Mid-Week Brain Teaser

Bloomfield said:
Because they can see them, if they couldn't see them they wouldn't be able to solve the puzzle at all.
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?

What if there was one blue hat in there that they didnt know about. How would the kids left (after the last red says he has a red hat) know they all had black hats?
 
AJK said:
They don't, they know that their hats are "not red". (We've just all been using the word "black" as a convenience.) The puzzle is phrased correctly and quite carefully.
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What are the first class told though, in terms of colours of their hats?

Are they just told: "you have hats in your head, you have to guess what colour it is?"
 
There are two colours of hat - red, and not red. Once the person wearing the last red has identified themselves, then logically (from solutions as given and discussed above) the remaining people know that their hats are not red.

It doesn't matter what colour they actually are.
 
Jono8 said:
What are the first class told though, in terms of colours of their hats?

Are they just told: "you have hats in your head, you have to guess what colour it is?"
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No, the puzzle clearly states: "The teacher then asks each child in turn whether her hat is red."
 
Psiko provided the answer. I'm glad I didn't ask the puzzle that is more complex than this one! :)

Frankly, we're presuming there are only red and black hats on. We also presume the students know they only have a red or a black hat on. To presume otherwise is just picky :p

I didn't write it for reference, it was in a lecture I did, and figured it would amuse people.

The reason the theory works, is because you add an extra level of information with the 'at least one' piece of information. It is a demonstration in the difference between near perfect common knowledge, and perfect common knowledge.

kd
 
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Jono8 said:
?

What if there was one blue hat in there that they didnt know about. How would the kids left (after the last red says he has a red hat) know they all had black hats?
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Oh, I misread it, I though it was the one in the red hat who worked it out saying that the rest of the hats are black :p.

The rest of the children just say that their hat "isn't red", not that they are black.
 
King Damager said:
How do people not understand this. Psiko has even provided the answer. I'm glad I didn't ask the puzzle that is more complex than this one! :)

Frankly, we're presuming there are only red and black hats on. We also presume the students know they only have a red or a black hat on. To presume otherwise is just picky :p

I didn't write it for reference, it was in a lecture I did, and figured it would amuse people.

The reason the theory works, is because you add an extra level of information with the 'at least one' piece of information. It is a demonstration in the difference between near perfect common knowledge, and perfect common knowledge.

kd
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How does "atleast one" change anything when they already know that there's atleast 6 or 7 depending on what hat they're wearing?
 
AJK said:
They don't, they know that their hats are "not red". (We've just all been using the word "black" as a convenience.) The puzzle is phrased correctly and quite carefully.
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Except for the unnecessary second trial that is carried out which introduces no new information.

I still want you to explain why this logic....

If I can see other children with red hats who haven't been asked the question by teacher yet I will always say "I don't know" otherwise I will say "red"

Can't be used by the first group of children.
 
estebanrey said:
I still want you to explain why this logic....
If I can see other children with red hats who haven't been asked the question by teacher yet I will always say "I don't know" otherwise I will say "red
Can't be used by the first group of children.
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For the same reason as this logic can't be used by the first group of children:
"I will say 'red' if there is a red hat to my left."

Or, "I will say 'red' if I am the last person and nobody else has spoken."
Or, "I will say 'red' if my name is AJK."

There is *not enough information* given to the first classroom to decide that that rule is true. We can only see that it's true objectively - a viewpoint that the kids do not have.

I have tried to explain in this post:
http://forums.overclockers.co.uk/showpost.php?p=23711588&postcount=59
 
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Psiko said:
This is very similar to the green eyes/blue eyes problem (you can search the forums for it).

Here are some hints:
First think what happens if there is only 1 kid with 1 red hat and no black hats. Then think what if there are 2 kids with 2 red hats and no black hats. Then what if there were 3 kids with red hats and no black hats. This continues for as many red hats as you like.

Then think about what happens if there some black hats as well.

Here is my solution:
Assume no black hats.
1 red hat: kid knows straight away that he must have a red hat because he is told there is at least 1.
2 red hats: first kid doesn't know because he sees another kid with a red hat. If the first kid had seen the other kid with a black hat he would have known he had a red hat. Because the first kid didn't know, the second now does know that he has a red hat and says so. Because the 2nd said he has a red hat, the first now knows he must also have a red hat. (otherwise the second kid would have said he had a black hat)
...
n red hats: this continues by iteration so that it takes n-1 kids with red hats to say they don't know before the nth knows.

Now introducing black hats, the same thing happens, but you can ignore every time a kid with a black hat says he doesn't know because it gives you no additional information. Once the nth kid with a red hat says he has a red hat, everyone looks around and if they see n-1 red hats then they have a red hat, and if they see n red hats they must have a black hat.

I hope I explained that vaguely clearly...
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Is this the one you're saying is the correct answer?

Because in this the children are using the logic of n-1, but the way the scenario is written the children don't know what n is do they?
 
AJK said:
For the same reason as this logic can't be used by the first group of children:
"I will say 'red' if there is a red hat to my left."

Or, "I will say 'red' if I am the last person and nobody else has spoken."
Or, "I will say 'red' if my name is AJK."

There is *not enough information* given to the first classroom to decide that that rule is true. We can only see that it's true objectively - a viewpoint that the kids do not have.

I have tried to explain in this post:
http://forums.overclockers.co.uk/showpost.php?p=23711588&postcount=59
Click to expand...

But both classrooms have the same information (I'm going to ignore the "atleast one" thing to make this more obvious).

Both classrooms know that there are atleast 6 (if they have a red hat) or 7 (if they have a black hat) red hats.

Why can't the first one work it out and the second could?
 
Bloomfield said:
But both classrooms have the same information (I'm going to ignore the "atleast one" thing to make this more obvious).
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Yeah, you can't just ignore the one thing that's different...

Why can't the first one work it out and the second could?
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Does this not make sense:

Consider the first classroom, who do NOT have the statement "there is at least one red hat":

Person 1 will ALWAYS say "I don't know", regardless of whether they see red hats, black hats, or a mixture of colours. They do not have enough information to deduce what colour their hat is.

Since person 1 will always answer "I don't know", person 2 can't draw any information from that answer, and thus can only base their own guess on the hats they can see - which, as for person 1, is not enough information to draw a conclusion and so person 2 will also always answer "I don't know".

This continues for persons 3, 4, 5, etc. It's impossible to reach a point where you can logically conclude "I see only black hats remaining, and nobody else knew the colour of theirs, so my hat must be red". It's not logically true.
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Probably been solved several time over by now, but surely it's simple?

As soon as you see 'all black hats', then you must be 'red'.

The reason for this, is that, if you weren't 'red' then the person before you would have solved it already.'

Start at the beginning, if person 1 sees all 'blacks' then they know they are the one 'red' hat. If they are unsure, then this indicates that they can still see at least 1 'red' hat. People keep being eliminated while they see 1 or more 'red' hats. As soon as someone can't see a 'red' hat, then they must be the last 'red' hat that was keeping the previous players unsure.

After that the remaining players know they are 'black'.
 
King Damager said:
The reason the theory works, is because you add an extra level of information with the 'at least one' piece of information. It is a demonstration in the difference between near perfect common knowledge, and perfect common knowledge.
Click to expand...

But again I don't see what new information has been added. Unless the children are blindfolded up until after the first child is asked then the rule will always work with or without the teacher telling them "at least one hat is red" because they already know that!


Let me approach this from a programming point of view. Let's say I have written the following code into my program...

HAT = Child's answer
X = Number of Unchecked Red Hats

If X > 0 then
HAT = "I don't know"
X = X - 1
else
HAT = "Red"
End If

Loop until HAT = "Red"
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Now that will always set HAT as "red" on the last child who has a red hat UNLESS you passed it a dataset that contained no red hats in which case it would loop forever and always produce "I don't know" each time. So from that point of view I can understand the need for an additional rule to be added.

But if I know the dataset I will pass to that definitely contains at least one red hat then I don't need to add any additional rules to it and it works every time.

The children in the first class know that their dataset contains at least one red hat because they can see it from the beginning, hence the 'program' still returns "Red" on the last child who is checked under it.

So I understand why you need to know there is "at least one red hat" for the solution to work, what I don't understand is why in the way you've written the question why the first group of children don't already have that knowledge without the teacher telling them as well.
 
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