Mid-Week Brain Teaser

estebanrey said:
But I agree it's badly worded because their is no new information being introduced in the second trial. The OP should have presented as one test or said the difference the between the first and the second test was that in the first the children can't see the other kids hats and in the second they can or something.
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Not at all. The new information in the second test that makes it possible is that they are told "there is at least one red hat".

In the first test, the children aren't told how many red or black hats there are, so you can't use that method of deduction to determine the colour of your hat.

However, if you know that there is at least one red hat and all of the other red hats you can see have said "I don't know" and been eliminated, the only possibility is that your hat is red.


Bloomfield said:
Someone still needs to explain why the first kids couldn't work it out :D.

The only difference between the situations is that the teacher in the second says "at least one hat is red", but that's completely irrelevant since the kids can already see that there are at least 6 (if they have a red hat) or 7 (if they have a black hat) red hats, so why even tell them that? :confused:.
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It's a process of elimination that can't be implemented in the first test because the vital piece of information isn't given.
 
Bloomfield said:
Someone still needs to explain why the first kids couldn't work it out :D
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See the spoiler tag in my post above yours.

The only difference between the situations is that the teacher in the second says "at least one hat is red", but that's completely irrelevant since the kids can already see that there are at least 6 (if they have a red hat) or 7 (if they have a black hat) red hats, so why even tell them that? :confused:.
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That piece of information isn't about what they can see, it's about what they can logically conclude from the non-responses of the people who've gone before them. Again, see the spoiler tag in my post above yours.
 
Irish_Tom said:
Not at all. The new information in the second test that makes it possible is that they are told "there is at least one red hat".

In the first test, the children aren't told how many red or black hats there are, so you can't use that method of deduction to determine the colour of your hat.

However, if you know that there is at least one red hat and all of the other red hats you can see have said "I don't know" and been eliminated, the only possibility is that your hat is red.
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That's still the situation for the first, whether you tell them there's at least one red hat or not they still know that. They can see that there are red hats.

When someone says "I don't know it" means that they can see people with red hats, that doesn't change in either situation.
 
AJK said:
It isn't badly worded. There is exactly one new piece of information introduced in the second trial, which is the statement "there is at least one red hat". Observationally, this doesn't tell you anything (every child can see at least 6 red hats), but logically it does.

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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Sorry but there is no NEW information in the second trial. In both the children can see there is "at least one rat hat" at the begining of the test so it is redundant information both observationally and logically.

If I told you "the sky is blue" would you regard that as 'new' information? No because you are already aware of that.

Again, please tell me why the first group of children can't use the following system....

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"

As long as every child uses that rule the last child with a red hat will be correct and it works in the first trial without the teacher telling something they already know.
 
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AJK said:
See the spoiler tag in my post above yours.


That piece of information isn't about what they can see, it's about what they can logically conclude from the non-responses of the people who've gone before them. Again, see the spoiler tag in my post above yours.
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I saw your spoiler after posting. I don't see how it makes any difference. Either way you know there are red hats, whether you're told or not :confused:.
 
Bloomfield said:
When someone says "I don't know it" means that they can see people with red hats, that doesn't change in either situation.
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This is the part which is NOT TRUE in the first classroom. You could see all black hats and still not know.
 
AJK said:
This is the part which is NOT TRUE in the first classroom. You could see all black hats and still not know.
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In that situation you wouldn't know, but that isn't the situation in the OP. In the OP you can see there are red hats, being told something you already know doesn't change anything.
 
I think I understand our differences in opinion here. Again if we agree the 'rule' each uses is...

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"

Then the only situation in which that wouldn't work would if there were no red hats. If they were all wearing black hats and following the rule then the first child would say "red" and be wrong.

So from a programming mindset this rule wouldn't suffice, you would have to add in another condition that states only to use the rule when at least one red exists. This is what I think AJK is getting at.

However, this isn't a computer program that is expecting a random data set, it's a fixed scenario where we know what the input contains both red and black hats in advance so that 'check' is redundant and that's what I and Bloomfield are saying.
 
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estebanrey said:
I think I understand our differences in opinion here. Again if we agree the 'rule' each uses is...

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"
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Where does that "rule" come from though? We can't just create it as outside observers and say "yes, that'll work". You need to derive it from the information in the puzzle, and you do not have enough information in the first classroom to create that rule. If you think you do, show your working?
 
Imagine there are seven red hats, you are wearing one of them but don't know there are seven. If the six red hats in front of you say I don't know, you still don't know. For all you know there are only six red hats.

However, if you are told there is at least one red hat and the other six say they don't know you can conclude that you are the red hat.

Just because you know there are six red hats in front of you does not mean you can work out that there are in fact seven red hats unless you are given that bit of information.
 
Irish_Tom said:
Imagine there are seven red hats, you are wearing one of them but don't know there are seven. If the six red hats in front of you say I don't know, you still don't know. For all you know there are only six red hats.

However, if you are told there is at least one red hat and the other six say they don't know you can conclude that you are the red hat.

Just because you know there are six red hats in front of you does not mean you can work out that there are in fact seven red hats unless you are given that bit of information.
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Why does being told there's at least one red hat make a difference when you already know that there is at least 6 red hats?
 
Bloomfield said:
Why does being told there's at least one red hat make a difference when you already know that there is at least 6 red hats?
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Because once the six red hats are eliminated (by saying they don't know) and all the other black hats you can see are eliminated by being black, there is only one possible outcome — you are wearing a red hat.
 
Irish_Tom said:
Imagine there are seven red hats, you are wearing one of them but don't know there are seven. If the six red hats in front of you say I don't know, you still don't know. For all you know there are only six red hats.

However, if you are told there is at least one red hat and the other six say they don't know you can conclude that you are the red hat.

Just because you know there are six red hats in front of you does not mean you can work out that there are in fact seven red hats unless you are given that bit of information.
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But that isn't the situation, in both classes all children can see a selection of red and black hats so the logic to determine your hat remains the same.

The extra information makes no difference in the circumstances described because all students know there are red and black hats.
 
AJK said:
Where does that "rule" come from though? We can't just create it as outside observers and say "yes, that'll work". You need to derive it from the information in the puzzle, and you do not have enough information in the first classroom to create that rule. If you think you do, show your working?
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The "rule" is the logic that has to be used by the children to solve the puzzle. If the children follow that rule then the last child wearing a red hat will always be correct and thus answers the OP question of “How do the children in the second class discover the answer?”.

We can work this rule out in the same way as the children can, it doesn’t matter that we are ‘outside observers’. The only thing different is we know there are 7 red hats in total whereas each child only knows there are 6 or 7 (if wearing red) or 7 or 8 (if wearing black). Us knowing the exact total isn’t necessary to us working out the rule and the children not knowing the exact total doesn’t prevent them from working out the rule because the rule doesn’t require you to know the exact number of hats; only that there are more than one (which they can all see from the beginning).
 
Irish_Tom said:
Imagine there are seven red hats, you are wearing one of them but don't know there are seven. If the six red hats in front of you say I don't know, you still don't know. For all you know there are only six red hats.

However, if you are told there is at least one red hat and the other six say they don't know you can conclude that you are the red hat.

Just because you know there are six red hats in front of you does not mean you can work out that there are in fact seven red hats unless you are given that bit of information.
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Well no because the sixth red hat would say "i'm red" if they saw only black hats left and there person before said "i don't know" due to there being at least one of each colour left.
 
Do the kids know there are only two colours though?

In the first classroom,what were they told by the teacher with regards to colours of their hats?
 
Jono8 said:
Do the kids know there are only two colours though?

In the first classroom,what were they told by the teacher with regards to colours of their hats?
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It's irrelevant, there could be any number of colours. The issue is red vs. not red.

The puzzle doesn't change if there are 7 red hats, one black, one white, one blue, one yellow and one green.
 
AJK said:
It's irrelevant, there could be any number of colours. The issue is red vs. not red.

The puzzle doesn't change if there are 7 red hats, one black, one white, one blue, one yellow and one green.
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It does, because how would the kids after the one who says " i have a red hat" know that all their hats were black?
 
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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