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. Real quick, this video is sponsored by MSCHF,
the internet explorers behind viral drops like the Man Eating Food channel -- he started
by eating noodles, and then he ate PewDiePie. And the Times Newer Roman font. It looks exactly like Times New Roman, but
it’s 8% wider, so it makes writing papers… 8% quicker. Or the MSCHF Box -- that sold out in under
5 seconds. It was like a Vsauce2 video in real life. The box had between $0-$7,000 of items inside. It cost $100. If you didn’t open it, they’d buy it back
for $1,000 after 100 days. Basically a massive experiment in expected
value and psychology. These are just 3 of over 20 drops. Every 2 weeks, never re-stocked or repeated…
like an AI bot that texts you pictures of feet, or a collar that swears when your dog
barks. Most of it’s free weird digital experiments
and a lot of it gets taken down pretty quickly. So to get early access to drops as well as
secret drops, download the free MSCHF app. M.S.C.H.F. No vowels. Okay let’s get back to our game. 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.
