P versus NP problem
Digging a little deeper, we see that all P problems are NP problems: it is easy to check that a solution is correct by solving the problem and comparing the two solutions. However, people want to know about the opposite: Are there any NP problems other than P problems, or are all NP problems just P problems? If NP problems are really not the same as P problems (P ≠ NP), it would mean that no general, fast ways to solve those NP problems can exist, no matter how hard we look. However, if all NP problems are P problems (P = NP), it would mean that new, very fast problem-solving methods do exist. We just have not found them yet.