Problems with difficult-to-see solutions that are nonetheless easy to grasp?

Question

Like the title says, I'm looking for math problems that are easy to understand, but difficult to see the solutions immediately or even after much thought; but when shown the solution, the solution is easy to understand. There's this "poisoned wine challenge" problem which is a pretty great example of this. (PBS infinite series made a video on that.) It could be an old problem, or a new one, doesn't matter.

P.S. I apologize if this question has been asked before.

Answer 1

Tanya Khovanova maintains a web site with problems of this type:

The Mathematics Department of Moscow State University, the most prestigious mathematics school in Russia, had at that time been actively trying to keep Jewish students (and other "undesirables") from enrolling in the department. One of the methods they used for doing this was to give the unwanted students a different set of problems on their oral exam. These problems were carefully designed to have elementary solutions (so that the Department could avoid scandals) that were nearly impossible to find. Any student who failed to answer could be easily rejected, so this system was an effective method of controlling admissions.

The introductory page to the site has a list of links to related material.

Answer 2

You can try to " Find out the average distance between two randomly selected points in a Circle " . As the problem looks innocent but , its quite tough . Here's a hint : , $$\begin{align}I(1)&=\frac{\int_{0}^{1}\int_{0}^{1}\int_{0}^{\pi}\sqrt{(\sqrt{r_1}-\sqrt{r_2}\cos\theta)^2+(\sqrt{r_2}\sin\theta)^2}⠀d\theta dr_1 dr_2}{\int_{0}^{1}\int_{0}^{1}\int_{0}^{\pi}d\theta dr_1 dr_2}\\&=\frac{1}{\pi}\int_{0}^{1}\int_{0}^{1}\int_{0}^{\pi}\sqrt{r_1+r_2+\sqrt{r_1r_2}\cos\theta}⠀d\theta dr_1 dr_2 \end{align} $$