The Dot Game That Breaks Your Brain

Vsauce! Kevin here, with a game you can’t possiblycomprehend.

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 forsponsoring this video and supporting Vsauce2.

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I'll explain more later but first let's explainour dots.

Alright, as you stare into these dots yourbrain 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 fora 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, analyzingwhat appears to be the simplest game in the world doesn’t just break your brain, computerscan’t even crunch the possibilities.

Here’s how it works.

The game of Sprouts starts with any numberof 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, youneed 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 dotto 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 newdot anywhere on that new line.

And in Sprouts, no dot can have more than3 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 todraw a line.

Or to put it another way, the player who can’tdraw another line loses.

Okay, now my friend and I will play a two-dotgame 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 onthe line.

Yes, yes, yes.

Invented this game, I did! Sprouts trained many Jedi minds, hundredsof years! Hundreds of years? No, No, No.

Sprouts was created in 1967 by Cambridge mathematiciansJohn 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.

.

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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 drawa line to get to this one.

*angry noises* Explain why I lost you must! Alright, the first player can always losea two dot game against a perfect opponent because, even though it’s complex — yourbrain can analyze two dot Sprouts.

I mean, you could literally just memorizethis whole game tree chart to make exactly the right moves as player two, rendering playerone helpless.

Player 2 can engineer the two-dot game sothat it ends on a 4th move win for them — but Conway and Paterson figured out when the gamehas to end.

Check it out.

They discovered that a game of Sprouts mustbe completed by 3n – 1 moves, where n = the number of starting dots.

So that means a two-dot game is concludedin no more than 5 moves because (3*2) – 1 = 5.

So problem solved, right? No.

Why? Because the game can play out in many differentways.

What’s interesting is that player 1 actuallyhas 11 ways of winning compared to player 2 having only 6.

It’s just that if player 2 knows exactlywhat 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-Toeit does! Adding a third dot at the beginning meansthat we could have up to 8 moves to determine a winner since (3*3) – 1 = 8, but we havemore possible moves to start.

It isn’t hard to figure out how many possibilitieswe begin with — it’s just [n(n + 1)] / 2.

So here we have our number of dots at startand number of initial possible moves.

[n(n + 1)] / 2.

And number of moves to determine a winnerthat's 3n -1.

So if we have 2 dots to start the game, theinitial 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 guaranteedstrategy for winning every time? There isn’t one.

Because since the game can develop in so manydifferent ways, especially once you start playing with 4 or 5 dots, players will haveto constantly re-analyze and adapt their moves to force their opponent into a loss.

You need to factor in which dots are stilllive 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 Sproutologycame from Denis Mollison, a Professor of Applied Probability at Heriot-Watt University.

Conway bet Mollison 10 shillings — beforethe 1971 decimalization of the British monetary system and equivalent to a little under $10today — that he couldn’t complete a full analysis of a 6-dot Sprouts game within amonth.

Well, he did.

And it only took 47 pages.

I'm not looking forward to picking those up.

Mollison’s analysis led to the conclusionthat Sprouts games with 0, 1, or 2 dots could always bewon by the second player.

Games with 3, 4, and 5 dots could always bewon 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 underthe 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 playerwouldn't be able to draw a line, so the second player wins.

One thing that’s really weird about Sproutsis… you’d think that playing the game would visually result in nothing but near-randomlines 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 lastsexactly 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, butthese 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 CarnegieMellon University fired up its computers in 1990.

Using some of the most advanced processorsof the era, computer scientists David Applegate, Guy Jacobson, and Daniel Sleator were ableto map Sprouts conclusively up to 11 dots.

They found the same pattern: 6, 7 and 8 favoredthe second player.

9, 10 and 11 favored the first player.

There appears to be an endless 3-loss-3-winpattern with a cycle length of 6 dots.

In 2001, Focardi and Luccio published “ANew Analysis Technique for the Sprouts Game” that showed a simpler proof of Sprouts to7 dots by hand.

Now we’re up to 11.

So, we’re making progress on the penciland 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 acomputer program called GLOP that could calculate Sprouts results more efficiently, and in 2011they were only able to process up to 44 dots consecutively.

Their results were in line with Carnegie Mellon’scycle of 6, but the computational power — and time — required to get us to proving resultswith, say, a million dots, is way beyond our reach.

It’s been over half a century since Conwayand Paterson were drinking tea in the Cambridge math department’s common room and playingaround with inventing a simple pencil and paper-based game.

They noticed that the game was spreading throughoutthe department and then the campus, seeing students hunched over tables and spottingthe discarded remnants of epic Sprouts battles.

They stumbled on something so big and so complexthat the human mind can’t fully fathom it beyond a very limited point — and it allstarted by just connecting a couple of dots.

And as always — thanks for watching.

Mmm, mmm.

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How do I change the wallpaper back? Yoda.

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