First, a little announcement.

I have produced a new kindle ebook containing many of the previous Friday Puzzles!  It is called PUZZLED (available in the UK here and USA here) and contains 101 puzzles and the solutions. If you enjoy the Friday Puzzles and the blog, feel free to show your appreciation by buying one of the books.

OK, ad over. Last week’s puzzles was easy. This week’s one is much trickier and was kindly sent to me by Milan V.

Imagine a monastery in which ten of the monks may have a disease which causes them to have blue spots on their foreheads but has no other symptoms. All the monks have taken a vow of silence, they meet just once a day, and there are no mirrors in the monastery, so nobody knows whether he has a blue spot on his forehead or not.

If a monk discovers that he has a blue spot on his forehead, he will have to leave the monastery by the end of the day. All the monks are perfect logicians – that is, they can instantly infer all the logical consequences of any statement made to them – and they all know that all the other monks are perfect logicians.

One day, the Guru, who is known to be truthful, gathers all the monks together and announces “At least one monk in this monastery has a blue spot on his forehead.” Nothing happens for nine days, but on the tenth day, all the monks with blue spots leave.

How many monks left and why?

As ever, please do NOT post your answers, but do say if you have solved it and how long it took! Solution on Monday.

1. significance says:

You gave away the answer in the first line. Might want to change the number to “some”.

1. Hah! You see I read that, so didn’t think that was part of the puzzle. And I went straight to the “why” part of the question.

2. Took me just under 1 minute (first time I looked at the clock and the end of reading the question, and then again when I got the answer… it was the same time!!!!), assuming I got the right answer. I think I’m reasonably happy with my answer (and, I must admit, with myself)

3. Laura says:

A bargain! And I’m not even confident enough to buy the 0.99 cent version.

4. AMWhy says:

Got it! And no, there isn’t a mistake in the wording – think about the ‘why’ part.

1. significance says:

I know the “why” answer, having seen this one before. But the “how many” is an important part of the puzzle, too.

2. AMWhy says:

If you understand the why part, then the how many takes care of itself. Don’t want to give too much away though…

5. Ersagun says:

6. nainfort says:

The guru broke the vow of silence. He should leave the monastery!

1. Berhard says:

Is the guru part of the set of monks?
No clear answer … but i think
even under the wow of silence you still have the chance of using singn language…
(but it is a nice argument!)

7. Rob says:

Yep, as other posters have said, there’s nothing wrong with the wording of the question. I cheated and Googled, but the solution is elegant and watertight. Great puzzle.

8. Tessa K says:

This is why I am not a mathematician. I immediately go to the real world and think that they would just point at each other.

1. JohnF says:

I agree with Tessa K. That’s what I think they would do, since it’s the easy and practical answer. Not sure why they’d want to put in all that extra working-it-out stuff when there’s a much simpler and more certain way of being sure who’s been blue-spotted.
Monks, huh? You just can’t live them…

9. Superdave says:

About two minutes following a logical progression. I also assume the disease isn’t contagious, and none of the monks have a playful sense of humour and a blue pen.

1. Berhard says:

Perfect answer for an april fools day question…

2. Berhard says:

I think the desease have to be contagious as otherwise it would not be possible to have an answer after 9 days…
(i.e. in case of a non-contagious desease all infected monks schould have left but instantaneously, as logic is usual independent of time… and no new information becomes available within the 9 days…)

10. Tom says:

Yeah, this one was easy.. under a minute.

I have probably heard it somewhere before, though.

11. JimC says:

I’ve seen this before, although framed as an isolated island race instead of a monastery. First time around it took me ages to get my head around it.

Afraid I don’t own a Kindle, otherwise I would certainly invest.

12. Dawn says:

Think I’ve got it. Don’t own a kindle either, prefer the satisfaction of owning and holding a book in my hands.

13. Daniel says:

Got it in about 1 or 2 minutes … explanation works like mathematical induction. But I think ‚some’ would have been a better wording than ‚10’

1. AMWhy says:

😉

14. jerome says:

the truthful guru was the only one left alone

1. Engywuck says:

April Fools’ day!

15. Berhard says:

I think the right trace to the answer resides in turn in the questions how many spots have been there in the beginning…

done it in about 5 minutes.

17. Savant says:

A mystery concocted by Agatha Christie and Ellis Peters.

18. Great puzzle

19. Joao Pedro Afonso says:

Solved in three minutes. Apply the unbulshit transformation and gives 4 minutes (little gain, I know, but I’m basically honest…. don’t apply the UT on that).

This is a tricky puzzle but it helps that I have solved similars in the past. The trick is to consider the several possible scenarios, starting from simples ones, with the goal to explain why only in tenth day there will be monks leaving.

20. David Mathew says:

I haven’t got a scooby doo!

21. Joao Pedro Afonso says:

But I have a question for you, @Richard: why call the second book, puzzled 2? How are you going to call the book with the next 101 puzzles? Puzzled 3 and 4?

And more important, are we going to see psychological considerations about the puzzles and your commentators in the books? It would be a deception for many if not, considering all the speculation about motives there was many times in this blog about Friday column. And personally, considering how easy or how well known are many of the puzzles, I would only buy them if I would learn more about us with them… blame on the quality of your other works, if I’m expecting much more from your new book than a simple puzzle book.

22. That was the example my math teacher gave us to understand what a recursion is !

23. Savant says:

“which causes them to …. has no other symptoms” (recte: “have no other symptom”)

“there are no mirrors” (recte: “there is no mirror”)

It isn’t that hard, surely?

1. ChivNick says:

OK, I think I get your first point – the individual has the symptoms, not the disease, but what’s wrong with saying, “there are no mirrors”?

2. Savant says:

Well, it’s a question of number; plurals are used when referring to two or more and the singular to one or none. In both cases here we are dealing with none.

Thus in the first sentence, ‘symptoms’ should be ‘symptom’ and in the second, ‘are no mirrors’ should be ‘is no mirror’.

3. Chris says:

I think you’re parsing the first bit wrong. I read it as saying that the disease:
a) causes them to…
and b) has no other symptoms.
This interpretation makes complete sense and is fine grammatically.

On the second point, what on Earth is wrong with “there are no mirrors”? Both singular and plural phrasings are perfectly common and correct.

24. Marion says:

I have an answer that I got fairly quickly, however after reading all the comments I am now confused!!
Must be to early in the morning for me, I will stick with my original answer and see how I get on!

25. neil j says:

Surely the guru should write on a blackboard

1. Engywuck says:

Heard this before but needed to read the commentary to remember the logic. I think this ones was my favourite puzzle in my first puzzle book a long time ago…

26. ivan says:

There is an implicit assumption in this kind of puzzle that there are specific moments at which current information is assessed and processed to its logical conclusion, and at that point everyone stops thinking. This is necessary, because for the puzzle to work you need to know where everyone else is their information processing, ie, when they would have concluded various things on the basis of various information sets. The puzzle tries to enforce this structure on us by saying “the monks meet once a day”. But the real world never works like that. You don’t actually have to meet up a second time to process more information, you can carry on thinking about it even when you are separated. You can never be sure where other people have got to in their own information processing. So this kind of staged iterative logic, which depends upon making conclusions about when other people concluded things, can never work in real life, even if you really did have a series of brief meetings.

27. Lazy T says:

Great, a puzzle to puzzle over!
I think I’ve got the answer but am still beset by Butwotifs,
thank you.

28. Super. Took me a couple of minutes – I wrote out the reasoning as I went. Seems solid. As many people, I still wonder if I’m missing something. But I’m fairly confident of the answer.

It still amazes me how many people complain pedantically about “real world” practicalities in your delightful little puzzles. Is it really so important to avoid suspending disbelief for the sake of a little game?

29. I don’t understand the premises.

“Imagine a monastery in which ten of the monks may have a disease…”

So there are n monks, and the disease hits exactly 0 or 10 monks at any time?

I’m sure I’m wrong, what is the right way to understand it?

1. To elaborate…
I want to read that there are 10 monks and 0-10 monks can be infected.

Guru is not a monk since he announces something and thus hasn’t taken a vow of silence.

2. Savant says:

It says: “At least one monk in this monastery has a blue spot on his forehead.” That rules out zero and only zero, I think.

3. Yeah true, when including Gurus statement.

Assuming there are 10 monks and 0-10 can be infected, I got a solution, which I think is the one Richard is seeking.

I’m still confused as to where in the puzzle it says that there are 10 monks though (it says 10 _of_ the monks, not 10 monks).

4. Just realised that it works for n monks as long as the max number of infected is 10 and they all know that.

Forget all of the above, I got a solution after 30 minutes 😉

30. DeepField says:

I think the monks are not the “perfect logicians” they think: if they were , they should have left on the second day.

31. That’s clever! I will look into purchasing your puzzles. Also, I think I got the answer!!

32. Pretty sure I got the answer after four and a half minutes. Actually, getting the answer was quicker, but it was becoming ‘pretty sure’ that took the time!

33. You must all be mathematicians – or nerds at the very least! – because I couldn’t understand it at all. I do think it weird that a monk in a silent order would be making an announcement. But I don’t really know what you meant about the ‘why’ part. I don’t know why it took so long for the monks to respond to the announcement, but I think they could’ve pointed to each other and used their fingers for numbers! They wouldn’t have needed numbers. But I haven’t solved it, so I know nothing . . .

34. Sorry, I meant ‘Why they wouldn’t have needed mirrors’! Blame it on the 3 glasses of wine.

35. The bit at the start which says 10 monks may… has changed since this morning. You actually don’t need to know it. The important facts are that they’re all perfect logicians, at least 1 has it, they see each other once a day, and they leave after 10 days. I could say that there are 100 monks who all have the potential to be infected, but that wouldn’t affect the result.

Hint: think what would happen if only one monk had the disease, the what if two had it, and so on.

With it being April Fools Day I was expecting some trick here but this puzzle is quite a well known one – it can be solved.

36. Chris says:

I got it – the clue was the number of monks and the number of days before the affected monks left – and why.

37. Chris Wonson says:

Excellent. A much more satisfying puzzle to solve.

Had a few minutes of thinking that no additional information was available after day 1, so what was special about day 10.
Then worked through a couple of simple scenarios, which made it all fall into place.

38. patthews says:

This is *very* similar to a puzzle from the website xkcd (noted for it’s math and engineering themes). http://xkcd.com/blue_eyes.html The solution isn’t on that page, but it is a description of the same problem with different variables.

39. shlomi ardan` says:

hey richard,

thx

40. Aha, I can thank my Distributed Computing professor for the solution to this one!

41. Ken Haley says:

This puzzle presents an interesting paradox, and I’m surprised no one else has mentioned it. I think I know how to resolve the paradox, but I’d like to hear others’ remarks. First of all, the solution seems irrefutable. The monks with blue dots can eventually deduce who they are, and must leave on a certain day–no sooner, and no later (see solution Monday for explanation why).

But consider this: Suppose the monks had been living for years under the rules of the puzzle (i.e., you must leave at the end of the day as soon as you know that you have a blue dot.). Before the guru uttered his statement, no one had any way to know whether he had one himself. But, when the guru spoke, the information he gave was no surprise. Every monk already knew that at least one monk had a blue dot, and furthermore, every monk knew that every other monk knew this. They could all see at least nine monks with blue dots. Therefore, when the guru spoke, he added nothing to the general knowledge. So, why should they behave any differently after he spoke than they did before? The only logical answer is that by saying what he said, the guru DID add information that wasn’t known before — but what?

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