### Brain Teasers

# Headbands I

Fun: (2.19)
Difficulty: (2.69)
Puzzle ID: #39479

Submitted By: zeepuzzlemaster Corrected By: zeepuzzlemaster

Submitted By: zeepuzzlemaster Corrected By: zeepuzzlemaster

Logic
Logic puzzles require you to think. You will have to be logical in your reasoning.

You are part of a group of 500 people who have been captured by an evil tribe. The tribe leader, who appreciates a person with good logic, decides that he will propose a challenge of many headband problems.

The first challenge is that each person will get a random headband of one of two colors, black or white. There are 500 of each (500 black and 500 white). If you look at yours or say a word after you have been given your headband and before you have guessed, the evil people will kill everybody. Also, if you make any movement with your body other than your eyes, you will ALL die.

At the count of 10, each person will shout out what color they think their headband is. If you or anyone else is right, you will survive for the next challenge. If not, you will be killed immediately. You are given some time to discuss your strategy; then you are given your headbands.

How many survivors can you guarantee, assuming the tribe leader is not lying and that each person will follow your instructions? For example, if you tell everybody to guess, you can guarantee zero survivors because you have a slight chance of getting zero survivors.

The first challenge is that each person will get a random headband of one of two colors, black or white. There are 500 of each (500 black and 500 white). If you look at yours or say a word after you have been given your headband and before you have guessed, the evil people will kill everybody. Also, if you make any movement with your body other than your eyes, you will ALL die.

At the count of 10, each person will shout out what color they think their headband is. If you or anyone else is right, you will survive for the next challenge. If not, you will be killed immediately. You are given some time to discuss your strategy; then you are given your headbands.

How many survivors can you guarantee, assuming the tribe leader is not lying and that each person will follow your instructions? For example, if you tell everybody to guess, you can guarantee zero survivors because you have a slight chance of getting zero survivors.

### Hint

You can guarantee a lot less than you think...### Answer

You can guarantee that 1/2 of the people will survive.What to do? Partner everybody up and tell one person to guess the same color as the other person's headband, and the other person to guess the color of the headband that they see.

For example if Person #1 guesses the same and Person #2 guesses opposite...

Person #1=Black Person #2=Black Person #1 Survives

Person #1= White Person #2=White Person #1 Survives

Person #1=Black Person #2=White Person #2 Survives

Person #1=White Person #2=Black Person #2 Survives

Note: You cannot tell person #1 to guess the same color as the second person's hat and then the second person to guess that color, because everybody has to guess at the same time.

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## Comments

I can guarantee 100% survival.

Simply pair everyone up and have each person signal their partner's headband colour (e.g. scratch face = black, scratch leg = white; or feet together = black, feet apart = white) while seeing what their partner is signalling to them.

Every pair of people can devise their own signals, so that it is not so obvious. (Although it doesn't actually say that hand signals aren't allowed).

The idea that a tribal leader would devise such games always amuses me; it reminds me of a Phantom comic

Simply pair everyone up and have each person signal their partner's headband colour (e.g. scratch face = black, scratch leg = white; or feet together = black, feet apart = white) while seeing what their partner is signalling to them.

Every pair of people can devise their own signals, so that it is not so obvious. (Although it doesn't actually say that hand signals aren't allowed).

The idea that a tribal leader would devise such games always amuses me; it reminds me of a Phantom comic

I believe your answer reads incorrectly - probably left out the word "opposite" for the second person inadvertantly.

The only downside of your answer is that you have also guaranteed that 1/2 of the people will lose/die. But it does answer the question posed.

The only downside of your answer is that you have also guaranteed that 1/2 of the people will lose/die. But it does answer the question posed.

tpg76, thanks, i didn't see that.

now all I need is an editor...

now all I need is an editor...

I am piggy backing on left clicks response, assuming that is allowed and you didnt state it wasnt so....thats what I would do as well....

That's brilliant! Took a LONG while to understand the logic but it's early on a Monday morning!

Or you could tell everyone to guess white (or black)

Oh, that is morbid! I was sure the answer would involve saving all and I couldn't come up with a solution.

Great job, LeftClick!

Great job, ZeePuzzleMaster on the teaser too!

Great job, LeftClick!

Great job, ZeePuzzleMaster on the teaser too!

Nomez, did you read the answer?

edit: did you read the teaser? i even stated that you CAN'T tell everyone to guess

i can guarantee 99% survival.just let somebody to line them up into black group and white group.

let all of them line up,you must touch everyone with black hat,and those who have been touched should move to left side.each of them have to shout the color of the hat in front of them.this is not wrong because the question did not mention that they cannot move.easier right??

let all of them line up,you must touch everyone with black hat,and those who have been touched should move to left side.each of them have to shout the color of the hat in front of them.this is not wrong because the question did not mention that they cannot move.easier right??

I was all for partnering up and holding up 1 finger if your partner had a black headband and 2 if they had a white headband -- that way everybody lives.

wussup evil tribe got to do with the hats?

ouch..

hurmmm..

ouch..

hurmmm..

It was ok the logic was kind of weird but ok. I saw a riddle sort of like this but they didn't have to call it out at the same time. In that case, you line everyone up, and designate the last person to be the one who has the possibility of losing. The last person would (by the way, in that riddle there were only 10 people so it's easier) count how many black hats he sees. If it is an even number, he will guess black for himself, if it is odd, he will guess white. The person in front of him will in turn count the number of black hats in front of him. Because he knows that the person behind him, say, saw an even number of hats, then if he too sees an even number, then the second person knows that he must be white. And so on and so forth. Each person just has to keep track of where they are in line, and how many blacks have been called. Does that make sense?

Overall, good teaser. I like these kinds.

Overall, good teaser. I like these kinds.

The only problem with the 100% and 99% survival rates is that he DOES say (and I quote) "Also, if you make any movement with your body other than your eyes, you will ALL die." so moving around, hand signal, etc is out of the question, HOWEVER blinking right eye for black and left eye for left would still gurantee 100% survival.

Every time there's a "figure out what color hat everyone is wearing using only logic" puzzle, even though they always state that you can't do anything to signal each other, there are always a ton of responses by people saying they would just signal everybody. Why would anyone keep posting logic puzzles like this if the answer is always "just secretly signal everybody!?" The point is to use LOGIC, not signals!

Sorry, I just had to rant.

Now that I've criticized everybody else's answers, I've got a possibly dumb alternative solution of my own to offer. You could guarantee that about 50% OR MORE would survive. I realize that the "about" part means that there are a few scenarios where you might have a little bit less than a 50% survival rate, and therefore the given answer is still the highest absolutely guaranteed survival rate...but this way would produce better than 50% survival in most scenarios, and only in about 1/5 of the possible scenarios would survival be as low as 50% (or close to it). Plus you're not guaranteeing that 50% will die.

There are too many headbands to count in the allotted time, but you would still be able to tell if there were a fairly clear majority of one color over the other (assuming you could see everybody else). So everyone should agree in advance to say the color that appears to be the majority (if there is a clear majority). That way the majority of the people would be right and survive. If it's close to 50/50, then that wouldn't be a safe option, so everyone should agree in advance that if the ratio appears to be close to 50/50 everyone says white. That way about 50% would survive (or whatever the actual "almost 50/50" ratio is).

If you're interested in my math on this, here it is: each individual person would actually have a better than 50/50 chance of survival, since most scenarios would result in a visible majority. Even if you could only see a majority if it were 60/40, there are still only 1 out of every 5 of the possible scenarios that would result in a ratio that would fall in that range. There are 501 possible combinations of headband colors: 1. zero whites, 2. one white, 3. two whites...all the way to 501, which would be 500 whites. So a ratio that would fall in the range of 60/40 to 40/60 would be 200 white and 300 blacks â€“ all the way to 300 whites and 200 blacks (scenarios 201 â€“ 301). So only those 100 out of the possible 501 scenarios would result in a survival rate of 40%-50%. But you would probably be able to discern a majority even if it were closer than 60/40. And if there happened to be 90% or even 100% of one color, then everyone would be REALLY grateful they didn't choose the guaranteed 50% survival rate!

Of course, since it doesn't GUARANTEE a 50% survival rate, the given answer is still technically the best fit for the stated conditions. But this is the strategy I would vote for, if I were captured by this particular evil tribe (although, personally I've decided to just stay away from evil tribes altogether, especially if I see a large stash of black and white hats or headbands)!

Sorry, I just had to rant.

Now that I've criticized everybody else's answers, I've got a possibly dumb alternative solution of my own to offer. You could guarantee that about 50% OR MORE would survive. I realize that the "about" part means that there are a few scenarios where you might have a little bit less than a 50% survival rate, and therefore the given answer is still the highest absolutely guaranteed survival rate...but this way would produce better than 50% survival in most scenarios, and only in about 1/5 of the possible scenarios would survival be as low as 50% (or close to it). Plus you're not guaranteeing that 50% will die.

There are too many headbands to count in the allotted time, but you would still be able to tell if there were a fairly clear majority of one color over the other (assuming you could see everybody else). So everyone should agree in advance to say the color that appears to be the majority (if there is a clear majority). That way the majority of the people would be right and survive. If it's close to 50/50, then that wouldn't be a safe option, so everyone should agree in advance that if the ratio appears to be close to 50/50 everyone says white. That way about 50% would survive (or whatever the actual "almost 50/50" ratio is).

If you're interested in my math on this, here it is: each individual person would actually have a better than 50/50 chance of survival, since most scenarios would result in a visible majority. Even if you could only see a majority if it were 60/40, there are still only 1 out of every 5 of the possible scenarios that would result in a ratio that would fall in that range. There are 501 possible combinations of headband colors: 1. zero whites, 2. one white, 3. two whites...all the way to 501, which would be 500 whites. So a ratio that would fall in the range of 60/40 to 40/60 would be 200 white and 300 blacks â€“ all the way to 300 whites and 200 blacks (scenarios 201 â€“ 301). So only those 100 out of the possible 501 scenarios would result in a survival rate of 40%-50%. But you would probably be able to discern a majority even if it were closer than 60/40. And if there happened to be 90% or even 100% of one color, then everyone would be REALLY grateful they didn't choose the guaranteed 50% survival rate!

Of course, since it doesn't GUARANTEE a 50% survival rate, the given answer is still technically the best fit for the stated conditions. But this is the strategy I would vote for, if I were captured by this particular evil tribe (although, personally I've decided to just stay away from evil tribes altogether, especially if I see a large stash of black and white hats or headbands)!

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