What are some "deep" math problems?

What are some more examples of “deep” math problems? By deep, I mean that the problem is easy to understand, but to find the answer you have to do some complex math in a field that does not intuitively have anything to do in the problem.

Here is an example-
You are playing a game with three piles of coins, each with a different number of coins in them. Each move, you can move some coins from a larger pile to double a smaller pile. You win when two of the piles have an equal number of coins in them. Is it always possible to win this game?

The answer is hard to find, and it has a lot to do with factorization, relative primes, powers of numbers, and greatest common factors, things that don’t intuitively go with the problem.