The problem is: There is a math class with some students such that no matter what four students you choose there is always at least one that knows the other 3. (Note that knowing is symmetric, if A knows B then B knows A). Show that there is a student that knows everybody.
And the solution starts by showing that every student does not know at most 2 people... and after this the problem becomes almost trivial.
How can someone think about something like this?
The beginning of a problem resolution, specially if the problem is somewhat difficult, often seems magic. Just like a supernatural being was giving clues to the person who solves it.
But this is not the truth, obviously. Behind this brilliant idea, there are always (or almost always, at least) many dead ways, many failed tries.
It's just like if you go to a unknown city and you ask for the Main Square. The peasant says you some perfect indications and, if you follow them, you arrive at your destiny. But, how could the peasant know that? The answer is simple: he knows his city, and he knows it because he has been walking in it for years.