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.

  • $\begingroup$ 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. $\endgroup$ – DanielWainfleet Oct 28 '17 at 11:40
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    $\begingroup$ 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. $\endgroup$ – DanielWainfleet Oct 28 '17 at 11:51
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    $\begingroup$ 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." $\endgroup$ – DanielWainfleet Oct 28 '17 at 12:19
  • $\begingroup$ Possibly also related: Yaglom and Yaglom, Challenging Mathematical Problems with Elementary Solutions, two volumes. Volume 1 is on sale this week. $\endgroup$ – MJD Nov 20 '17 at 16:48

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.


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