Question

Why are some competitive programming problems difficult? What makes them difficult and why do we find them hard to solve?

Answer 1

What you’re really asking is: why do difficult problems exist, in general? There is nothing special about competitive programming that gives rise to hard problems. Every technical field has hard problems. (Problems with a social element are also hard, but for a different reason.)

The reason why problems are difficult has to do with the limitations of our ability to search for solutions. The easiest problems are ones that we’ve seen before. The next easiest problems are ones that can be solved using a minor variation of a technique we already know is useful for solving other, related problems. When a problem doesn’t fall into one of those categories, it may be easy to other people (or, perhaps, to some extraterrestrial race that happens to have already developed the techniques required to solve it) but it’s not easy for us, because there’s no clear path toward constructing a solution inside our minds.

The fact that humans are able to solve some complex problems of an abstract nature is already remarkable.