Vsauce!
Kevin here with a game that starts out easy, but quickly becomes impossible.
You can be the best in the world at it and still never get very far.
Here, I’ll show you -- the goal is to get one of these checkers to the other side of
this line.
Haha!
I WIN.
I just won an impossible game, right?
WRONG!
Kinda.
I won the baby level.
But it gets a lot more complex than this.
Your goal is to get these checkers as far over the line as possible.
There are hardly any rules.
You can use as many checkers as you want… and actually.
I got some empty spaces here.
My checker set only came with 30 and that won’t be enough for my needs so I’m uh
I'm not gonna use these.
I'm gonna use poker chips.
There’s no limit to how many you can use, you just have to place them under the line.
Chips can only jump other chips -- they can’t just, like, slide forward to an unoccupied
space.
You can't just do that.
They also can’t jump diagonally, but they can jump both vertically and horizontally.
So, at least we’ve got some of that sweet sideways action.
And when a chip does get jumped, it gets removed from the board.
Like that.
That’s it.
Welcome to Conway’s Soldiers.
Oh -- and this is a one-player solitaire game.
Solitaire is any one-player game, not just the card game.
So get ready to social distance your little heart out.
In the beginning of the video, I used 2 checkers to get to Level 1.
It’s almost just as easy to get to Level 2 using 4 checkers... it only takes 3 quick
moves and we have ascended to new heights.
Well done!
Yet, it’s still not all that impressive.
Getting to Level 3 is where it starts to get interesting.
In 1961, mathematician John Conway invented this simple chip game to prove it wasn’t
as trivial as it appears.
We can think of the chips as soldiers trying to figure out how to penetrate as deep as
possible into enemy territory.
Getting to Levels 1 and level 2 wouldn’t take a whole lot of soldiers or sophistication,
but the deeper those soldiers try to go, the harder it’ll be to arrange.
Actually, it doesn’t just become hard.
It becomes mathematically impossible.
Which seems impossible.
If we have infinite soldiers, why can’t we make it to infinite levels?
First, let’s figure out how to maneuver one soldier from our tiny chip battalion into
Level 3.
You’ll need at least 8 soldiers -- but you can do it in just 7 moves.
In this configuration, you essentially have two separate forces.
The left wing repeats what we did in the last game to get a soldier to Level 2, and the
right wing sets up two vaults to get our little Rambo to Level 3.
You’ve got to think about it a little bit, but… ehhhh, you’re gonna get it.
For Levels 1, 2, and 3, it took 2, 4, and uh hang on.
Gotta make some room.
Hang on.
I'm gonna try to move this without ruining everything in the world.
For Levels 1, 2, and 3, it took 2, 4, and 8 soldiers and we’ve got a nice 2 to the
nth power pattern here.
Achieving Level 4 should be 2 to the 4th power for 16 soldiers.
There seems to be a pattern on the number of moves it’ll take, as well.
If we call the initial number of moves to get to Level 1 “n,” then the Level 2 seems
to be (2n + 1) -- 2*1 + 1 = 3 moves.
Then we’ll make n that number of moves and put it into our (2n + 1) algorithm for 2*3
+ 1 = 7 moves.
And yeah!Yeah!
All of this works perfectly.
Our 2^4 = 16 soldiers will take (2*7) + 1 = 15 moves to reach Level 4.
Finally, we’ve got a clean, easy system to get to Level 4 and beyond.
Right?
WRONG.
This is where the simulation breaks.
We actually need.
Get this.
We actually need 20 soldiers to make it to Level 4.
Look.
Slide this back on into here.
I don't know why I did that voice.
Oh well.
Can't take it back.
We create the same starting pattern as we used to get to Level 3, but it’ll end up
one level higher.
Then we play it out just as before and launch our soldier to save Private Ryan in Level
4 of enemy territory.
And it takes 19 moves… not the 15 we expected.
EVERYTHING I ASSUMED WAS WRONG.
Our easy rule on soldiers stopped working.
Yet, we still found a way.
We didn’t have a formula for how many moves we needed, but we got there anyway with a
little ingenuity from our five-star general.
Me.
We conquered Level 4.
I'm ambitious, though, like Genghis Khan and Alexander the Great before me.
What'll it take to extend our Vsauce2 empire into Level 5?
Five?
This board ends at four.
I guess I gotta make my own board, SAYONARA PUNY CHECKER BOARD!
Okay, okay.
Alright.
I did it.
Here's my custom board.
And the first move that you need to make to get to Level 5 is...
Nothing.
Nothing works for Level 5.
It’s like trying to invade Russia in winter.
Seriously -- it’s impossible.
IMPOSSIBLE.
I’ll even link you to a website where you can try for yourself.
It’s so weird.
Having a limit of 4 rows seems to make no sense.
When I first learned about this I thought, “this makes no sense!”
We have an infinite number of soldiers and a board of theoretically infinite size.
It only took 20 chips to get to Level 4, and Level 5 isn’t even that far!
How can we not EVER even get to Level 5?
The answer is... beauty.
And taxis.
If we mark our target on Level Five with a T, we can mark all the moves around it with
an exponent that shows how many steps away that position is -- it’s the sum of the
horizontal and vertical steps to get there.
It’s the “manhattan number,” which stems from taxicab geometry -- just like a taxi
in New York City has to drive across a grid, so do we.
And as we map out the moves away from our target, we think about the three types of
possible jumps: ones that get us closer to the goal, ones that get us further, and ones
that are neutral.
Conway used this weighting of each cell to… uh hang on a second... to calculate…
I NEED NUMBERPHILE.
The mathematical proof of why Conway’s Soldiers can never make it to Level Five is explained
perfectly by Berkeley math professor Zvezdelina Stankova over at Numberphile2 -- and it’s
42 minutes long.
It’s surprisingly easy to follow, and along the way you’ll see how Conway’s game depends
on the uh.
Oh no.
Depends on.
I gotta destroy my beautiful board.
Oh well!
Depends on the golden ratio, the proportions we see in the natural world from flowers to
the composition of pretty faces.
And here it pops up out of nowhere in the middle of the game.
But… it turns out we can reach the impossible Level 5.
Simon Tatham and Gareth Taylor proved that we can get to the 5th row with a board of
infinite space, infinite soldiers, and infinite moves -- provided we bend the rules just a
tiny bit.
All you gotta do is manipulate TIME.
You really can go as far down this mathematical rabbit hole as you want but what started as
a simple solo lattice game morphed into a metaphor for both expertise and hope.
With nearly everything we do, the first few levels are… actually pretty easy.
But as we aspire to the most elite tiers, it takes more soldiers, more moves, and a
more sophisticated configuration to reach the top.
It takes hard work and understanding.
And just when we think we can’t go any further, ingenuity and creativity take us to a place
we thought was impossible.
On April 12, 2020, we lost one of the greatest mathematical minds of the 20th and 21st centuries
to the Covid-19 virus.
John Horton Conway was 82 years old.
He talked once about an object that existed
not just in 3 or 4 dimensions, but in 27-digit-number
dimensional space.
He called it “the Monster.”
He said:
“These things are so beautiful.
It’s such a pity that people can’t see them.”
Well, maybe we can’t.
I can’t.
But John Conway helped us see the beauty of math in everything -- even something as simple
as a checker.
And as always, thanks for watching.