Vsauce!
Kevin here, with…
100 pieces of paper.
And I’ll teach you how to hire the right plumber, choose the best parking space, and
even find the love of your life... by quitting.
Each of these 100 slips of paper has a number on it.
The numbers can be anywhere from 1 to a Googol -- which is 1 followed by 100 zeroes.
The Game of Googol is to turn them over one by one and stop when you think that you’ve
got the highest number.
Here’s the important part.
Your final flipped paper is your choice.
You can’t, like, flip five papers over and then retroactively choose the second paper
because it had the biggest number.
No, no, no.
That’s against the rules.
So the trick to winning The Game Of Googol is knowing when to stop.
Alright, let’s play.
Paper number one.
100,028.
Ok, that’s a pretty big number, but it’s not even close to a Googol.
It can’t possibly be the highest number out of all of these.
So let’s flip another.
94 trillion, 288 billion, 381 million, 109 thousand 276.
That’s huge.
But it’s far from a Googol, so I should probably keep going.
I think?
Honestly, I don’t know.
How am I supposed to choose the highest numbered slip when I don’t even know what the numbers
are?
The lowest number could be…
7, or 10,000.
The highest number could be... well, anything up to or including a Googol.
So, the odds of me picking the highest number are what, like 1% or something?
No.
The odds of me choosing correctly are actually about 1 in 3 -- because I’m going to employ
Optimal Stopping Theory, the branch of mathematics that deals with knowing when to quit.
That way you get the highest reward or the lowest cost.
The Game of Googol was invented in 1958 and it didn’t take long for mathematicians to
figure out an elegant approach.
The key is… e.
The number e. Uhh.. excuse me, papers.
I gotta talk about some e here.
Euler’s Number is an irrational number that’s roughly 2.71828, a mathematical constant that
shows up in exponential growth and even calculating
compound interest.
e is one of the most important numbers in mathematics along with 0, 1, pi, and i -- and
it’s a component of Euler’s Identity, the gold standard of mathematical beauty.
And to win The Game of Googol, we need to flip Euler upside down.
Not like that.
Like this!
The reciprocal of Euler’s number -- 1/e, or 1/2.71828, or… .367879… or approximately
37%, is the lowest our probability of winning the Game of Googol will ever be as long as
we stick to the optimal strategy.
We don’t need to know the numbers on the slips and it doesn’t matter if we’re playing
with 10 slips, 100 slips, or… 642 duodecillion slips.
We just need to answer one question...
When, exactly, mathematically, do we start stopping?
Ok, so I can just go through this solution to the Game of Googol.
Alright.
Here we go.
Yeah, lookin' good.
Mmm Hmmm.
Yup.
No problems here.
And, forget it.
We’re not gonna do that.
We’re gonna do instead is we're gonna use e and basic optimal stopping theory, to find
a quick, accurate point at which we need to stop playing to maximize our chance of winning.
Let’s solve The Game of Googol with just 10 slips of paper.
The trick is to take a sample of numbers, and then decide to commit to the very next
number we pull that’s larger than any in that previous sample.
So, with 10 slips total, do we decide to take the highest number after…what?
The third?
The 5th?
It can’t be too late in the game, because if we wait until, like, slip nine we’ll
likely have already revealed and passed over our biggest number.
Let's not guess.
Let's get Eule-y.
If we divide our number of slips, 10, by e, we get 3.67889 by rounding up to the next
integer, we can stop collecting our sample on the 4th slip.
Then we’ll keep choosing slips and stop when we find one that’s higher than any
of those first four -- and there’s around a 1/e chance that we’ll be right.
This works for 10, 100, or 3,781 slips.
Because that’s just 3,781 divided by e, so 1,390.95 -- we’d choose until we pulled
a number higher than any in the 1,391st slip.
We’ll be exactly right about a third of the time… and when we aren’t, we’d still
be really close.
Look, I’ll mix these up and I'll play a totally random game using this strategy, and
we’ll just see what happens.
So there's my sample of four slips.
And now I'll just choose the next number that's bigger than these four.
Alright, my choice is number six.
Let's see if that's the highest number here.
Oops.
So the strategy didn't work.
Six wasn't the highest number, it was actually slip number nine.
A 1 out 3 success rate of this strategy might not sound impressive.
You're still going to lose about 2 out of every 3 times.
But imagine how low your success rate would be if you had no system to help you decide...
and as the number of slips in The Game of Googol becomes larger, the likelihood that
your uninformed guess is actually right trends toward zero.
Thirty-something percent is a lot better than, say, .003%.
We, as humans, don’t calculate probabilities with Euler’s number 10 times a day, but
our brain does seem to have a general heuristic -- a series of mental shortcuts and basic
rules -- that helps us decide when to stop.
Researchers at the Laboratory of Neuropsychology at the National Institute of Mental Health
ran fMRI scans on brains that were trying to decide whether to keep searching for the
best choice or commit to the current option… and they found that several parts of the brain
light up when faced with an optimal stopping problem.
Euler’s number and optimal stopping theory is how, in a sea of possibilities, you decide,
“this is the best parking spot I’m gonna find,” or, “this is the house that I’m
gonna buy,” or, “this is the YouTube video I’m gonna watch.”
Without even realizing it, you played The Game of Googol in your brain to watch this
video about The Game of Googol.
You didn’t scroll through every single video on YouTube and go back to determine that this
was the one to click on.
You scanned some options, clicked me, and here we are.
You’re Johannes Kepler and I’m bachelorette number five.
In 1611, Kepler interviewed eleven women in the hopes of finding a wife.
He passed on eleven women.
Kepler was left alone.
He rejected all of his options and thought he'd missed his chance.
Luckily, he went back to bachelorette number 5, she said yes, and they lived happily ever
after.
Kepler found the love of his life, but he did it the hard way.
All he needed to do was divide 11 by e to get a value around 4, then taken the very
next option better than those -- which would’ve been bachelorette number 5, the love of his
life, hidden in plain math all along.
And as always, thanks for watching.