In a prison, every prisoner has either a red dot or a blue dot on their back. Th

ID: 3728564 • Letter: I

Question

In a prison, every prisoner has either a red dot or a blue dot on their back. They can see everybody else's dot except their own. They can't exchange information by any means, nor do they have a mirror. On day one, the director told them that those who will tell him they are wearing a blue dot at noon will be let go if it is true (else they will be executed). As it so happens, all 50 prisoners are wearing a blue dot. One afternoon, a journalist comes in and publicly asks the director, "Why are some prisoners wearing blue dots?" (publicly means that everyone can hear the question). Prove by induction that every prisoner will leave 50 days after the journalist's talk.

Explanation / Answer

Lets simplify things by imagining just two prisoners A and B.
Day 1: Each see one person with blue dot and for all they know that could be the only one for the first noon, each stays put.
Day2 : But when they see each other still there in the morning, they gain new information.

A realizes that if B had seen a non blue dot person next to him he would have left the first noon after concluding the statement could only refer to himself.
B simultaneously realizes the same thing about A.
The fact that the other person waited tells each prisoner that his dot color is blue.
And on the second noon they're both gone.

Now imagine a third prisoner C
A, B & C each see two blue dot people but not sure if each of the others is also seeing 2 blue dot people, or just one. They wait out the first night as before, but the next morning, they still can't be sure.

C thinks if I have red dot then A and B were just watching each other, and will now both leave on the second noon. But when he sees both of them the third morning, he realizes they must have been watching him too.

A and B have each been going through the same process, and they all leave on the third noon.

NOTE: Refer the video below for better understanding the concept.
https://ed.ted.com/lessons/the-famously-difficult-green-eyed-logic-puzzle-alex-gendler

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In a prison, every prisoner has either a red dot or a blue dot on their back. Th

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