[Daily puzzle] Eye-land problem \ stacker news ~science

520 sats \ 13 comments \ @south_korea_ln 11 Nov science

On an island, there are 100 people with blue eyes and 100 people with brown eyes. They don’t know their own eye color, but they can see everyone else's eyes. The islanders follow these strict rules:

If you know you have blue eyes, you must leave the island at midnight.
No one can communicate their eye color to others.
If anyone deduces their own eye color, they must leave that night.

One day, a visitor announces, "At least one person on this island has blue eyes."

What happens?

Previous iteration: #761484 (answer in #761856).

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20 sats \ 0 replies \ @IamSINGLE 12 Nov

This puzzle is interesting. The math puzzle is not for everyone.

I think the visitor will leave the island on that night because he's the one who talked about eyes.

300 sats \ 0 replies \ @SimpleStacker 11 Nov

Assuming that the people know that they are either blue or brown eyed, but not which, then we can solve this by induction.

If there was 1 blue-eyed and 199 brown eyed people, the blue eyed person would leave on the day of the announcement, and everyone else would leave the next day.
- This is because the blue-eyed person immediately sees that everyone else has brown eyes so he must have blue, so he leaves as soon as he can.
- The rest of the brown-eyes don't know immediately whether they had brown or blue eyes, but after they see that the blue-eyed person left, they realize they must have brown, so they can also leave. This happens on the second day after the announcement.
If there were 2 blues and 198 browns, the blue eyes would leave on the second day and the brown eyes on the third day.
- On day 1, the two blue eyes know that there's one other blue but they don't know their own color.
- However, they know that if they weren't blue, the one blue-eye would leave on day 1.
- However, since no one leaves on day 1, they realize they must also have blue, so they both leave on day 2.
- Seeing this on day 3, the browns realize they cannot have blue eyes and they all leave as well.
Using the same logic, we can predict that the blue eyes will all leave on day 100 (with day 1 being the day of the announcement), and the brown eyes will leave on day 101.

This is an interesting puzzle because it seems to show that a statement containing no new information (everyone knows at least one person has blue eyes) can lead to a change in outcomes.

However, the statement does contain new information in one hypothetical state of the world (the state of 1 blue 199 brown.) So, information about hypothetical, non-actualized states of the world can actually impact outcomes in the real world. Fascinating.

101 sats \ 0 replies \ @Cotton 11 Nov

After 100 nights, all 100 blue-eyed people will realize they have blue eyes and leave the island together. :/

0 sats \ 0 replies \ @gatto 11 Nov

as soon as a person with blue eyes counts 100 people with brown eyes, he/she will leave the island. So in the island there won't be anymore people with blue eyes. But when the visitor arrives and announces the false thing (there is someone with blue eyes) the people with brown eyes will be forced to believe someone with blue eyes is hiding, so they will continue to live in the island.

0 sats \ 6 replies \ @0xbitcoiner 11 Nov

Do people know that 100 + 100 people are on the island? If they all get together in one place, everyone can tell what color their eyes are.

0 sats \ 5 replies \ @senf 11 Nov

They know there are 200 people on the island, and each person knows the eye color of 199 of them.

0 sats \ 4 replies \ @south_korea_ln OP 11 Nov

That's right

0 sats \ 3 replies \ @0xbitcoiner 11 Nov

I must not be understanding something! When the visitor goes there, no one is there because everyone has left.

0 sats \ 2 replies \ @Scroogey 11 Nov

They do NOT know about the 100:100 ratio (only we do). Furthermore, they do not know that the only eye colors present are blue and brown. Each one must consider the possibility that he/she has an arbitrary color (e.g. green).

0 sats \ 1 reply \ @0xbitcoiner 11 Nov

ah! ok. If that's the case, I can only see one possibility: using water as a mirror! Ahhah

0 sats \ 0 replies \ @senf 11 Nov

Think about the case of just one person having blue eyes. What would they know and do?

0 sats \ 1 reply \ @Scroogey 11 Nov

0 sats \ 0 replies \ @south_korea_ln OP 11 Nov

I know that he knows that i know...