend. You've done so much for us.

[dorukefeozkan]

every actual time i watch a vsauce2 video:
kevin: yes
me: yes?
kevin: no
me: no?
kevin: WRONG

[arcurifamily]

Level 1, dirt, level 2, wood, level 3, Stone, level 4, bedrock

[xarim]

Kevin: mentions the golden ratio
Me, an idiot that read JoJo: I’ve seen this one!

[kabirbatra]

I'm all sad now..I remember when I was super fascinated with Conway's game of life in middle school, and I remember when my 8th grade teacher told me she had actually met Conway in person and they were friends..and I was so amazed and wished I could meet him some day..what a weird feeling it is to know that he was just human, vulnerable to corona like the rest of us. May his beautiful lessons live forever in the mathematical community, and may his games continue to appeal to the interests of people of all ages, from cs students programming the game of life at school, to nerds watching Vsauce and Numberphile. Thanks for watching? More like thanks for making this video Kevin.