Funproblems for REU

Editor's remarks: This is an ever growing set of funproblems.
You are invited to propose your own (send them to reu@math.rutgers.edu).
All our REU students are encouraged to think about these problems - alone or in groups -, but discouraged to simply tell the others about their wonderful solutions (and thus destroying the fun of others).

What are funproblems?

Not all fun  problems are funproblems, and definitely not all problems are funproblems.

Here are some vague criteria: A funproblem should have either an interesting naïve formulation (like L2 below) or an element of surprise (I2 or I3). Also, they should be fairly elementary and not too technical.

Illustration; the following problems certainly don't qualify:

If a sequence of continuous functions is pointwise convergent on a compact set $C$ then it is uniformly convergent on $C$. (False, by the way.)

If $R$ is a semi-simple ring then $R$ is left-Artinian and left-Noetherian. (True, by the way.)