Nine brilliant most students were summoned by an inspecting professor. He gave them a situation and told them that he is having nine hats. The hats are either red colored or blue colored as told by the professor. He also tells them that he have at least one red colored hat and the number of blue hats are greater than the red hats.
He places one hat on each of their heads. The students are not allowed to talk with each other and no means of communication is feasible. He asks them how many hats are blue and how many are red. He gives them half an hour to deduce and moves out of the room. Nobody is able to answer when he returns back and thus he gives them another fifteen minutes. But when he returns, no one can answer again. Thus he gives them final five minutes. On returning back this time, everyone was available with an accurate answer.
How could the students have deduced the right answer? What is the right answer?
Solution:
In the first wave, let us make an assumption that there is only 1 red hat and other 8 hats are blue. The student who can see all the other 8 blue hats on others' head will certainly be clear that there are 8 blue hats and 1 red hat since the professor told them that there is at least 1 red hat among the nine. But no one was able to answer after the first half an hour. Thus this can't be possible and our assumption is wrong.
Now let us assume that there are 2 red hats and 7 blue hats. The student who is having a red hat on his head can definitely see 7 blue hats and one red hat. He can deduce that he is wearing a red hat. But nobody was able to answer in the second interval as well. Thus our assumption is again wrong.
In the last wave of final five minutes, let us assume that there are 3 hats and 6 blue hats. The student with a red hat can see 2 other red hats and 6 blue hats and he may deduce that he is wearing a red hat. In this interval, everyone was able to answer which states that our assumption is correct.
Thus there are 3 red hats and 6 blue hats.