Monks With Hats

Usually, after solving a logic problem, I like to take some sort of generalization of it just to complicate the issue. A typical example of that is to take a problem that only applies to natural numbers and solve it assuming its parameters can take on real values. This process isn't always very well defined, but I am usually pleased with the result anyway. Before I can start that though, I suppose I'll introduce the logic problem that triggered me to actually start this blog. I first learned this problem from Bart.

Consider a monastery with many hat-wearing monks in it. The hats can be either white or black, and the monks have taken vows not to look at their own hat or point out others hats or stuff like that (standard hat rules apply). Further, the monks have taken a vow that any monk who discovers their own hat colour is to kill themselves at midnight that night. When a monk dies this way, all the other monks are aware of it the next day. Finally, all the monks are perfectly logical and are all aware of all information in this paragraph.

Let us assume for definiteness that there are 50 monks, 25 with black hats and 25 with white hats. One day a rabbi, who does not particularly like these crazy hat-wearing monks, comes by to speak with the monks. He gathers all the monks together and announces to them, "I see that at least one of you is wearing a white hat today." The rabbi then leaves the monastery, what is the fate of the monks?

If you think the answer is "they happily live out many years of life", I should tell you the easier version of the problem where I end off the problem by saying this: Two months later the rabbi returns to the monastery, and all the monks are dead. On which day did the monks die (specifically, for each monk, on which day did that monk die)?

I won't state the solution in this post, just because some people might not want it spoiled for them, so I'll give it a day before making the next post.


Anonymous said...

I was thinking, if the 50 and 25/25 split is known to the monks, then they all die the first day. Before the Rabbi gets there.

So it is imperative that the split not be known to the monks.

Or am I just wrong?

- !Bob

kstevens said...

Yeah, if the monks know the distribution, then they all die some very long time ago (presumably when they learned the distribution). Thats why I specify the statement "are all aware of all information in this paragraph" in a separate paragraph from "25 with black hats and 25 with white hats".