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

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.

• I don't think it is obvious that there is no largest prime. Euclid's method of showing that if $S$ is a non-empty finite set of primes, then $S$ is not all the primes by observing that $1+\prod_{p\in S}p$ is divisible by a prime not in $S$ is clear. – DanielWainfleet Oct 28 '17 at 11:40
• The Monty Hall problem: Suppose you and your twin play together by jointly choosing the same initial door (at random, by dice). Suppose you never switch and your twin always does. Your chance of winning is unaffected by Monty Hall's opening of another door . You just wait until all doors are opened. So you win 1/3 of the time. But every time you don't win, your twin, who always switches, $does$ win....... A great deal of ink, much of it nonsense, has been spilled on this problem. – DanielWainfleet Oct 28 '17 at 11:51
• From "Mathematical Recreations And Essays" by Rousse-Ball & Coxeter (11th edition, which is Coxeter's updated revision of R.-B.'s old classic): "It is amusing to ask someone for a proof that if $p, q$ are consecutive odd primes then $p+q$ is the product of $3$ integers, each greater than $1.$ At first glance this seems very hard. In fact it is quite easy." – DanielWainfleet Oct 28 '17 at 12:19
• Possibly also related: Yaglom and Yaglom, Challenging Mathematical Problems with Elementary Solutions, two volumes. Volume 1 is on sale this week. – MJD Nov 20 '17 at 16:48