Three top math students with equally great achievements were subjected to the following task: They were shown 5 hats. Three of them were red, two were black. The students were given a hat to wear each and while they could see the hats the two other students wore they could not see their own and not the two spares. The task was to find out which colour their own hat was without any help and without talking or signing to each other. After ten minutes one of them said: "I wear a red hat". And it was true. How could she know?

I'm going to go out on a limb here and guess that, because there are THREE of them, and THREE red hats, they probably thought, with two of the hats being spares after all, and two of them being black, that they in some way would "trick" themselves into stupidly guessing they were wearing black.

No doubt this sounds clearer in my mind than on the page but here goes...

The other two students had already guessed that they were wearing black hats. That meant they each saw a red hat on the other two students and, so, probability suggested to each of them that they would be wearing black hats. However, the third student saw each of the others with red hats and realized that they saw she was wearing a red hat as well.

magicalfreddiemercury wrote:
No doubt this sounds clearer in my mind than on the page but here goes...
The other two students had already guessed that they were wearing black hats. That meant they each saw a red hat on the other two students and, so, probability suggested to each of them that they would be wearing black hats. However, the third student saw each of the others with red hats and realized that they saw she was wearing a red hat as well.
???

I don't think so...that's not much different from my shorter answer but still, nice explanation.

magicalfreddiemercury wrote:
No doubt this sounds clearer in my mind than on the page but here goes...
The other two students had already guessed that they were wearing black hats. That meant they each saw a red hat on the other two students and, so, probability suggested to each of them that they would be wearing black hats. However, the third student saw each of the others with red hats and realized that they saw she was wearing a red hat as well.
???

I don't think so...that's not much different from my shorter answer but still, nice explanation.

Ah, but in my scenario, the other two wore RED hats, not black.

Let me try again: The third student DEFINITELY wears red(given).

The first student looks at the second and third ones. He sees the second one wearing black and the third one red. He can't come to a conclusion. The second student looks at the first and third ones. He sees the first one wearing red and the third one red as well. He can't deduct the colour of his own hat either.
The third student looks at the first student and the second student. He sees that the first one has a red hat and the second one a black one. He sees that both of them are silent. He realizes that the first student could
easily have deduced that he himself was wearing a red hat had his(third student's) hat been black since there were two black hats in total and the second student was wearing one already. So the third student realized that he must be wearing a red hat...
Does it make sense? I don't know if I'm right...

Here is the solution: Just imagine to be the student who has the solution first:
You see the other two students - which are the possibilities?

1. both wear a black hat

In this case you know you must wear a red hat and you say it immediately.

2. one of them wears a red hat, the second one wears a black hat.

Now you can wear either black or red but if you wear a black hat the guy with the red hat would say "red" immediately because he would see two black hats and would know immediately. When he does not say "I wear a red hat" you know that he does not see 2 black hats and you wear a red hat. This cannot take 10 minutes because the other two are very intelligent and if they do not say anything, you know right away that you wear a red hat.

3. both wear red hats.

Now you can again wear a black or red hat but if you wear a black hat, the other two see a black and a red hat each and are in the situation Nr. 2. They would guess pretty soon that they wear red because the other one is silent. Since none of the two says he wears red, it's obvious that you all wear red hats and the first who has thought this to the end wins.