Are Asperger's really smart?

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I don't mean to offend. I read about a guy in the first half of the 19th century who was suspected to have this syndrom... he knew about 200 languages, got into harvard at the age of 11, was reading at 6 months, total genious in math...

I know this is an extreme case, but do you guys have better logic abilities, even slightly? Could you solve this logic question in a few seconds or something?

"Three children are in a line. From a collection of 2 red hats and 3 black hats, the teacher places a hat on each child's head. The third child sees the hat on the first two, the the second child sees the hat on the first, and the first child sees no hats. The children, who reason carefully (and for clarity, perfectly) are told to speak out as soon as they can determine the color of the hat they're wearing. After 30 seconds, the front child correctly names the color of her hat. What color is it, and why?"

Before I answer your question, I have one for you:

What is your purpose in being here, exactly?

Now, on to the problem...

The first (front) child's hat is red. The third child saw that the first and second children both have red hats, and therefore determined he had a black one (because there were only two red hats). The third child called out that he had a black hat. The second child and first child realized that they must have red hats, because otherwise, the third child would not have known that the color of his hat was black.

This admittedly took me awhile, I got caught up on the details of the question when visualizing it:

1. Are the children allowed to converse with eachother?
2. Can the child see the brim of her hat?
3. Are the "first" and "front" child the same child?
4. Is there a mirror in front of them?
5. Can they see the left over hats that the teacher did not give them?

Then I realized that the front child did not have to be the first one to call out the color of her hat. I had a hard time realizing that the knowledge did not have to come directly from the front child, but that it could be inferred from what the third child saw. At first I thought that the answer was black, because there were only two red hats, but then I realized there was no way for the front child to know that the two red hats were in use.

As for having superior logic skills, I'm often told that I'm very logical, and I often see simple solutions to problems that others seem to miss. I also analyze everything to a fault, which is why I had some difficulty with this question you've posed.

There are also some "theory of mind" issues here, I'm not sure if you are familiar with that term, but this question does seem to require a lot of thought about the mental states of others, something many aspies have difficulty with. (Which may be the reason why I intuitively thought the front child could have a black hat and know the other two were wearing red). Logic, for me, is a way of compensating for this difficulty I have with intuition. And no, it is not always faster. In fact, it is more often a slower process. But it's also more often correct.

I don't think i'm particularly smart. I'm not dumb either.

I would have trouble figuring this out, in fact i can't even be bothered

Most people would say that I am very smart and I scored in the 99th percentile on standardized tests. What is odd is that the last time I took an IQ test, I think I got a score of about 100. When I was in third grade, I scored so poorly on an assessment test that it was believed that I would need to be in special ed and was given a bunch of other diagnostic tests to make sure. I scored so well on those tests that I was told to go to a summer school program for gifted kids.

Asperger's Syndrome is defined in the DSM-IV diagnostic manual usually of an intelligence quotient of normal to above average. Not all individuals with Asperger's Syndrome display savant abilities. Nor do you need to have Asperger's Syndrome or on the autism spectrum to have savant abilities.

As far as answering that question, I would agree with the questions that Civet asked in relation to the question, I'm not always good at visualizing things as far as problem solving questions without some visual supports, like I'd likely draw out a picture to represent it.

That's a very difficult problem. Civet says she called out red hat, but I really don't know. The third child may see the two other children wearing black hats, so he could be wearing either a black or a red hat. He can only know for sure that he's wearing a black hat if the two in front of him are wearing red hats. If the third person is able to figure out he's wearing a red hat by this logic, the other two can figure they're wearing red hats. For any other combination of hats, I don't know how this problem could be solved.

I think that Civet's answer is correct and her reasoning valid. The problem is not difficult, but the key is to realize that the children were able to determine the colors of their hats. This information is explicitely provided in the problem description. The combination red-red-black is the only one that children can determine with certainty, and therefore is a solution to the problem. Any other combination would leave children not knowing what they were wearing and therefore not speaking out colors.

My best guess is that the child has a black hat on becasue none of the red hats were used, since someone would have to have on a black hat if one, or both of the red hats were used. This would mean that each child could guess either red or black, and have a 50/50 chance of being right.

I have a feeling that is one of the those "thinking outside the box" kind of problems like this one...

**using only four straight lines, draw a path so that the lines go through each number**

1-----2-----3

4-----5-----6

7-----8-----9

edit: An hour and a half has passed and I don't think I explained myself as well as I should have. In the problem, I assume that

- the children can't communicate with each other
- the children are lined up in a single file line so they can't look at the person behind him

To me, the fact the child #3 and #2 know what the colors are of the hats on the people ahead of them is irrelavant. What IS relavant to me is the fact that there are 3 children and 3 black hats, and that the "front" or 1st child in the line knows correctly what color hat he has on, but yet he can't see the colors of the hats on the people behind him. Is says nowhere in the problem that the red hats were used at all. And with out the red hats in the equation, all doubt is removed from the first child as to the color of his hat since there would only then be three black hats for three children.

Last edited by Scoots5012 on 25 Oct 2004, 1:16 pm, edited 1 time in total.

All right, this is a literally "outside of the box" problem! Unfortunately I knew it, so I just recalled the image of solution. It is here (one of possibilities). Don't click unless you're totally stuck, exercise is good for your brain!

Last edited by magic on 12 Mar 2005, 12:51 pm, edited 1 time in total.

Scoots wins, the first child has a black hat!

Civet assumed the other kids had already called out? The question doesn't state that!

More likely the third kid said nothing, as, unless the two red hats were out front, he coudn't, tell what he had on.

The second kid would reason that two red hats weren't therefore an option, so if the first kid didn't have a red hat on then he couldn't know if he had red or black.

The first kid therefore reasons that if the second kid couldn't work it out, it's because he had the only option which left the second kid unsure.

So the first kid has a black hat on!

As for Scoots' question, it took longer to work out if the lines were supposed to be joined, than it took to subsequently draw the solution!

Yes, the child in front must have a black hat. The question is confusing, and it is not clearly stated that the child in front is the first to respond.
This is interesting though, because if that is the case, it then becomes a "theory of mind" problem. And I too must ask, what is your purpose in being here?

This is just the sort of assumption that what some call "theory of mind" usually relies on.

Basically if the child in back could see 2 red hats then he would speak up.
He does not, therefore one of the front 2 must have black.
Given this knowledge, the middle child would speak up if he saw a red hat.
He does not, therefore the first child reasons after 30 seconds that she must be wearing a black hat.

But how does the first child know that the red hats weren't used? All three children think that they could have either a red hat or a black hat, I don't think you can just eliminate a color.



I'm sorry, but I don't understand part of your explanation. Why would the middle child speak up if he saw a red hat on the first child?

Civet, I must confess that after I read your original response I agreed with you. The question is not clear about who speaks first so it appeared that it might be a trick question.
It was only after reading Gwynfryn's explanation that I considered this more thoroughly. The whole problem is based on the fact that two children are looking at the first child's hat, and reacting to what they see.
No one is looking at the third childs hat, and only one child can see the middle hat.

The middle child has two clues.
1 the child in back did not speak up right away so he does not see two red hats in front of him. This means that either, child one or two or both are wearing black.
2 he knows the color of child one's hat in front of him. If it were red then he could deduce that his own had could not be red. If it is black then he can't be sure about his own hat, it could be either black or red.

Wow, that is rather complicated for me to wrap my head around, so to speak. I think I understand it now, and agree that you are correct.

It was this fact that I was having the most trouble with, before. I am having a very hard time keeping track of who is thinking what and who is aware of what, even as I go over your explanation. But it does make sense, now that I understand what you've said. And I don't know if I ever would have come up with that explanation.

I guess I was wrong when I said this:



It was not really about the number of hats as much as it was about the observations the other children were able to make about the hats, and the thoughts that their behavior (silence) implied. This is quite a few steps above the Sally-Anne test, I wonder how NTs generally do with this question. I guess since it's also about logic (not just theory of mind) it would be hard to tell what the results imply.

There are 7 possibilities of color distribution, listed below.
B - black hat, R - red hat, S - silence (1st/front child - left, 2nd child - middle, 3rd child - right)

Kudos to Gwynfryn for finding that silence can carry meaning and that possibilities 4-7 are valid. I think that Civet's solution is also valid. The problem description does not state whether children 2 and 3 said something or not. We cannot infer that "after 30 seconds" means "after silence". Please note that solutions 2 and 3 include silence, which may well last 30 seconds, yet 1st child speaks "red".

Anyway, we seem to have answered Ferno's question. No, most of us can't solve this problem within a few seconds.