Solution to April's Puzzle: Naming an Heir

A mighty king had three sons and wanted to declare the wisest of them as his heir. He decided to give them a logic puzzle to test their wisdom. He placed the sons in a triangular room, each in a different corner, and placed a hat on each son's head. The king said, "You need to determine the color of your hat. You can't take your hat off to look at it, and you can't communicate in any way. The hat on your head is either green or red. At least one of you is wearing a green hat. I'll be waiting outside the door and will ring a bell every five minutes. You can't leave the room until you know the color of your hat. If you know the answer, you must wait for the next bell ring, then come tell me the answer." At the third bell ring, one of the sons opened the door and told the king the answer. The king said, "You're correct, and I'm naming you my heir. However, I'm disappointed in you. You have still much to learn." What was that son's answer, and why was the king disappointed?

That son's answer was green, based on the following logical deduction:

  • If there were two red hats and one green hat, the son with the green hat would have realized it immediately (by seeing both his brothers wearing red hats) and approached the king at the first bell ring. Because this didn't happen, there's—at most—one red hat among the sons.
  • If there was one red hat and two green hats, each of the two sons wearing green hats should have seen his brothers wearing one red and one green hat; therefore, both these brothers could have deduced that they were wearing green hats (since no one approached the king after the first bell ring, and there's at most one red hat in such a case) and thus approached the king at the second bell ring.
  • The son that ultimately figured out the answer reasoned that his brothers weren't stupid, so if no one approached the king at the second bell ring, they must all be wearing green hats. Of course, this tells you that he saw both his brothers wearing green hats. So, he approached the king at the third bell ring to say that he was wearing a green hat.

Why was the king disappointed in his son? The answer involves true wisdom. The son should have reasoned that any setting in which (at minimum) one of the hats is green and not all of them are green is an unfair contest. If at least one hat is green, and not all hats are green, different sons can figure out their own hat color at different points in time.

For example, if two of the hats are red, the son that wears a green hat can figure out the answer immediately and approach the king at the first bell ring, while the other two must wait to see whether someone approaches the king at the first bell ring (in which case it will be too late for them). Similarly, if one of the hats is red, the two sons wearing green hats can know the answer after the first bell ring and approach the king at the second bell ring, while the son with the red hat must wait to see whether someone approaches the king at the second bell ring (in which case it's too late for him).

If the king had favored one of the sons, he would have named that son his heir without a contest. Because he wanted to put their wisdom to test, you would expect the contest to be fair. The only way for the contest to be fair while having a minimum of one green hat is to have three green hats. The king expected one of his sons to approach him at the first bell ring with this logic.

May's Puzzle: The Next element in a Series

I got this nice puzzle from Adi Dafni. Given the following series of elements, can you determine the next element?

1, 11, 21, 1211, 111221, 312211, ?