School teacher has a bag of 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]

white
Because I like to answer white.

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.

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".

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.

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.

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 ).

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~

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]

Just before the blindfold of the fourth pupil is removed the pupil shouts 'STOP. I know what colour my hat is'.

The Bob (2004 ©)

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 don't think so, we can wait for his solution to his own joke.

 

[Njorl]

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.

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]

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.

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]

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.

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 ?

 

[The Bob]

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 ?

Sorry but what is QED?

I cannot show any of you how until tomorrow at about 1 P.M. GMT time I am afraid because I cannot remember it. I am disappointed that it was up and finished in a day. I will post another one for you all though that I do know the answer right now to.

Have fun with my posts

The Bob (2004 ©)

 

[Njorl]

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 ©)

If you do not embrace the possibility that your math teachers are wrong, you will never surpass them.

Njorl

 

[The Bob]

A good point. What annoys me more is that I did it myself a while ago and it was right. I believe (hint here) you have to assume the different possible combinations that the hats could be in. Then imagine what the pupil is thinking and then work it out.

Hope this helps a bit

The Bob (2004 ©)

 

[The Bob]

If it turns out I am wrong then I will start a new thread and remove the hats thread.

The Bob (2004 ©)

 

[The Bob]

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 ?

I think Njorl has used the CTRL + V command 8 times in this thread

The Bob (2004 ©)

 

[BobG]

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 ?

Three times. (An alternate question might have been how many times did Njorl hit CTRL+C.)

In his last posting, Njorl copied, pasted, and edited the first half of paragraph 1, as needed, to make a longer paragraph in which the word "yields" was mispelled. Njorl corrected his mispelling in the pasted portion of the paragraph, but did not correct the original mispelling. (Normally, the photocopier catches most mispellings, right?)

Njorl then copied the entire first paragraph and edited it as needed for the second paragraph.

Repeat for the third paragraph.

What Njorl did not realize, is that he used the "American" spelling of anouncement in the first instance, then, in respect to The Bob, reverted to inserting the ever popular British "u" into subsequent spellings.

I would have to see the eyes of at least three other posters before venturing a guess as to how many times Njorl has hit CTRL+V in his lifetime.

 

[The Bob]

I would have to see the eyes of at least three other posters before venturing a guess as to how many times Njorl has hit CTRL+V in his lifetime.

Well I will say about 40 times a day.

Therefore 14,610 times a year and then multiple it by Njorl's age and add the necessary days and binjo. There is my random guess

The Bob (2004 ©)

 

[Njorl]

I use "shift insert" and "shift delete", thank you very much. Control V indeed!

Njorl

 

[The Bob]

I use "shift insert" and "shift delete", thank you very much. Control V indeed!

Njorl

--~~$LOL$~~--

 

[Gokul43201]

I use "shift insert" and "shift delete", thank you very much. Control V indeed!

Njorl

I know you can CUT using shift+delete, but how do you COPY ? Always wanted to know if there was one for that too !

 

[Njorl]

I never figured it out. I just cut and paste when I want to copy.
Njorl

 

[The Bob]

If you do not embrace the possibility that your math teachers are wrong, you will never surpass them.

Njorl

The Bob will show us how.

Well thanks to everyone for trying my impossible question. I would like to thank Gokul43201 for his/her support and encouragement and to Njorl for allowing me to realize my mistake. It turns out I was wrong so I apologise for my mistake (I admit I was obviously hoping it was right but it wasn't).

Therefore here it is (the one that works ):

Professor Pots has set a tough challenge for his four brightest students, who are sat facing each other. (Sorry I got this wrong in the original). He says "I have seven hats here, four black and three white. I will blindfold you all and then give you each a hat to wear. I will then remove the blindfolds and ask each of you in turn if you are able to work out the colour of your hat."

He does this. Each student thinks hard before he speaks and this is what each one says:
First Student: "I don't know".
Second Student: "Nor do I".
Third Student: "I don't know either".

Before his blindfold is removed, the fourth student announces the colour of his hat.

What is it and how does he know?

Well there it is. I hope it is easier now for you to do.

I apologise again.

The Bob (2004 ©)

 

[Gokul43201]

This is what Njorl solved in post#11.