The riddle of the 4 trolls

[Bas Hamstra]

Once upon a time there were 4 smart trolls, they were captured and buried till their head in the ground, see picture below. The fourth troll is behind a brick wall and can not be seen by the others. Neither troll knows the colors of their own hat, but they know red and blue hats are evenly distributed between the four of them.

One troll must guess the color of his hat. If guessed wrong, they will all be shot. If right, they are all free. The first word spoken must be the color of the speaker's own hat (or they will all be shot). No other communication possible. Do they have a fair chance?

Bas

 

[M87]

Once upon a time there were 4 smart trolls, they were captured and buried till their head in the ground, see picture below. The fourth troll is behind a brick wall and can not be seen by the others. Neither troll knows the colors of their own hat, but they know red and blue hats are evenly distributed between the four of them.

One troll must guess the color of his hat. If guessed wrong, they will all be shot. If right, they are all free. The first word spoken must be the color of the speaker's own hat (or they will all be shot). No other communication possible. Do they have a fair chance?

Bas

Yes, there is only one who can make the logic work, but I will not reveal until others have had a chance to look.

--
Andy
http://www.flickr.com/photos/29788088@N07/

 

[Kevin Coppalotti]

If they are dp review trolls they have no chance as logic plays no part in the life of a dp review troll, - its all about emotion and irrational fears/prejudices. But the blue capped troll has the best chance, given the phrase 'evenly distributed' and being able to see the two red hats of his mates he knows there are two blue hats left, his own and the one behind the wall.

too easy, now, which is better, the s100fs or the S200 exr, or a DSLR?...heh heh.....purchase your flame suit now......
--
Kevin Coppalotti
http://maxhr.zenfolio.com/
http://razorsharp.smugmug.com

 

[Bas Hamstra]

I see your point, but...they can only see in 1 direction... so the blue capped troll you refer to sees only the guy in front, he cannot look backward... The solution has to work for every configuration, the picture is just an example.

Bas

If they are dp review trolls they have no chance as logic plays no part in the life of a dp review troll, - its all about emotion and irrational fears/prejudices. But the blue capped troll has the best chance, given the phrase 'evenly distributed' and being able to see the two red hats of his mates he knows there are two blue hats left, his own and the one behind the wall.

too easy, now, which is better, the s100fs or the S200 exr, or a DSLR?...heh heh.....purchase your flame suit now......
--
Kevin Coppalotti
http://maxhr.zenfolio.com/
http://razorsharp.smugmug.com

 

[NIK11]

Try this - any troll spits (generously) on the ground, then leans head forward slightly to observe which colour is reflected is his spit - alternatively the hat falls off. Either way he knows the colour of his own hat before speaking.

Nick

 

[rolleiman]

The troll in the middle knows that if the guy behind him doesn't say anything his hat must be different from the red hat in front of him, thus his is blue.

 

[GlennAC]

Bas, I promise I have not read any of the other posts here, but...

It makes sense that, though each troll can't see their own hat, they can see each other's hats. So it makes sense that the 2nd Troll from the left state that he is wearing a blue hat since the only other two trolls he can see are wearing red hats.

Once upon a time there were 4 smart trolls, they were captured and buried till their head in the ground, see picture below. The fourth troll is behind a brick wall and can not be seen by the others. Neither troll knows the colors of their own hat, but they know red and blue hats are evenly distributed between the four of them.

One troll must guess the color of his hat. If guessed wrong, they will all be shot. If right, they are all free. The first word spoken must be the color of the speaker's own hat (or they will all be shot). No other communication possible. Do they have a fair chance?

Bas

 

[Conrad Birdie]

Once upon a time there were 4 smart trolls, they were captured and buried till their head in the ground, see picture below. The fourth troll is behind a brick wall and can not be seen by the others. Neither troll knows the colors of their own hat, but they know red and blue hats are evenly distributed between the four of them.

One troll must guess the color of his hat. If guessed wrong, they will all be shot. If right, they are all free. The first word spoken must be the color of the speaker's own hat (or they will all be shot). No other communication possible. Do they have a fair chance?

No, because the troll behind the wall wins regardless. If he guesses blue, they all are O.K. If he guesses RED, he is still safe because he is behind the wall and the others are shot....lol.

Bas

--
Conrad 'Bye Bye' Birdie
'Aspire to inspire before you expire'.

 

[MattRat]

The troll in the middle knows that if the guy behind him doesn't say anything his hat must be different from the red hat in front of him, thus his is blue.

Ha, that's really smart! Of course, he'd want to make sure he doesn't speak too soon...

But, it could also depend on what ' every configuration' means. If it means that they can be buried looking in either direction, then a configuration is possible where none of them can see more than one other e.g. ( ) ) I (.

Even so, they have a 'fair chance' - depends how long it takes them to be shot. If they all say the same thing, then they will be free. So, I guess 'fair chance' means 50/50...

Btw, are they Fuji trolls?

matt

 

[Fotonut]

Hiya Bas,

Number one can see the 2 colors ahead of him. If the requirement is that the hat colors are enenly distrubuted, he can deduce that the one behind the wall must be blue and thus his own must be red.

Also the riddle does not state which troll has to speak first.

Greets.
Fotonut.
--
Snap snap - click click.

FujiLinks - http://fujilinks.110mb.com/index.html (Main Site)
FujiLinks - http://fujilinks.fizz.me.uk/index.html (Mirror Site)

 

[Adam-T]

Number one can see the 2 colors ahead of him. If the requirement is that the hat colors are evenly distrubuted

I took it that all that means is that there are two of each, not the order of the distribution - Troll 1 can see one of each so the one behind the wall could be either red or blue

The question is "Do they have a fair chance" the answer is "Who gives a Smeg, the less Trolls the better! "

--
Please ignore the Typos, I'm the world's worst Typist

 

[azumimm]

Hiya Bas,

Number one can see the 2 colors ahead of him. If the requirement is that the hat colors are enenly distrubuted, he can deduce that the one behind the wall must be blue and thus his own must be red.

Or he can deduce guess that the hat of the one behind the wall is red and hence his hat is blue.

So they only have a 50/50 chance.

A fair chance the will get if either there is a mirror at the wall or a whole or the second troll switchs the place with the first troll or the third troll switchs with the fourth troll (the one behind the wall).

 

[azumimm]

I got the solution!

The first troll safes them all.

He just moves, with some force, his head toward the front whereby the hat falls off, he sees the color.. bingo!

What camera did I win? :O

 

[NIK11]

The troll in the middle knows that if the guy behind him doesn't say anything his hat must be different from the red hat in front of him, thus his is blue.

Interesting. Yours sounds like the smart answer, based on logic. However, in terms of the question of 'fair chance', perhaps another answer is that only the 2nd troll has a better than 50% chance of guessing right- there is 66% chance of it being blue, having accounted for one red in front.

If you get this right, I shall be seeking your assessment on the best WA ultracompact!!

Nick

 

[azumimm]

The troll in the middle knows that if the guy behind him doesn't say anything his hat must be different from the red hat in front of him, thus his is blue.

But only if the first troll can see the third troll as well. See also my solution.

 

[M87]

Once upon a time there were 4 smart trolls, they were captured and buried till their head in the ground, see picture below. The fourth troll is behind a brick wall and can not be seen by the others. Neither troll knows the colors of their own hat, but they know red and blue hats are evenly distributed between the four of them.

One troll must guess the color of his hat. If guessed wrong, they will all be shot. If right, they are all free. The first word spoken must be the color of the speaker's own hat (or they will all be shot). No other communication possible. Do they have a fair chance?

Bas

Ok

Evenly distributed means two of each. A fair chance means “is there a solution” and there is.

The troll on the extreme right is behind the wall and is facing the wrong way – a troll trait that you might recognise. He must guess with a fifty-fifty chance or keep quiet.

The troll on the other side of the wall is facing the wall and can therefore also see nothing, another quiet troll.

The troll on the extreme left can see two hats, but one of each colour. He does not know the colour of the hat behind the wall, so he must also keep quiet.

Three simultaneously quiet trolls – must be a record.

That leaves the troll in the middle of the three. He can see only one hat but he knows that the troll behind him can see two hats. These two hats can only be in one of two configurations, either both the same colour or of differing colours. Now if the troll behind him could see two of the same colour then the troll behind him would have spoken out because his would therefore necessarily be a different colour from the two he can see. However, that troll is silent, therefore the middle troll can therefore deduce with certainty that the troll behind him can see two hats of different colours. Therefore the hat worn by the middle troll must, because of the silence of the troll behind him, be different from the colour in front of him. He can say for certainty that his is blue.

Four trolls and only allowed one word between them and one of them needs to be smart enough to figure it out. Chances are all four will speak at once ‘cause trolls are just not bright enough, but they have a fair chance

--
Andy
http://www.flickr.com/photos/29788088@N07/

 

[MattRat]

Well, you wouldn't trust a troll, no matter he said - would you?

But, don't trolls just start irrelevant threads with something provocative in the hope of getting everyone agitated enough to reply and getting lots of posts, before everyone realises they've been, well, trolled...?

:0

matt

 

[Paul De Bra]

Your solution is of course correct but it has one tiny snag: the second troll (from the left) must know how long to wait before deciding that the first troll is keeping silent. So normally this (rather well known) puzzle is accompanied by a clock pulse. With the given configuration at the first clock pulse all trolls remain silent. This tells the second troll that the first troll sees two different hats, so at the second clock pulse the second troll will speak.
Without a clock the second troll may speak just too early...

--
Slowly learning to use the 450D and and the Canon G6.
Public pictures at http://debra.zenfolio.com/ .

 

[NIK11]

Evenly distributed means two of each. A fair chance means “is there a solution” and there is.

The troll on the extreme right is behind the wall and is facing the wrong way – a troll trait that you might recognise. He must guess with a fifty-fifty chance or keep quiet.

The troll on the other side of the wall is facing the wall and can therefore also see nothing, another quiet troll.

The troll on the extreme left can see two hats, but one of each colour. He does not know the colour of the hat behind the wall, so he must also keep quiet.

Three simultaneously quiet trolls – must be a record.

That leaves the troll in the middle of the three. He can see only one hat but he knows that the troll behind him can see two hats. These two hats can only be in one of two configurations, either both the same colour or of differing colours. Now if the troll behind him could see two of the same colour then the troll behind him would have spoken out because his would therefore necessarily be a different colour from the two he can see. However, that troll is silent, therefore the middle troll can therefore deduce with certainty that the troll behind him can see two hats of different colours. Therefore the hat worn by the middle troll must, because of the silence of the troll behind him, be different from the colour in front of him. He can say for certainty that his is blue.

Four trolls and only allowed one word between them and one of them needs to be smart enough to figure it out. Chances are all four will speak at once ‘cause trolls are just not bright enough, but they have a fair chance

Thank you for the explanation, but 'rolleiman' expalined the same solution 13hrs ago, and more concisely.

Of course this solution assumes the first troll can see past the 2nd troll to observe the third hat colour. To some, 'evenly distributed' could mean every other is the same colour etc, etc.These teasers are bit like golf, you need too many qualifying rules.

Nick

 

[M87]

Evenly distributed means two of each. A fair chance means “is there a solution” and there is.

The troll on the extreme right is behind the wall and is facing the wrong way – a troll trait that you might recognise. He must guess with a fifty-fifty chance or keep quiet.

The troll on the other side of the wall is facing the wall and can therefore also see nothing, another quiet troll.

The troll on the extreme left can see two hats, but one of each colour. He does not know the colour of the hat behind the wall, so he must also keep quiet.

Three simultaneously quiet trolls – must be a record.

That leaves the troll in the middle of the three. He can see only one hat but he knows that the troll behind him can see two hats. These two hats can only be in one of two configurations, either both the same colour or of differing colours. Now if the troll behind him could see two of the same colour then the troll behind him would have spoken out because his would therefore necessarily be a different colour from the two he can see. However, that troll is silent, therefore the middle troll can therefore deduce with certainty that the troll behind him can see two hats of different colours. Therefore the hat worn by the middle troll must, because of the silence of the troll behind him, be different from the colour in front of him. He can say for certainty that his is blue.

Four trolls and only allowed one word between them and one of them needs to be smart enough to figure it out. Chances are all four will speak at once ‘cause trolls are just not bright enough, but they have a fair chance

Thank you for the explanation, but 'rolleiman' expalined the same solution 13hrs ago, and more concisely.

He did, and thereby showed that he understood the question and could work out or already knew the answer.

His one-liner did not provide an explanation of why for the less logically minded, which would appear to include you given your response to his solution which suggested that you were not sure if his answer was in fact correct or complete.

I offered to provide a solution some two hours before his answer, a commitment I made and that I have now fulfilled.

Of course this solution assumes the first troll can see past the 2nd troll to observe the third hat colour. To some, 'evenly distributed' could mean every other is the same colour etc, etc.These teasers are bit like golf, you need too many qualifying rules.

Nick

--
Andy
http://www.flickr.com/photos/29788088@N07/