Dot Game

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

all right 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 could draw

out

all the possible moves for a game this

simple

look i'll show you

[Music]

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 sprout

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 you 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

three lines coming from it or going to

it

once a dot has three lines it's an

unplayable

dead dot the winner of sprouse 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

all right yoda dude okay hang on all

right fine

just go right all right 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

sprouts was created in 1967 by cambridge

mathematicians john conway and michael

martin alternators

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 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

[Music]

explain why i lost

you must

all right 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 move as player 2

rendering player 1 helpless player 2 can

engineer the 2 dot game so that it ends

on a 4th move win for them

but conway and patterson figured out

when the game

has to end check it out they discovered

that a game of sprouts

must be completed by three n minus one

moves

where n equals the number of starting

dots so that means

a two dot game is concluded in no more

than

five moves because three times two minus

one

equals five so problem solved right

no why because the game can play out in

so 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 six winning outcomes

what's amazing to me about sprouts is

this is all

just with 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 eight

moves to determine a winner

since three times three minus one equals

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 times n plus 1 over 2.

so here we have our number of dots at

start and number of initial possible

moves

n times n plus one over two and number

of moves to determine

a winner that's three n minus one so if

you have two dots to start the game

the initial possible moves would be

three with three dots to start that

jumps to

six for four it's ten for five it's

fifteen

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

four or five 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 dennis mollison a professor of

applied probability

at harriet watt university conway bet

mollison

10 shillings before the 1971

decimalization of the british monetary

system

and equivalent to a little under 10

bucks today that he couldn't complete

a full analysis of a six sprout

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 zero

one or two dots could always be won by

the second player

games with three four and five dots

could always be won by the first player

the second player can always win with

six 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 zero

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

photosem the fundamental theorem of

zeroth order

morabundity which states that any

sprouts game of

n dots must last at least two and moves

and

if it lasts exactly 2n moves

the final board will consist of one of

five

insect patterns laos beetle

cockroach earwig and scorpion surrounded

by

any number of lice scorpions are

arachnids not insects but

these guys don't have no time for no

biology and that's

fatasm 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 slater 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

three loss three win pattern with a

cycle length

of six dots in 2001

ficardi and luccio published a new

analysis technique for the sprouts game

that showed a simpler proof

of sprouts to seven dots by hand

now we're up to eleven so we're making

progress on the

pencil and paper front but what about

1 272 dots or

a billion dots were not even close

like really not close

julianne lamont and simone vienno

created a computer program

called glop that could calculate

sprout's 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 patterson 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