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# Head Bands

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Head Bands - Back to the Logic Puzzles

Three Palefaces were taken captive by a hostile Indian tribe. According to tribe’s custom they had to pass an intelligence test, or die. The chieftain showed 5 headbands – 2 red and 3 white. The 3 men were blindfolded and positioned one after another, face to back. The chief put a headband on each of their heads, hid two remaining headbands, and removed their blindfolds. So the third man could see the headbands on the two men in front of him, the second man could see the headband on the first, and the first could not see any headbands at all.

According to the rules any one of the three men could speak first and try to guess his headband color. And if he guessed correctly – they passed the test and could go free, if not – they failed. It so happened that all 3 Palefaces were prominent logicians from a nearby academy. So after a few minutes of silence, the first man in the line said: "My headband is ...".

What color was his head band? Why?

This old topic is locked since it was answered many times. You can check solution in the Spoiler below.

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Head Bands - solution

The first one (he did not see any head bands) thought this way:

The last one is silent, which means, he does not know, ergo at least one of head bands he sees is white. The one in the middle is silent too even though he knows what I already mentioned. If I had a red head band, the second one would have known that he had a white head band. However, nobody says anything, so my head band is not red – my head band is white.

Three white men were taken captive by a hostile Indian tribe. The chieftain was willing to let them go so he took them to a tepee, where there was no light. He put one head band on each of their heads (he had 3 white and 2 red - so 2 head bands were not used - captives were informed about these numbers). Then they went out in a queue so that each man saw the head-band of those standing in front of him (the first one did not see any head band, the second one saw the first one's head band, and the third one saw the head bands of the other two ). If somebody said the color of his head-band, they all would be free. After a few minutes of silence one of them said: "My head-band is ...".

What color was his head band? And how would you reason it?

Edited:

You have to assume that all the prisoners are fairly intelligent and have confidence in the intelligence of their fellow prisoners.

An incorrect guess deems them to imprisonment.

Only one guess can be made by the group.

All the captives want to be free and live.

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• 3 weeks later...
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Couldn't the last person in the queue also be able to guess this correctly? Assuming the 3 men knew the amount of white and red bands (3 and 2) the last man could see that the first two men both have red bands. The 3rd man knowing there are only 2 red bands could then realize that he must have a white band and state that thus setting them free.

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I sort of disagree with this, or I am missing something. The second one only sees in front of him. So, if the first one's band has a red, then the second one could not be sure the third has a white and would remain silent. The third would only see one white and would be unsure and would remain silent. That scenario would be the same if the first had a white and the second a red. Both second and third would remain silent in that case as well. Am I missing something?

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Try to think as if you were the first one - as I mentioned above. Of course, we have to assume that all 3 are quite intelligent to pass the test that way.

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cheizmann13 and Nitevizn:

The key to both of your questions is that there was no immediate response from anyone.

Cheizmann13, your answer doesn't work because the guy in back would see two red head bands and respond without delay.

Nitevizn, the second guy would see one red head band, and he would know that if he also had a red head band, the third guy would respond without delay. When this doesn't happen, the second guy would know his head band was white.

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• 4 weeks later...

Actually, whether or not they knew the number of each color is irrelevant. "He took them to a tepee, where there was no light" - Therefore, without stating that the headbands were in the possession of the Chief PRIOR to taking them out to the tepee, this puzzle cannot be solved as they would not be able to see the colors (or anything else for that matter) of the headbands nor the quantities.

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• 3 weeks later...

I don't know whether i'm right or wrong.Pls clarify.

If the 1st guy sees 2 white bands, he might think whether he has a white or red.

So the 2nd guy has 2 options

1)either both of them are white

or

2)one of the 2 headbands is white.

So he can't answer even if the first guy(who doesn't see anything) is red or white.

If that's the case how come the first guy(who doesn't see anything) can confirm he is white.

It might be red too, becoz i see the solution to be justified for only option 2 for the 2nd guy.

Correct me if have missed something.

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the clues we have are as follows:

1st and 2nd can't both have red because 3rd person would have immediately known he had a white headband. Therefore 1st and 2nd either both have white or have one red and one white.

The only logical conclusion is that the first person has a white headband then. Why? Because if the first person had a red headband, the 2nd person would have immediately said that he himself had a white headband if the 3rd person didn't say anything. Since the 2nd person didn't say anything (due to there being a "quiet while" - long silence), the 1st person can conclude that the 2nd person isn't seeing a red headband, so the 1st person must have a white headband.

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like senti_2k2 said....what if third sees 2 white headbands?

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i am going to post my explanation again, for the benefit of those confused. this time, there will be colors denoting the important parts...

the clues we have are as follows:

1st and 2nd can't both have red because 3rd person would have immediately known he had a white headband. Therefore 1st and 2nd either both have white or have one red and one white. (Note: if 1st and 2nd both have white then the 3rd sees 2 white headbands, which is what everyone seems confused about...)

The only logical conclusion is that the first person has a white headband then. Why? Because if the first person had a red headband, the 2nd person would have immediately said that he himself had a white headband if the 3rd person didn't say anything. Since the 2nd person didn't say anything (due to there being a "quiet while" - long silence), the 1st person can conclude that the 2nd person isn't seeing a red headband, so the 1st person must have a white headband.

Obviously, if the 3rd person sees 2 white headbands, the 2nd person wouldn't see a red headband on the 1st person. I guess I really needed to point that out? ><

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There was no light so i don't think this riddle is even a riddle. One of the men must have randomly guessed and hoped for the best.

Edit: oh i understand, he put their headbands on in the room with no light, then took them outside >< sorry!

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oh ok i understand larryhl, thanx for the explanation.

If 3rd is confused and 2nd is seeing a red on 1st - then 2nd has to be wearing white and 2nd would not have stayed silent or confused.

if 3rd is confused and 2nd is seeing a white on 1st, then 2nd could be wearing white or red. Therefore 2nd would be confused and silent, so 1st would know that he is wearing white.

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• 2 weeks later...

I get it!

The one thing that people don't understand is that, the 2nd guy is quiet because

the the 1st guy is white AND that the 3rd guy is quiet! If the 3rd guy spoke up, then the first two would have to be red,

but since the 3rd guy is quiet, the 2nd guy knows that the combinations possible are:

2nd=White 1st=White

2nd=Red 1st=White

2nd=White 1st=Red

But since #2 is quiet, the 1st must be white, because if the first was red,

2 would have to be white, but if #1 is white then 2 is either Red or White and he doesn't

want to take that 50/50 chance. The first notices this and speaks up. He has to be white!

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No way! This answer is BOGUS! The question only states that one of them spoke up, it doesn't say the first one spoke up. It solves the puzzle equaly well to have per1 and per2 wearing white, and per3 saying he has red because he can see the two whites. And the idea that someone might pause to double check in his head how many of each color headbands there were before speaking up when his life is on the line I think is perfectly reasonable.

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I understand the solution as given, but I don't feel that it is a complete solution. It only explains the case of the 4 of the seven possibilities. It implies potential solutions for the other possibilities, but I think all possibilities should be explicitly covered in the solution.

The seven possibilities are as follows:

P 1 2 3

1 R R W

2 R W R

3 R W W

4 W R R

5 W R W

6 W W R

7 W W W

In case 1, the third prisoner would (or at least should) immediately know that his headband is White because the two red headbands are on the other prisoners. In cases 2 and three, the second prisoner should know that his headband is white because he can see that the first prisoner's headband is red and would expect the third prisoner to respond immediately if he saw two red headbands. The color of the third prisoner's headband would be irrelevant. In cases 4 through 7, the first prisoner could eliminate that the first three possibilities by the lack of response from the other prisoners, leaving only possibilities where his own headband is white. In all cases, you have to assume that all the prisoners are fairly intelligent and have confidence in the intelligence of their fellow prisoners.

A couple people above seem to be confused about how the prisoners would know how many of each headband were available. I think it is reasonable to accept that the chief was testing the intelligence of the prisoners and would give them the information they needed to succeed. Otherwise, there would not be much point in the exercise.

Another possible solution would be for all three prisoners to state that they had white headbands. The instructions simply state that "If somebody said the colour of his head-band, they all would be free." Nothing says that an incorrect guess deems them to imprisonment. So, since they all know there are only two red headbands, at most only two of them could be wrong, making at least one of them correct. In the same light, one of the prisoners could claim to have a white headband and then if he is incorrect immediately state that he has a red headband. Again, this does not violate the instructions. I think both of these are cheesy solutions, though, and can both be eliminated by simply adding a condition to the puzzle stating that only one guess can be made by the group.

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second person will say that he has white. the first one has red. third's hat is indeterminate

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rather it depends upon who speaks out. if 2nd and third are quiet then the first one can know that he is wearing white

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I've got it. It's much simpler than any of you think.

Assuming they saw the headbands before the darkness:

The 2 unused headbands were red and the third man saw two white headbands.

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oops... so he said mine is white lol

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Since there were only 2 red bands, The last person could confidently assume that his is WHITE because he saw that the other two people had(the only two) red ones.

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• 2 weeks later...

As demonstrated by the replies to this quiz, none of the three men could make any assumptions about how quickly the other two could make any conclusions. The length of silence would not reveal anything to any of them. The quiz does not state that the three white men are smart either. In fact, since they were captured, one could possibly assume that they're not very smart at all.

So, one or more of the following are true:

1. There is not enough information provided in the quiz to come up with a logically difinitive answer.

2. The 3rd man says, "I'm wearing white", seeing the first two are wearing red.

3. The quiz should be reworded to provide more information... perhaps saying that the 3 white men are highly intelligent. Even then, we're still relying on the length of quiet time. What's the correct length of quit time for the 2nd man to determine the 3rd doesn't know? What's the length of time for the 1st man to determine that both 3 and 2 don't know? Perhaps 2 over estimates the intelligence of #3 or #1 over estimates the intelligence of #'s 2 and 3 and responds too quickly?

I believe, with the wording of the quiz, there's only one scenario that can yield a truly correct answer and that is the scenario with 1 and 2 wearing red, leaving #3 to figure out he's wearing white (and the unstated assumption being added to the quiz; "the chiefton tells the 3 men of the 5 bandannas and the distribution of colors").

Any other scenario relies on timing of silence, which is totally unreliable with 3 men who allowed themselves to be captured and who's intelligence is not mentioned.

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read the edits before you go saying anything, it actually says that they are all inteligent and rely on the intelligence of the other 2

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if all three said my headband is white then one of them HAD to be right...hence they are all free...

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wrzesinski: but two of them would be wrong and they would be imprisoned.

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I had a weird idea

It never said their were any consequences to saying the wrong head band color.

So someone could just guess a color and if he got it wrong then he would just say the other color and they would be free.

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