[roolstar]

You are on a steam train going for a very long trip accross China.

On this train, there are with you 50 monks going back to their temple on the terminal of the train. (I assume you are not a monk, and if you are, you are a very cool one to be on this forum!)

When the controler came in to check their tickets, he noticed that SOME of them had soot on their faces (their faces were dirty). And of course they cannot feel it.

So he said: "You know, there are cleaning areas at every stop we will make until we arrive. So anybody with soot on his face can have time to get off at any of them, get cleaned up, and come back before the train resumes its trip."

Now these monks are very wise, but they are also EXTREMELY lazy! So lazy in fact that they don't want to get off the train to get cleaned up unless its necessary (if they have dirty faces). They also took vows of silence untill they arrive to their destination, so they cannot say anything to each other. They don't even bother to look for a mirror or something to see their faces with. They don't even want to signal each other in anyway. They can't see their own faces in any of the windows in the train. They donb't even want to put their hands on their faces to check!

In brief, THEY DON'T WANT TO MAKE ANY PHYSICAL EFFORT UNLESS IT'S ABSOLUTELY NECESSARY NOR THEY CAN COMMUNICATE WITH EACH OTHER NOR SEE THEIR OWN FACES in any way! They can only THINK and REASON! (Sometimes a more ardeous act for some people BTW)

They can only see each other's faces while maintaining a poker face.

In this particular case there were 5 monks with soot on their faces.

The question for this puzzle is:

WHAT DO YOU THINK WILL HAPPEN?

Will they remain dirty? (I promise you they won't)

If not what?

Now I know this is not a very clear question, but remember this is the same situation these monks are in. Plus I will post future hints later on (If you need them of course)...

PS: This is not a lateral thinking puzzle nor a play on words. Just not my style...

Edited February 8, 2008 by roolstar

[bonanova]

[1] Do the Monks know [or do only we know] that only five of them have soot?

[2] Do the Monks know that at least one of them has soot?

[roolstar]

[1] Do the Monks know [or do only we know] that only five of them have soot?

[2] Do the Monks know that at least one of them has soot?

[1] NO they don't: It would be very easy in this case, and It certainly won't help me regain my respect as I promised in the Hole in a Sphere earlier...

[2] YES: After all, the controler told them about cleaning areas when he saw their faces.

I'm just glad it's not a very well known puzzle and that the Great Bonanova is working on it!!

[Guest]

I must be missing something. If they can see each other's faces, anyone seeing only 4 sooty faces should exit next stop and wash.

[Guest]

I must be missing something. If they can see each other's faces, anyone seeing only 4 sooty faces should exit next stop and wash.

They DO NOT know that only 5 have dirty faces.

[EventHorizon]

This is very similar to another puzzle that I believe I saw on here a while back.

First stop: If a monk doesn't see any monks with dirty faces, then he would know he was the only one since at least one monk has soot on his face. Nobody gets off because there are 5 people with dirty faces. So they know there is more than one monk with soot on his face.

Second stop: If there were two monks with soot on their faces, then they would both see each other as having soot on their face and since they only see one other, they know they both have soot on their faces. Nobody gets off. So they know there is at least three monks with soot on their faces.

Third stop: If there were three monks with soot on their faces, then they would see only two other monks with soot on their faces and know that there are three. All three would get off then. Nobody gets off the train, so they know at least four monks have soot on their faces.

Fourth stop: If there were four monks with soot on their faces, one of the ones with a dirty face would see three other people with soot on their faces and know there is one more so he would know that he had a dirty face. Nobody gets off the train, so all the monks know there are at least 5 monks with dirty faces.

Fifth stop: The monks with dirty faces look around and see four dirty faces, but know there are at least 5 monks with dirty faces. So now the monks with dirty faces know who they are and wash their faces at the fifth stop.

[roolstar]

I must be missing something. If they can see each other's faces, anyone seeing only 4 sooty faces should exit next stop and wash.

Your reasonning will be true if they KNEW that there are 5 monks with dirty faces. But they don't: I already answered Bonanova's question about it.

Now the fact that there are 5 dirty monks is just given to make this a little easier to answer!

The fact is, that I could have just as easily said nothing about the number of Monks with dirty faces. (<<< This is also a hint)!!!

[roolstar]

This is very similar to another puzzle that I believe I saw on here a while back.

First stop: If a monk doesn't see any monks with dirty faces, then he would know he was the only one since at least one monk has soot on his face. Nobody gets off because there are 5 people with dirty faces. So they know there is more than one monk with soot on his face.

Second stop: If there were two monks with soot on their faces, then they would both see each other as having soot on their face and since they only see one other, they know they both have soot on their faces. Nobody gets off. So they know there is at least three monks with soot on their faces.

Third stop: If there were three monks with soot on their faces, then they would see only two other monks with soot on their faces and know that there are three. All three would get off then. Nobody gets off the train, so they know at least four monks have soot on their faces.

Fourth stop: If there were four monks with soot on their faces, one of the ones with a dirty face would see three other people with soot on their faces and know there is one more so he would know that he had a dirty face. Nobody gets off the train, so all the monks know there are at least 5 monks with dirty faces.

Fifth stop: The monks with dirty faces look around and see four dirty faces, but know there are at least 5 monks with dirty faces. So now the monks with dirty faces know who they are and wash their faces at the fifth stop.

Good job!

Which puzzle is that??

What if for some reason, nobody got off the train on the fifth stop??

Edited February 9, 2008 by roolstar

[Guest]

This is very similar to another puzzle that I believe I saw on here a while back.

First stop: If a monk doesn't see any monks with dirty faces, then he would know he was the only one since at least one monk has soot on his face. Nobody gets off because there are 5 people with dirty faces. So they know there is more than one monk with soot on his face.

Second stop: If there were two monks with soot on their faces, then they would both see each other as having soot on their face and since they only see one other, they know they both have soot on their faces. Nobody gets off. So they know there is at least three monks with soot on their faces.

Third stop: If there were three monks with soot on their faces, then they would see only two other monks with soot on their faces and know that there are three. All three would get off then. Nobody gets off the train, so they know at least four monks have soot on their faces.

Fourth stop: If there were four monks with soot on their faces, one of the ones with a dirty face would see three other people with soot on their faces and know there is one more so he would know that he had a dirty face. Nobody gets off the train, so all the monks know there are at least 5 monks with dirty faces.

Fifth stop: The monks with dirty faces look around and see four dirty faces, but know there are at least 5 monks with dirty faces. So now the monks with dirty faces know who they are and wash their faces at the fifth stop.

What about:

All monks will look around and know that their are at least 4 monks with soot on their faces. 45 monks will look around and know there are at least 5 with soot on their faces. (And they all know that they all know this.) The only thing they do not know is the status of their own faces.

So at the first stop everyone knows there are at least 4 and some know there are at least 5. No one gets up. This tells the 5 who thought that there were 4 that there are actually 5 and since they only count 4, they know they are the fifths. Then they all wash up at the second stop.

[Guest]

Good job!

Which puzzle is that??

What if for some reason, nobody got off the train on the fifth stop??

They would all get off the bus at the 6th station thinking there were 6 and they were the sixth.

[EventHorizon]

The puzzle went something like ....

"Queen XXXXXX (can't remember her name) of atlantis said that at least one man was cheating on his wife in atlantis. It was common knowledge that all wives know about whether or not other men were faithful, but not about their husband. She said that once you know your husband cheats, then shoot him at midnight. After 49 nights of silence, gunshots were heard. How many men were cheating? What was Queen XXXXXX's great accomplishment?"

Obviously not verbatim, but that's the gist of it.

As for your follow-up...

Since no one got off on the fifth stop, everyone "knows" there are at least 6 people with soot on their faces. At the sixth stop, everyone with clean faces will get off and wash their faces because they see 5 dirty faces and know there is a sixth (, and the ones with dirty faces will be confused/daydreaming?)

Though the commotion would probably make them all think that someone wasn't paying attention to their logic.

[roolstar]

They would all get off the bus at the 6th station thinking there were 6 and they were the sixth.

Now that's what I call PLAYING IT DIRTY!!!

Just to make everybody go off the train as well and not be the only unlucky ones who will do the effort!!

[Guest]

Don't the five with dirty faces get off after the second stop. Isn't the number of faces completely irrelevant and all monks with dirty faces will get off at the second stop?

[roolstar]

What about:

All monks will look around and know that their are at least 4 monks with soot on their faces. 45 monks will look around and know there are at least 5 with soot on their faces. (And they all know that they all know this.) The only thing they do not know is the status of their own faces.

So at the first stop everyone knows there are at least 4 and some know there are at least 5. No one gets up. This tells the 5 who thought that there were 4 that there are actually 5 and since they only count 4, they know they are the fifths. Then they all wash up at the second stop.

The colored part in your quote isn't as obvious as it seems.

let me explain:

Why would the 5 monks get off thinking they are number 5 and not the rest of them get off thinking they are number 6?

What is the difference between the 5 monks and the rest? NOTHING

For all they know (the 5 monks), the other 4 may be wondering if they are number 4!

Remember they don't know what everybody else is thinking!

Edited February 9, 2008 by roolstar

[roolstar]

As for your follow-up...

Since no one got off on the fifth stop, everyone "knows" there are at least 6 people with soot on their faces. At the sixth stop, everyone with clean faces will get off and wash their faces because they see 5 dirty faces and know there is a sixth (, and the ones with dirty faces will be confused/daydreaming?)

Though the commotion would probably make them all think that someone wasn't paying attention to their logic.

In fact it's not enough that someone didn't pay attention to the logic, ALL 5 MONKS didn't: because if only one of them was stupid, the other 4 would still get off on the 5th stop.

So the commotion would make them realize the other five were playing it DIRTY!!!

[Guest]

This is very similar to another puzzle that I believe I saw on here a while back.

First stop: If a monk doesn't see any monks with dirty faces, then he would know he was the only one since at least one monk has soot on his face. Nobody gets off because there are 5 people with dirty faces. So they know there is more than one monk with soot on his face.

Second stop: If there were two monks with soot on their faces, then they would both see each other as having soot on their face and since they only see one other, they know they both have soot on their faces. Nobody gets off. So they know there is at least three monks with soot on their faces.

Third stop: If there were three monks with soot on their faces, then they would see only two other monks with soot on their faces and know that there are three. All three would get off then. Nobody gets off the train, so they know at least four monks have soot on their faces.

Fourth stop: If there were four monks with soot on their faces, one of the ones with a dirty face would see three other people with soot on their faces and know there is one more so he would know that he had a dirty face. Nobody gets off the train, so all the monks know there are at least 5 monks with dirty faces.

Fifth stop: The monks with dirty faces look around and see four dirty faces, but know there are at least 5 monks with dirty faces. So now the monks with dirty faces know who they are and wash their faces at the fifth stop.

The 1st - 3rd stops above tell the monks nothing new. When they get to the first stop they all already know that there are at least 4 with dirty faces. So you can remove the first three stops in the above answer and rename "Fourth stop" and "Fifth stop" as "First stop" and "Second stop".

"Why would the 5 monks get off thinking they are number 5 and not the rest of them get off thinking they are number 6? (a)

What is the difference between the 5 monks and the rest? NOTHING (b)

For all they know (the 5 monks), the other 4 may be wondering if they are number 4! ©

Remember they don't know what everybody else is thinking!" (d)

(a) Good question. Let me come back to that.

(b) The difference is that some know there are at least 4 dirty faces and some know that there are at least 5.

© All 5 dirty faces will be wondering if they are number 5 because all 5 will see the other 4.

(d) All of the monks are thinking, "I count x amount of dirty faces. The total possible amount of dirty faces is x+1, because I do not know about mine." Since they are all supposed to be smart enough to figure this out, I assume they all know that every monk is thinking this same thing.

(a, again) Hmmm...let me get back to you. Roadblock.

[Guest]

Simple....

If the controller told me so, & if i was one amongst them, I'd have just covered my face with my cloth...

Since everyone out there is smart, we can expect everyone to cover their faces...

The legend says that monks expose only the face & palms while everything else is covered.... hence covering the face wouldnt be an issue...

Another possiblity is that, I'd just close the windows....

Possibility 3, I will sit in the direction opp to the wind flow....

In the worst case scenario, I'll turn my back to the wind/window.... whatever.... Hence face is still clean....

[roolstar]

The 1st - 3rd stops above tell the monks nothing new. When they get to the first stop they all already know that there are at least 4 with dirty faces. So you can remove the first three stops in the above answer and rename "Fourth stop" and "Fifth stop" as "First stop" and "Second stop".

"Why would the 5 monks get off thinking they are number 5 and not the rest of them get off thinking they are number 6? (a)

What is the difference between the 5 monks and the rest? NOTHING (b)

For all they know (the 5 monks), the other 4 may be wondering if they are number 4! ©

Remember they don't know what everybody else is thinking!" (d)

(a) Good question. Let me come back to that.

(b) The difference is that some know there are at least 4 dirty faces and some know that there are at least 5.

© All 5 dirty faces will be wondering if they are number 5 because all 5 will see the other 4.

(d) All of the monks are thinking, "I count x amount of dirty faces. The total possible amount of dirty faces is x+1, because I do not know about mine." Since they are all supposed to be smart enough to figure this out, I assume they all know that every monk is thinking this same thing.

(a, again) Hmmm...let me get back to you. Roadblock.

(a) The answer lies in the small flaws in (b) (c ) & (d) will come back to that later!

(b) Even if some of them think there may be 5 & others think there might be 6, Nobody knows what everybody else is seeing or thinking (d)

==> There will be no way of agreeing on deleting the stops from 1 - 4 and starting from 5. They may be deleting 1 - 3 and starting at 4. See (a) below for illustration.

(c ) This means that if there were only 4 faces dirty, 4 people will see 3 dirty faces and will be wondering if they are number 4 and 36 see that there are 4 dirty faces and will be wondering if they are number 5!

==> This scenario is no different than the 5 dirty faces scenario and the monks cannot tell if it's this scenario or the other. See (a) below for illustration.

(a) Let me put it this way:

The different scenarios all haved a possibility in common (Two by Two) and the doubt cannot be eliminated except by missing the stops.

Scenario $: 4 Dirty Faces

- 4 people see 3 and wonder if they are number 4

- 36 people see 4 dirty faces and wonder if they are number 5 <=

Scenario 5: 4 Dirty Faces

- 5 people see 4 and wonder if they are number 5 <=

- 35 people see 5 dirty faces and wonder if they are number 6

And as 1 monk not knowing the other people thinking, you cannot tell if you are part of the group of 36 in scenario 4 or part of the group of 5 in scenario 5

The same in Scenario 5 & Scenario 6...

Ans that's why they cannot do it by stop #2

[unreality]

This is similar to the Josephine problem good one though ;D

[Guest]

(a) The answer lies in the small flaws in (b) (c ) & (d) will come back to that later!

(b) Even if some of them think there may be 5 & others think there might be 6, Nobody knows what everybody else is seeing or thinking (d)

==> There will be no way of agreeing on deleting the stops from 1 - 4 and starting from 5. They may be deleting 1 - 3 and starting at 4. See (a) below for illustration.

(c ) This means that if there were only 4 faces dirty, 4 people will see 3 dirty faces and will be wondering if they are number 4 and 36 see that there are 4 dirty faces and will be wondering if they are number 5!

==> This scenario is no different than the 5 dirty faces scenario and the monks cannot tell if it's this scenario or the other. See (a) below for illustration.

(a) Let me put it this way:

The different scenarios all haved a possibility in common (Two by Two) and the doubt cannot be eliminated except by missing the stops.

Scenario $: 4 Dirty Faces

- 4 people see 3 and wonder if they are number 4

- 36 people see 4 dirty faces and wonder if they are number 5 <=

Scenario 5: 4 Dirty Faces

- 5 people see 4 and wonder if they are number 5 <=

- 35 people see 5 dirty faces and wonder if they are number 6

And as 1 monk not knowing the other people thinking, you cannot tell if you are part of the group of 36 in scenario 4 or part of the group of 5 in scenario 5

The same in Scenario 5 & Scenario 6...

Ans that's why they cannot do it by stop #2

In reality, they can't do it at stop 5 either...these monks are WISE, so why in the world would the thought "If a monk saw no other dirty faces, he'd get off at the first stop" EVER cross his mind? He knows FOR A FACT that there are only two possibilities...either a) there are 4 dirty faces, and mine isn't one of them, or b) there are 5 dirty faces, and mine is one of them. There is not a single way to logically deduce which scenario it is (and the same goes for the Josephine one)

[bonanova]

I'm late to the game on this ... sorry.

Monks with dirty faces see four other dirty faces.

Each reasons that if their face were clean, the others would see three dirty faces, and they would have washed on the 4th stop.

Since that didn't happen, they all wash on the 5th stop.

Interestingly, if the ones with dirty faces don't reason correctly, all the monks who do reason correctly - including the 45 clean ones - get off to wash at the following stop!

[roolstar]

I'm late to the game on this ... sorry.

Interestingly, if the ones with dirty faces don't reason correctly, all the monks who do reason correctly - including the 45 clean ones - get off to wash at the following stop!

Nice of you to join in Bonanova...

I don't totally agree with the red part of your above quote.

Here's why:

Now if ALL of the 5 dirty faces miss the reasonning and miss the stop #5, this would make the SMART ONES among the 45 get off on stop #6!

(Here we can debate if being SMARTER is always a good thing!! )

But even if ONLY ONE monk reasonned correctly, he would be off on stop #5!. And that would make the rest of the monks know that the other 4 missed the reasonning, and this would make the rest of the 45 remain in the train for the remainder of the trip.

Now for the monk (or monks) who got off, seeing that he's ALONE going to the cleaning area, he will think about two possible scenarios:

[A]:

The other 4 monks missed the reasonning AND he is dirty.

:

The other 4 missed the reasonning (and they missed stop # 4!)

AND the rest of the 45 monks missed the reasonning as well (they didn't go off with him on stop #5)

AND he's clean

Now if I'm that monk I would easily pick possibility A as beeing the most likely to happen! And would continue my ardeous journey to the cleaning area!

What do you guys think??

[roolstar]

Now for the monk (or monks) who got off, seeing that he's ALONE going to the cleaning area, he will think about two possible scenarios:

[A]:

The other 4 monks missed the reasonning AND he is dirty.

:

The other 4 missed the reasonning (and they missed stop # 4!)

AND the rest of the 45 monks missed the reasonning as well (they didn't go off with him on stop #5)

AND he's clean

Now if I'm that monk I would easily pick possibility A as beeing the most likely to happen! And would continue my ardeous journey to the cleaning area!

What do you guys think??

Just to make the above easier to read:

Now for the monk (or monks) who got off, seeing that he's ALONE going to the cleaning area, he will think about two possible scenarios:

[A]:

The other 4 monks missed the reasonning AND he is dirty.

[b ]:

The other 4 missed the reasonning (and they missed stop # 4!)

AND the rest of the 45 monks missed the reasonning as well (they didn't go off with him on stop #5)

AND he's clean

Now if I'm that monk I would easily pick possibility A as beeing the most likely to happen! And would continue my ardeous journey to the cleaning area!

What do you guys think??