The Dot Game That Breaks Your Brain

Vsauce! Kevin here, with a game you can’t possibly
comprehend. Really, it’s too hard for you. Your brain can’t take it. Look, I’ll show you: That’s it. Are you sweating yet? You should be. Real quick, huge thanks to ExpressVPN for
sponsoring this video and supporting Vsauce2. If your device is unsecured you’re gonna want
to get ExpressVPN to take care of that. I’ll explain more later but first let’s explain
our dots. Alright, as you stare into these dots your
brain starts to short circuit, doesn’t it? No. Why would it? I mean… it’s just two dots! I can draw out all the possible moves for
a game this simple. Look, I’ll show you. Okay, my award-winning handwriting aside,
this was a lot more complicated than I thought it was gonna be. And the thing is… as it scales, analyzing
what appears to be the simplest game in the world doesn’t just break your brain, computers
can’t even crunch the possibilities. Here’s how it works. The game of Sprouts starts with any number
of dots placed… anywhere. The boundaries of the game board are limitless,
so put the dots wherever you want. We’ll play with two dots. But ya can’t just play with yourself, you
need an opponent. Yes, yes. A worthy adversary, you need. Let’s go over the three rules of Sprouts. First, a player draws a line from one dot
to another, or from one dot back to itself. Lines can be curved or they can be straight…
they just can’t cross another line or themselves. When you draw a line, you get to place a new
dot anywhere on that new line. And in Sprouts, no dot can have more than
3 lines coming from it or going to it. Once a dot has 3 lines — it’s an unplayable,
dead dot. The winner of Sprouts is the last person to
draw a line. Or to put it another way, the player who can’t
draw another line loses. Okay, now my friend and I will play a two-dot
game of Sprouts. Go first, I will! Alright, Yoda. Dude. Okay Hang on! Alright fine. Just go. Alright, alright. Great job. You gotta make sure you draw a new dot on
the line. Yes, yes, yes. Invented this game, I did! Sprouts trained many Jedi minds, hundreds
of years! Hundreds of years? No, No, No. Sprouts was created in 1967 by Cambridge mathematicians
John Conway and Michael Paterson. My turn it is! Dots lead to lines. Lines lead to dots. Sprouts is the path to the light side of the… And I just won. Alive this dot still is! Yeah but you can’t connect it to anything. Look. Dead, dead, dead, dead and you can’t draw
a line to get to this one. *angry noises* Explain why I lost you must! Alright, the first player can always lose
a two dot game against a perfect opponent because, even though it’s complex — your
brain can analyze two dot Sprouts. I mean, you could literally just memorize
this whole game tree chart to make exactly the right moves as player two, rendering player
one helpless. Player 2 can engineer the two-dot game so
that it ends on a 4th move win for them — but Conway and Paterson figured out when the game
has to end. Check it out. They discovered that a game of Sprouts must
be completed by 3n – 1 moves, where n=the number of starting dots. So that means a two-dot game is concluded
in no more than 5 moves because (3*2) – 1=5. So problem solved, right? No. Why? Because the game can play out in many different
ways. What’s interesting is that player 1 actually
has 11 ways of winning compared to player 2 having only 6. It’s just that if player 2 knows exactly
what they’re doing they can always facilitate one of their 6 winning outcomes. What’s amazing to me about Sprouts is…
this is all with just two dots! As soon as we add a third dot to the game… Become more difficult to analyze than Tic-Tac-Toe
it does! Adding a third dot at the beginning means
that we could have up to 8 moves to determine a winner since (3*3) – 1=8, but we have
more possible moves to start. It isn’t hard to figure out how many possibilities
we begin with — it’s just [n(n + 1)] / 2. So here we have our number of dots at start
and number of initial possible moves. [n(n + 1)] / 2. And number of moves to determine a winner
that’s 3n -1. So if we have 2 dots to start the game, the
initial possible moves would be 3. With 3 dots to start that jumps to 6. For 4, it’s 10. For 5 it’s 15. And so on. Now that we know this, what’s the guaranteed
strategy for winning every time? There isn’t one. Because since the game can develop in so many
different ways, especially once you start playing with 4 or 5 dots, players will have
to constantly re-analyze and adapt their moves to force their opponent into a loss. You need to factor in which dots are still
live and which ones are dead. You need to force your opponent into bad moves
— and eventually no moves at all. There’s just no formula for this. Adapt and overcome, you must! What we do know — kind of — is who can win. The first real glimpse into dominant Sproutology
came from Denis Mollison, a Professor of Applied Probability at Heriot-Watt University. Conway bet Mollison 10 shillings — before
the 1971 decimalization of the British monetary system and equivalent to a little under $10
today — that he couldn’t complete a full analysis of a 6-dot Sprouts game within a
month. Well, he did. And it only took 47 pages. I’m not looking forward to picking those up. Mollison’s analysis led to the conclusion
that Sprouts games with 0, 1, or 2 dots could always be
won by the second player. Games with 3, 4, and 5 dots could always be
won by the first player. The second player can always win with 6 dots,
but that’s where the computational power of the human mind started to strain under
the weight of the Sprout. There were just too many scenarios to compute. WAIT — how can you have a game with 0 dots? Well, if there are zero dots, the first player
wouldn’t be able to draw a line, so the second player wins. One thing that’s really weird about Sprouts
is… you’d think that playing the game would visually result in nothing but near-random
lines and patterns but Conway and Mollison unearthed something: bugs. They call this.. FTOZOM! The Fundamental Theorem of Zeroth Order Moribundity,
which states that any Sprouts game of n dots must last at least 2n moves, and if it lasts
exactly 2n moves, the final board will consist of one of five insect patterns: louse, beetle,
cockroach, earwig, and scorpion, surrounded by any number of lice. Scorpions are arachnids, not insects, but
these guys don’t have time for biology. And that’s the FTOZOM for you. But this was all 50 years ago. How has Sproutology progressed since? Well, it lay dormant for decades until Carnegie
Mellon University fired up its computers in 1990. Using some of the most advanced processors
of the era, computer scientists David Applegate, Guy Jacobson, and Daniel Sleator were able
to map Sprouts conclusively up to 11 dots. They found the same pattern: 6, 7 and 8 favored
the second player. 9, 10 and 11 favored the first player. There appears to be an endless 3-loss-3-win
pattern with a cycle length of 6 dots. In 2001, Focardi and Luccio published “A
New Analysis Technique for the Sprouts Game” that showed a simpler proof of Sprouts to
7 dots by hand. Now we’re up to 11. So, we’re making progress on the pencil
and paper front. But what about…1,272 dots? Or a billion dots? We’re not even close. Like really… not close. Julien Lemoine and Simon Viennot created a
computer program called GLOP that could calculate Sprouts results more efficiently, and in 2011
they were only able to process up to 44 dots consecutively. Their results were in line with Carnegie Mellon’s
cycle of 6, but the computational power — and time — required to get us to proving results
with, say, a million dots, is way beyond our reach. It’s been over half a century since Conway
and Paterson were drinking tea in the Cambridge math department’s common room and playing
around with inventing a simple pencil and paper-based game. They noticed that the game was spreading throughout
the department and then the campus, seeing students hunched over tables and spotting
the discarded remnants of epic Sprouts battles. They stumbled on something so big and so complex
that the human mind can’t fully fathom it beyond a very limited point — and it all
started by just connecting a couple of dots. And as always — thanks for watching. Mmm, mmm. Perfect! What are you doing to my phone? Oh, great! Listen! If you use unsecured public wifi like at a
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