Past open problems with sudden and easy-to-understand solutions What are some examples of mathematical facts that had once been open problems for a significant amount of time and thought hard or unsolvable by contemporary methods, but were then unexpectedly solved thanks to some out-of-the-box flash of genius, and the proof is actually short (say, one page or so) and uses elementary mathematics only?

From "Polygonal Rooms Not Illuminable from Every Point" by George W. Tokarsky, The American Mathematical Monthly Vol. 102, No. 10 (Dec., 1995), pp. 867-879:

Imagine two people in a dark room with many turns and cul-de-sacs.  Assuming that the walls, floors and ceilings are constructed of reflective material, can one person strike a match and be seen by the other after repeated reflections, no matter where the two are located?
This problem has been attributed to Ernst Strauss in the early 1950's, and has remained open for over forty years. It was first published by Victor Klee in 1969. [...]
In this article, we will settle the above problem in the negative.  We will as well give elementary techniques for constructing rooms, both in the plane and in three-space, which are not illuminable from every point.

