WEBVTT

00:00:10.714 --> 00:00:13.842
You and nine other individuals 
have been captured

00:00:13.842 --> 00:00:17.495
by super intelligent alien overlords.

00:00:17.495 --> 00:00:20.807
The aliens think humans look quite tasty,

00:00:20.807 --> 00:00:26.133
but their civilization forbids eating
highly logical and cooperative beings.

00:00:26.133 --> 00:00:29.831
Unfortunately, they're not sure 
whether you qualify,

00:00:29.831 --> 00:00:32.753
so they decide to give you all a test.

00:00:32.753 --> 00:00:34.398
Through its universal translator,

00:00:34.398 --> 00:00:37.962
the alien guarding you 
tells you the following:

00:00:37.962 --> 00:00:43.423
You will be placed in a single-file line
facing forward in size order

00:00:43.423 --> 00:00:47.344
so that each of you can see 
everyone lined up ahead of you.

00:00:47.344 --> 00:00:51.073
You will not be able to look behind you
or step out of line.

00:00:51.073 --> 00:00:55.304
Each of you will have either a black
or a white hat on your head

00:00:55.304 --> 00:00:57.238
assigned randomly,

00:00:57.238 --> 00:01:01.255
and I won't tell you 
how many of each color there are.

00:01:01.255 --> 00:01:05.074
When I say to begin, each of you must
guess the color of your hat

00:01:05.074 --> 00:01:09.474
starting with the person in the back
and moving up the line.

00:01:09.474 --> 00:01:12.522
And don't even try saying words
other than black or white

00:01:12.522 --> 00:01:16.987
or signaling some other way,
like intonation or volume;

00:01:16.987 --> 00:01:19.749
you'll all be eaten immediately.

00:01:19.749 --> 00:01:24.117
If at least nine of you guess correctly,
you'll all be spared.

00:01:24.117 --> 00:01:27.243
You have five minutes to discuss
and come up with a plan,

00:01:27.243 --> 00:01:31.991
and then I'll line you up,
assign your hats, and we'll begin.

00:01:31.991 --> 00:01:36.042
Can you think of a strategy guaranteed
to save everyone?

00:01:36.042 --> 00:01:39.137
Pause the video now 
to figure it out for yourself.

00:01:39.137 --> 00:01:40.154
Answer in: 3

00:01:40.154 --> 00:01:41.246
Answer in: 2

00:01:41.246 --> 00:01:43.111
Answer in: 1

00:01:43.111 --> 00:01:45.940
The key is that the person 
at the back of the line

00:01:45.940 --> 00:01:50.494
who can see everyone else's hats
can use the words "black" or "white"

00:01:50.494 --> 00:01:53.872
to communicate some coded information.

00:01:53.872 --> 00:01:56.540
So what meaning can be 
assigned to those words

00:01:56.540 --> 00:02:00.742
that will allow everyone else 
to deduce their hat colors?

00:02:00.742 --> 00:02:04.109
It can't be the total number
of black or white hats.

00:02:04.109 --> 00:02:06.872
There are more than two possible values,

00:02:06.872 --> 00:02:11.888
but what does have two possible values
is that number's parity,

00:02:11.888 --> 00:02:15.192
that is whether it's odd or even.

00:02:15.192 --> 00:02:19.008
So the solution is to agree
that whoever goes first will,

00:02:19.008 --> 00:02:23.355
for example, say "black" if he sees
an odd number of black hats

00:02:23.355 --> 00:02:27.787
and "white" if he sees 
an even number of black hats.

00:02:27.787 --> 00:02:32.411
Let's see how it would play out
if the hats were distributed like this.

00:02:32.411 --> 00:02:35.429
The tallest captive sees three black
hats in front of him,

00:02:35.429 --> 00:02:40.634
so he says "black," telling everyone else
he sees an odd number of black hats.

00:02:40.634 --> 00:02:44.006
He gets his own hat color wrong,
but that's okay

00:02:44.006 --> 00:02:48.286
since you're collectively allowed 
to have one wrong answer.

00:02:48.286 --> 00:02:51.040
Prisoner two also sees an odd
number of black hats,

00:02:51.040 --> 00:02:55.251
so she knows hers is white,
and answers correctly.

00:02:55.251 --> 00:02:58.044
Prisoner three sees 
an even number of black hats,

00:02:58.044 --> 00:03:01.256
so he knows that his must be 
one of the black hats

00:03:01.256 --> 00:03:03.601
the first two prisoners saw.

00:03:03.601 --> 00:03:05.337
Prisoner four hears that and knows

00:03:05.337 --> 00:03:08.578
that she should be looking for 
an even number of black hats

00:03:08.578 --> 00:03:10.510
since one was behind her.

00:03:10.510 --> 00:03:16.165
But she only sees one, so she deduces
that her hat is also black.

00:03:16.165 --> 00:03:20.499
Prisoners five through nine are each
looking for an odd number of black hats,

00:03:20.499 --> 00:03:25.227
which they see, so they figure out
that their hats are white.

00:03:25.227 --> 00:03:29.054
Now it all comes down to you
at the front of the line.

00:03:29.054 --> 00:03:32.235
If the ninth prisoner saw 
an odd number of black hats,

00:03:32.235 --> 00:03:35.026
that can only mean one thing.

00:03:35.026 --> 00:03:39.398
You'll find that this strategy works
for any possible arrangement of the hats.

00:03:39.398 --> 00:03:44.512
The first prisoner has a 50% chance of
giving a wrong answer about his own hat,

00:03:44.512 --> 00:03:46.821
but the parity information he conveys

00:03:46.821 --> 00:03:52.020
allows everyone else 
to guess theirs with absolute certainty.

00:03:52.020 --> 00:03:56.035
Each begins by expecting to see an odd
or even number of hats

00:03:56.035 --> 00:03:58.066
of the specified color.

00:03:58.066 --> 00:04:02.502
If what they count doesn't match,
that means their own hat is that color.

00:04:02.502 --> 00:04:04.136
And everytime this happens,

00:04:04.136 --> 00:04:08.883
the next person in line will switch
the parity they expect to see.

00:04:08.883 --> 00:04:10.624
So that's it, you're free to go.

00:04:10.624 --> 00:04:13.032
It looks like these aliens 
will have to go hungry,

00:04:13.032 --> 00:04:16.461
or find some less logical 
organisms to abduct.