Hey, this is Presh Talwalkar.
Here's a fun logic puzzle. Four friends
claim they had been wrongly imprisoned and they are just mere logicians,
and they have been confused for criminals.
The warden says if you are in fact
logicians, I'm going to give you a chance to prove it.
So he lines up the four logicians as follows.
We have 1, 2, 3 all facing forward, then a brick wall.
And then on the other side of the brick wall is logician 4.
The warden says, “I'm going to place
2 blue hats and 2 red hats on the 4 of you.”
They all know these rules.
They're only allowed to look in front of them.
There's no cheating.
They're not allowed to take their hats off.
They're not allowed to talk to each other.
And they're not allowed to do anything else.
They have to figure out the answer using pure logical reasoning.
The warden says, “I’m going to give you some time.
If any one of you can state the color of your hat with certainty,
I will let you all out for free.”
Being perfect logicians, none of them will speak out of turn and just guess.
They will only speak if they know the answer for sure.
So the question is can the four logicians be freed?
And how would they do it?
Pause the video if you'd like to give this problem a try.
And when you're ready, keep watching to learn
how to solve this problem.
So how can we work it out?
First, logician 3 only looks at a brick wall.
So there's no information that the third person could have.
There's no way this person could shout out with certainty.
The same thing is true for logician 4.
The logician only sees a brick wall and doesn't have any information.
So 4 is not going to shout out any answers.
Now, each of them is thinking
there are 2 blue hats and 2 red hats.
So there are, of course, 4 choose 2, which is 4 factorial over 2 factorial,
2 factorial, equals 6 different ways for the warden to arrange
2 blue hats and 2 red hats on the 4 of them.
So they start thinking, what are the possibilities?
It could be red, red, blue, blue, red, blue, red, blue,
red, blue, blue, red, blue, blue, red, red,
blue, red, blue, red or blue, red, red, blue.
Now logician 2 sees
a red hat on the third person.
So logician 2 looks at which possibilities have a blue
hat on the third option and says these cannot be the possible arrangements.
So we are down to 3 possibilities.
Now, logician 1 is also thinking the same thing
because logician 1 can see the hats on logicians 2 and 3.
So logician 1 not only knows they're down to these 3 possibilities,
but logician 1 also knows that there's a blue hat and a red hat.
So that leaves only 2 possibilities
But still in each of these possibilities,
logician 1 says, I don't know if I'm wearing a red hat or blue hat
because either one is possible in these two possibilities,
And that's the clue!
So after a while, logician 2
knows that logician 1 is confused.
Now, if logician 1 had seen
two red hats, that is red hats on 2 and 3,
logician 1 would know that his hat was blue.
But since logician 1 is confused,
logician 2 says this possibility cannot be true.
So we are down to 2 possibilities
And logician 2 thinks in each of these possibilities.
There is a blue hat in the second position.
Therefore, logician 2 says, I know my hat is blue!
And all 4 of the logicians get set free.
What an incredible logic puzzle!
Thanks for making us one of the best communities on YouTube.
See you next episode of Mind Your Decisions while we solve the world's
problems, one video at a time.