Hats

The Bob

Right. Here I go.

A school teacher sits four pupils around a table so that two are facing another two. The pupils are not allowed to look side-ways.

The school teacher has a bag of hats (4 are black and 3 are white). He blindfolds the pupils and randomly places hats on the pupils heads.

One at a time the teacher asks the pupils if they know what colour hat they have one. The first pupil says no and removes his blindfold and is then asked again. He still states that he does not know. The second pupil says no and removes her blindfold and still does not know. The third pupil does the same. Just before the blindfold of the fourth pupil is removed the pupil shouts 'STOP. I know what colour my hat is'.

What is the colour of this pupils hat and how does he know?

The Bob (2004 ©)

TenNen

white
Because I like to answer white.

The Bob

I asked how he knows not why you want to say white.

If I am not going to get sensable answers then what is the point in me posts these messages? They are here to make people think not to make fun of me.

Sorry to sound a bit mean.

The Bob (2004 ©)

TenNen

I am sorry, I didn't mean to offend you, not at all, really!
I didn't complete the whole sentence, I meant "I don't know how, I just answer white Because I like to answer white".

So I will now leave it blank waiting for some others who will join to have fun with you, not to make fun of you.

You didn't sound mean at all, even a little. It is I who now have to say "thank you, Bob" for your excuse...

Does that clear up everything ??? ~lol~

The Bob

No Problem.

It is just I thought you may have been having ago at me because of my age. That is all. I may be young but no stupid.

I do not want you to leave it blank at all. I want people to try to have fun at it (and because off hand I have forgotten but I will remember ).

Everything is clear except what is my excuse?

Cya Around

The Bob (2004 ©)

P.S. Back to the point in hand. What is the colour of the pupils hat and why is it? How does the person know?

Njorl

First assumption - The fourth pupil is smarter than the third pupil. This must be the case. The third pupil has all the information the fourth pupil has and more, yet fails to solve the problem. If the problem is solvable by the fourth pupil, it is solvable by the third pupil. The alternative is an awful answer like "The teacher told the fourth pupil ahead of time."

Second assumption - Hats can be seen in the reflections of the eyes of the pupils across the table. I believe the author is giving a hint by using "pupil" instead of "student"

Third assumption - The nature of the reflected image is poor, such that one could only be sure of the color if one had two different reflected hats for comparisson.

Fourth assumption - This is the assumption that only the fourth sudent is clever enough to make. A white hat would have a stark, identifiable reflection. Anyone who only saw black hat reflections would remain uncertain, but if someone saw a white hat reflected, he would know it for sure. A black hat reflected from the eyes black pupil would be an indefinite thing.

Fifth assumption - the pupil next to the fianal pupil did not go first, and the final pupil knows this from hearing the voices of the pupils.

Since the pupil next to the final pupil did not see a white hat reflection, neither he nor the student next to him had a white hat on. The final pupil figures this out, and knows he has a black hat.

Njorl

vikasj007

It clearly says that the fourth pupil shouts before his blindfold is removed, so there is no chance that he sees any sort of reflection in the eyes of the pupil sitting in front of him.

Also stop making these vague assumptions.

TenNen

I dont think so, we can wait for his solution to his own joke.

Njorl

My solution does not require the fourth pupil to see anything. It only requires that he knows no other pupil has seen a white hat reflected.

Assumptions are required. The puzzle is not solvable from a strictly logical viewpoint. If it were, the third student could solve it. Therefore, it is a logical imperitive that assumptions be made.

Njorl

Gokul43201

hey, the bob,

could you please confirm that #1 and #2 sit on the same side of the table ? If that's true, this one's a real toughie. I can't seem to find any arrangement that would make #3 certain of his color. But the problem suggests that there would have to be more than one such configuration, for #4 to be able to eliminate them on the basis that #3 is unsure.

Also it would seem that #1 and #2 have access to the same information, as do #3 and #4. So, as Njorl suggested, there must be something that breaks the symmetry. I'm not sure if the sequence of being asked can provide the breaking of this symmetry. But I can see why Njorl chose to resort to "additional cleverness" !

Njorl

If the students were allowed to look sideways, it would be logically solvable this way -

First student sees something other than 3 white hats. If three white were seen, the pupil would know his own hat was black.

Second pupil knows that the first pupil saw at least one black hat. If pupils 3 and 4 both have white hats, his own hat is black. 3 and 4 do not both have white hats.

Pupil 3 knows that pupil 2 saw at least one black hat between pupil 3 and 4. Pupil 3 sees a black hat on pupil 4, so he does not know if he has a black or white hat.

Pupil 4 knows that pupil 3 would know his own hat color if he saw a white hat on top of pupil 4's head. Pupil 4 knows his hat is black.

This solution requires looking sideways.

He could have put the students in a line, and disallowed looking backwards. Pupil 1 in back, pupil 4 in front.

Njorl

Njorl

Consider the information you get given that no sideways looking is allowed.

Pupil 1 sees 2 pupils across from him. He announces he does not know his own hat color. We know that he does not see either 3 white hats or 4 black hats. Considering that we know he only sees two hats, we have gained no information at all. We do not even know which two students he sees, only that he does not see himself.

The same for pupils 2 and 3.

Njorl

The Bob

I can confirm that pupil 1 and 2 are on the same side and pupil 3 and 4 are on the same side. Pupil 1 and 2 can see pupil 3 and 4's hats but pupil 1 cannot see pupil 2's hat. This is the same for pupils 3 and 4.

The Bob

Pupil 1 can see pupil 3 and 4's hats.
Pupil 2 can see pupil 3 and 4's hats.
Pupil 3 can see pupil 1 and 2's hats.
Pupil 4 can see pupil 1 and 2's hats.

It is pupil 4 that shouts stop and the order of calling is (obviously) 1, 2, 3 and 4.

Njorl

Pupil one states he does not know his hat color blindfolded.
All possibilities available prior to the annoncement are equally possible after the announcement.
The announcement yirlds no information.
Pupil one sees pupil 3&4's hats. He still does not know his own hat color.
All possibilities available prior to the annoncement are equally possible after the announcement.
The announcement yields no information.

Pupil two has gained no information.
Pupil two states he does not know his hat color blindfolded.
All possibilities available prior to the annoncement are equally possible after the announcement.
The announcement yields no information.
Pupil two sees pupil 3&4's hats. He still does not know his own hat color.
All possibilities available prior to the annoncement are equally possible after the announcement.
The announcement yields no information.

Pupil three has gained no information.
He announces he does not know his own hat color blindfolded.
All possibilities available prior to the annoncement are equally possible after the announcement.
The announcement yields no information.
He then sees pupil 1&2's hats. He states he still does not know his own hat color.
All possibilities available prior to the annoncement are equally possible after the announcement.
The announcement yields no information.


Pupil 4 has gained no information. He claims he knows his own hat color. From the information provided, this is not possible.

Njorl

The Bob

The question is possible. I know it is because it is outside my maths classroom at school. I cannot, however, off hand remember how it is done. The question is all I can remember. I will check and see if any other information is given, but I do not think there is.

The Bob (2004 ©)

Gokul43201

May I assume that we have all conceded the floor to the bob ?

Njorl, you forgot to add "QED", so your proof is incomplete. The Bob will show us how.

Bonus question : How many times did Njorl hit CTRL+V ?