## Sunday, February 7, 2010

### The red-eyed monks

Time for more classic puzzles!

There is an island whose only inhabitants are perfectly logical monks with strange practices. Every monk has either red eyes or brown eyes. Red eyes are considered a curse, and any monk who discovers that he has red eyes must kill himself next time it is midnight.

However, the monks do not have any way to discover if they have red eyes. There are no reflective surfaces on the island. While monks can see each other's eyes, they have taken a vow of silence and are not allowed to communicate with each other.

One day, a tourist came by and loudly noted that at least one monk on the island had red eyes. At first, all the monks just silently regarded this fact, but many nights later there were a bunch of suicides. What happened and how?

Bonus question: How do the writers of these puzzles come up with such disturbing situations?

See the solution

schrodingasdawg said...

If there is only one monk with red eyes, then he sees all the others are brown-eyed, so he must be the red-eyed one. He kills himself the first night.

If there are two monks with red eyes, then each sees one monk with red eyes and reasons that if this other monk is the only monk with red eyes, he will kill himself the first night. Neither monk kills himself the first night, so they each reason that they must have red eyes too. Both kill themselves the second night.

If there are three, each expects the other two to commit suicide the second night. This doesn't happen, so each deducts that he must be a third, and the suicides happen the third night. Extends to four, five, etc.

If the suicides happened n midnights after the tourist's remark, then there are n monks with red eyes.

I haven't figured out the bonus question yet.

miller said...

Correct answer, and very nice explanation.

I always thought this problem was very counterintuitive, since all the monks already know, without the tourist's help, that there is at least one monk on the island with red eyes.

Secret SquÃ¯rrel said...

Miller, there is something very subtle going on here as to why the process of red-eyed monks deducing that they themselves have red eyes doesn't start spontaneously.

It's clear to see that if there were only one red-eyed monk, he/she would be blissfully unaware until the pronouncement that there is at least one red-eyed monk.

If there were two red-eyed monks, each would think that the other was blissfully unaware of their red-eyed status, since they would each assume that the other couldn't see any red eyes which is the same case as in the previous paragraph.

If there were three red-eyed monks, each one would think that the other two were unable to discern their own red-eyedness, because they were assuming that that there was only one red-eyed monk (A would think that B would think that C was the only red-eyed monk AND that C would think that B was the only red-eyed monk; B and C would think similarly about the others).

As with the original puzzle, this can be extended to how ever many red-eyed monks there actually are.

Anonymous said...

Here's a simple version of this classic puzzle (from Littlewood’s Miscellany):

Three ladies, A, B, C in a railway carriage all have dirty faces and are all laughing. It suddenly flashes on A: why doesn’t B realize C is laughing at her? – Heavens! I must be laughable.

(Formally: If I, A, am not laughable, B will be arguing: If I, B, not laughable, C has nothing to laugh at. Since B doesn’t argue, I, A, must be laughable.)

miller said...

I often hear that problem posed as "three wise men". They wake up with markings on their faces, and start laughing at each other. But then they stop when they realize that they too have markings on their own face.

Of course, the problem with that puzzle is that you'd think wise men would be able to laugh at themselves. :)

Sam said...

They live on an island and are therefore surrounded by water, why don't they look at their reflection?

khaley4 said...

OK, but here's the paradox: Suppose 10 monks have red eyes. Every monk has seen at least 9 monks with red eyes. So, it's not exactly earth-shattering news when the visitor announces that one monk has red eyes. Everyone already knew that! So why should they behave any differently than before, since no new knowledge was gained? (BTW, I have an answer, but I want to see others' comments on this.)

miller said...

All of them are red eyed.......
if there was only one red eyed monk ,on the first night he already commites suicide given that he can see the other monks eye.

if there were two or more red eyed monks no suicide would take place; red eyed monks would easily assume that the other monks( red eyed) dont know that they have red eyes.

but if all of the monks were red eyed, each would think that everyone has red eyes except their own.thats y at first night no one commites suicide.and slowly each of them would realize that they also have red eyes.....

miller said...

I've waited 3 years, can you give me the answer yet?

miller said...

See SecretSquirrel's answer above. The tourist adds the extra degree of freedom necessary to change the outcome.