This is a nice list of famous unsolved math problems from Wolfram MathWorld:
There are some well-known problems here, like the Goldbach Conjecture and the Collatz Conjecture, and some lesser-known open problems like finding an Euler Brick with an integral space diagonal.
It’s especially nice that several of these challenges are easy to explain to non-mathematicians. For example, the Goldbach Conjecture asks “Can every even number be written as the sum of two prime numbers?” Somewhat surprisingly, after nearly 300 years, the best answer we have is probably.
I think I’ll make this page next year’s summer homework assignment.