Can you solve the prisoner hat riddle? - Alex Gendler

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You and nine other individuals have been captured by super-intelligent alien overlords. The aliens think humans look quite tasty, but their civilization forbids eating highly logical and cooperative beings. Unfortunately, they’re not sure whether you qualify, so they decide to give you all a test. Can you solve this hat riddle? Alex Gendler shows how.

Lesson by Alex Gendler, animation by Artrake Studio.

You and nine other individuals
have been captured

by super intelligent alien overlords.

The aliens think humans look quite tasty,

but their civilization forbids eating
highly logical and cooperative beings.

Unfortunately, they're not sure
whether you qualify,

so they decide to give you all a test.

Through its universal translator,

the alien guarding you
tells you the following:

You will be placed in a single-file line
facing forward in size order

so that each of you can see
everyone lined up ahead of you.

You will not be able to look behind you
or step out of line.

Each of you will have either a black
or a white hat on your head

assigned randomly,

and I won't tell you
how many of each color there are.

When I say to begin, each of you must
guess the color of your hat

starting with the person in the back
and moving up the line.

And don't even try saying words
other than black or white

or signaling some other way,
like intonation or volume;

you'll all be eaten immediately.

If at least nine of you guess correctly,
you'll all be spared.

You have five minutes to discuss
and come up with a plan,

and then I'll line you up,
assign your hats, and we'll begin.

Can you think of a strategy guaranteed
to save everyone?

Pause the video now
to figure it out for yourself.

Answer in: 3

Answer in: 2

Answer in: 1

The key is that the person
at the back of the line

who can see everyone else's hats
can use the words "black" or "white"

to communicate some coded information.

So what meaning can be
assigned to those words

that will allow everyone else
to deduce their hat colors?

It can't be the total number
of black or white hats.

There are more than two possible values,

but what does have two possible values
is that number's parity,

that is whether it's odd or even.

So the solution is to agree
that whoever goes first will,

for example, say "black" if he sees
an odd number of black hats

and "white" if he sees
an even number of black hats.

Let's see how it would play out
if the hats were distributed like this.

The tallest captive sees three black
hats in front of him,

so he says "black," telling everyone else
he sees an odd number of black hats.

He gets his own hat color wrong,
but that's okay

since you're collectively allowed
to have one wrong answer.

Prisoner two also sees an odd
number of black hats,

so she knows hers is white,
and answers correctly.

Prisoner three sees
an even number of black hats,

so he knows that his must be
one of the black hats

the first two prisoners saw.

Prisoner four hears that and knows

that she should be looking for
an even number of black hats

since one was behind her.

But she only sees one, so she deduces
that her hat is also black.

Prisoners five through nine are each
looking for an odd number of black hats,

which they see, so they figure out
that their hats are white.

Now it all comes down to you
at the front of the line.

If the ninth prisoner saw
an odd number of black hats,

that can only mean one thing.

You'll find that this strategy works
for any possible arrangement of the hats.

The first prisoner has a 50% chance of
giving a wrong answer about his own hat,

but the parity information he conveys

allows everyone else
to guess theirs with absolute certainty.

Each begins by expecting to see an odd
or even number of hats

of the specified color.

If what they count doesn't match,
that means their own hat is that color.

And everytime this happens,

the next person in line will switch
the parity they expect to see.

So that's it, you're free to go.

It looks like these aliens
will have to go hungry,

or find some less logical
organisms to abduct.