The Problem
A wealthy man has three sons, and wishes to divide up his inheritance between them. He wants to give the majority of his money to his most clever son, so he devises a test to figure out who his most clever son is. He calls all three sons and tells them:
You shall all close your eyes and then I will put either a green or a purple dot upon each of your foreheads. Then you may open your eyes. If you see a green dot on the head of either, or both, of your brothers, raise your right hand into the air. Whichever of you can determine what color the dot upon your own head is quickest shall have the majority of my inheritance.
The man then has his sons sit down and close their eyes. He puts a green dot on all three of their heads. He then tells them to open their eyes, and leaves the room.
After about 15 minutes, one of his sons comes out of the room and says to him, "The dot on my head is green."
How did the son know that the dot on his head was green?
The Solution
To solve this problem, let's reverse the question for a second: why wouldn't the son know that the dot on his head was green? In the case we are given, with all three sons having green dots on their heads, all three sons raise their hands. If you think about it, there is one other possibility where all three sons would raise their hands. If two of the sons had green dots and one had a purple dot, all three of them would end up raising their hands.
If it were the case where one son had a purple dot and two sons had green dots, everything would look exactly the same from the point of view of the son with a purple dot on his head. He would see that both his brothers had green dots and both of them were raising their hands. So now the question becomes, how did the clever son know that it was in fact the case where all three of them had green dots and that he did not have a purple dot on his own head?
We have to continue to think from the perspectives of each the people involved in the situation. Imagine if it were the case where one son had a purple dot and the other two had green dots. Think from the perspective of one of the sons who had a green dot on his head. He would see one of his brothers had a purple dot, one had a green dot, and both of them were raising their hands. For the second brother with the green dot to be raising his hand, at least one of the other brothers would have to have a green dot. The first brother with the green dot would know this, and would also know that his third brother had a purple dot. It would be more or less immediately obvious to him that he had a green dot on his own head. To summarize, in the case where one son has a purple dot and two have green dots, both the sons with green dots can know almost instantaneously that they have green dots.
If you refer back to the original problem, you will recall that the clever son took about 15 minutes to figure out that he had a green dot. You can conclude that his thought process must have been something like this:
I have either a purple dot or a green dot on my head. However, if I had a purple dot on my head, then both my brothers with green dots would know very quickly that they had green dots. If it has been this long and they have not concluded that they have green dots, then I must not have a purple dot on my head. I have a green dot on my head.