Confidence is quite important in mathematics. You would usually go further if you are confident than if you are not. That said, most unsolved problems in mathematics require more knowledge to state than these simple problems.

I think for most people, not knowing a problem is unsolved will lead them to try out many more approaches than if they knew. The problem is that most of the approaches will be approaches that have already been tried before and the rest will be useless. This does nothing for the solving of the problem but greatly aids your own understanding.