# The Demonetization Game

Vsauce!

Kevin here with 100 YouTubers.

Some are friends, some are colleagues — all

are about to be demonetized forever.

Unless… each one chooses their own channel

out of these 100 eggs.

I’ll put one YouTuber in each egg and every

player gets only 50 guesses.

If just one of the 100 YouTubers fails to

find their channel in the egg, everyone loses.

Here’s the setup.

All 100 YouTubers are in the same waiting

room, they’re each assigned a number and

then… one at a time, brought into the egg

room to play the game alone.

Once a YouTuber completes their round, they’re

immediately escorted out the back door so

they can’t go back to the waiting room and

say, like “hey Rhett & Link, you’re hidden

in Egg #93!” or whatever.

No.

No.

That’s cheating.

This game is 50 blind guesses for each YouTuber,

and they all have to find their egg or everyone

is demonetized.

After each round, the eggs are all reset and

re-sealed so that the next YouTuber can’t

see which has been chosen.

Every player has to start fresh.

So, how do you win when you can’t see what’s

in the eggs and you can’t communicate with

any other players?

The most obvious option is for everyone to

just choose randomly and hope for the best.

I mean, you have 50 tries — that’s a lot,

right?

50 out of a 100 eggs gives you 50/50 odds

of finding your egg.

Not bad!

But…

The odds of all 100 YouTubers pulling this

off successfully are virtually impossible

— let’s crunch the numbers.

It’s pretty simple: with 100 eggs and 50

guesses, the odds of selecting the right egg

randomly are 50% — or .5.

One hundred successful attempts at odds of

.5 comes out to…

0.0000000000000000000000000000008.

To get the percentage odds, we just shift

the decimal over two places… and uh, yeah,

there are still 28 zeros.

The odds of me getting drafted in the NBA

or winning the lottery are unfathomably higher

than this demonetization egg game working

out for these YouTubers.

These odds are literally the same as correctly

predicting 100 coin flips in a row.

Try it.

Try to successively predict coin flips.

You may get 5 or 6 in a row right, maybe 10!

But you almost certainly won’t get 50 — let

alone the 100 we need to save our YouTubers.

So sorry, 3Blue1Brown.

And sorry, EpicRapBattles.

There’s just no hope of winning this game.

Your channels, like everyone else’s here,

are cancelled.

Unless…

What if there was a mathematical strategy

that our 100 YouTubers could employ to give

their channels a realistic chance of survival?

Let’s actually play this game and try to

figure it out.

I’ve given each of these YouTubers a number

between 1 and 100, and each one will randomly

go in a numbered egg.

Now we’ve got 100 YouTubers hidden in 100

eggs.

And the secret solution to saving their channels

goes like this…

When it’s their turn to play, each YouTuber

just needs to start by choosing the numbered

egg that matches their personally assigned

number.

If they aren’t in that egg, rather than

choosing another egg randomly, the next egg

to be opened is the number of the YouTuber

inside the one that they just opened.

Each player will repeat this strategy until

they find their egg or until they’ve opened

50 eggs unsuccessfully — in which case everyone

loses.

MaxMoeFoe is player #1, so I will play as

MaxMoeFoe.

Here we go.

Okay, Egg #1 is…

Gus Johnson.

Who’s #62, so we’ll just open up Egg #62.

And?

PhysicsGirl.

And she is #12, so now we just open up Egg

#12…

Nerd City.

#77, so let’s open up Egg #77.

Numberphile, #35.

Okay, let’s check out Egg #35.

And here we have Carson, number #69 — this

is very, very important.

Whatever you do, do not forget 69.

We’ll talk about that later.

But for now let’s open up Egg #69.

AspectScience #7.

Y’know?

This might actually take a little while.

So while I find the MaxMoeFoe channel, let’s

montage.

#20?

#20.

Is Max!

Is me!

MaxMoeFoe!

We did it!

We found, me!

Well, we found Max.

And it only took 31 guesses.

Way below our 50 guess threshold.

Now we need to reset our eggs.

Okay, we’re all set for YouTuber #2 — RedLetterMedia.

Let’s do this.

RedLetterMedia!

I found it already!

That didn’t take long at all.

Okay, now let’s reset and go with YouTuber

#3.

Okay, now It is time for Danny Gonzalez.

Here we go.

Danny Gonzalez!

#3!

Did it again!

Another winner!

That is awesome.

Okay, I am going to reset this and then let’s

talk about what just happened.

Okay, our strategy is working — three YouTubers

have found their eggs.

The odds of the first three YouTubers choosing

successfully with random guessing would have

been .5 x .5 x .5 which equals 12.5%.

Four in a row drops down to 6.25%.

Five in a row… you’re looking at 3.125%

and by seven in a row you’re below 1%.

So how much better is our system?

Well, remember random guessing has the odds

of all 100 YouTubers winning at 0.000 28 zeros

and then an 8 percent..

To put it in perspective, our system increases

the odds more powerfully than turning a penny

into all the money in the world.

We can increase the odds of all saving all

100 YouTubers from effectively 0% to 31%.

And I’ll explain how it works but, here’s

a question.

Did I just make this whole thing up?

No.

Usually, I like to examine old, longstanding

problems in math and science, but this one

is actually pretty new.

In 2003, computer scientists Peter Bro Miltersen

and Anna Gál, in a paper titled, The Cell

Probe Complexity of Succinct Data Structures,

created a version of this game in which red

or blue slips were dropped in 100 little drawers.

To win, every single one of the 100 players

had to correctly guess whether their drawer

contained a red or a blue slip.

The game is a little different, but the odds

of guessing red or blue vs. choosing 50 eggs

out of 100 are exactly the same.

Miltersen’s initial thought was that as

the number of players increases, the probability

of winning the game would trend to zero.

But by starting with the egg matching their

own number, each player is guaranteed that

they’re on a path to an egg that contains

them, and it’s just a question of whether

their egg is within a cycle of 50 attempts.

Okay.

Why?

Because each egg contains one unique number

that points to another unique number and that

creates, as Nick Berry of Datagenetics described,

a circular chain.

One number points into the chain and one number

points out.

So if the YouTuber uses their own number to

point into the chain, they will eventually

find their own number.

Rather than blindly guessing, our system is

a way of tapping into that circular chain.

To get math-y about it, by giving each YouTuber

a number, we’ve created a permutation of

the set that’s a one-to-one mapping of all

100 numbers to itself.

Our strategy makes a cycle so each number

returns to itself, and it’s successful when

all 100 YouTubers find a cycle length of 50

or less.

With 100 boxes, there can only, mathematically,

be one cycle longer than 50.

But that’s all it takes for us to lose the

game, anyway.

So what are the chances of a permutation in

which there’s one cycle longer than 50?

Let’s find out.

The solution to our chance of success equals

1 minus the probability of getting a cycle

longer than 50.

Let’s call the cycle length L. That gives

us 100 choose L possible sets, times L!/L

permutations of the cycle within the set,

times (100 – L)! permutations of the remaining

YouTubers… and that comes out to 100!/L,

which is really, really easy to work with.

We know that there are 100! ways to arrange

the tickets, so the probability of chain length

L is just…

1/L.

Our chance of success is 1 minus all those

long-chain failures combined.

Since we also know that only 1 cycle of 51

or longer is possible in each set, we can

calculate the probability of losing by 1 – (1/51

+ 1/52 + 1/53…

+ 1/98 + 1/99 + 1/100), which comes out to

about .69.

I told you to remember 69.

There’s a 69% chance that we encounter a

cycle longer than 50.

So, the probability of all 100 YouTubers having

a cycle of 50 or under — which means they’d

win the demonetization game using this strategy

— is the remaining .31, or 31%.

31% isn’t amazing, but just like The Game

of Googol, around 30% is waaaay better than…

basically zero.

The weird thing is that even with our solution,

each individual YouTuber’s chance of success

is still 50% just like if they’d guessed

randomly.

The strategy changes the group’s chance

of success.

So, that’s that.

Problem solved.

Until… you tweak the problem.

Miltersen’s game continues to evolve.

Like… what happens if we have more than

100 eggs but some of them are empty?

Reaching an empty egg would stop the cycle

and completely ruin our system.

But even this seemingly impossible version

of the game has inspired computer scientists

to develop a strategy that defies our expectations

of failure.

Which is what we do.

Whether we’re pretending the floor is lava,

whether we’re watching superheroes defeat

galaxy-destroying villains, or we’re trying

to cure diseases that seem impossible to overcome.

We test, push, and expand our limits because

challenges are only impossible until they

aren’t.

And as always, thanks for watching.

That makes a weird sound.

Hey there!

If you haven’t yet taken advantage of my deal

with CuriosityStream, you better do that.

There’s a link down below for you to click

and you get 30 days for free.

Just go to CuriosityStream.com/Vsauce2.

And fill your brain with amazing education,

and stimulating in the mind content.

You will absolutely love it.

If you want to love yourself more Vsauce2,

first of all you should.

You should subscribe.

I don’t know what happened to my voice here.

But, y’know, the battery is dying on this

computer, it’s not a computer, it’s a camera.

Okay, this is going south really quickly.

Subscribe to Vsauce2, ring the notification

bell so you know when I upload a new video.

And uh, I’m just going to.

Uh.

I guess see ya later, bye.