# 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!

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coffee shop or airport then your phone is

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# How do I change the wallpaper back?

# Yoda...