Just because we can't find a solution it doesn't mean that there isn't one.
The greatest problem for mathematicians now is probably the Riemann Hypothesis.
But the best problem I ever found, I found in my local public library.
I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days.
However impenetrable it seems, if you don't try it, then you can never do it.
Pure mathematicians just love to try unsolved problems - they love a challenge.