An "Impossible?" Math Problem

As in, a problem from the math book "Impossible?" by Julian Havil:

There are a group of people in a room, and 15 people are given red hats and the rest are given blue hats. They can each see the color of everyone else's hats but can't see the color of their own hat. Also, they are all geniuses (and not the self-proclaimed kind, the real kind).  In other words, they all know logic perfectly (and they all know that they all know logic perfectly). A clock strikes every hour, and each person is told to leave when the clock strikes and they know they are wearing a red hat. In this situation, no one would leave, but if someone walked in and said, "At least one person is wearing a red hat," then everyone would leave after 15 strikes of the clock.  Sounds impossible, right?  (Explanation after the break)

If I were to write a story...

World Description

It was a world full of fighters, the strong and those who aspired to be.  A world ranking showed the relative skill levels of all those who participated in the system.  In this world, it was a generally accepted rule that open battles were not allowed; battles were to take place in magically created protected battle zones, in which no one would die.  However, there was no true law.  As a replacement, power became “law”.  The strongest were free to do as they pleased; the weak had to take care not to oppose those stronger on their journey to become strong themselves.

Reasons to Hate: Getting too Much Credit

Some people have really big egos.  I mean, I bet there are people out there who would go so far as to proclaim themselves a genius with no basis.  Whenever you give yourself too much credit, there are bound to be people who hate you for no other reason than that.  But I think that's not very surprising news to most people.

The Master of Everything

I was the master of everything.  I had the power to learn, which essentially meant that I could become good at everything I had the opportunity to do.  With an unyielding spirit, I would never lose a battle I cared about.    Basically, I was a true genius, and I would show the world why I deserved its respect.