What are people's favorite examples of problems in which one is looking for needles in a superexponential haystack that we suspect contains exponentially many needles but can't prove contains any at all? Here I am adopting Richard Guy's nice way of describing such problems, which he used last summer when he taught me about a beautiful example of this phenomenon (which I hope I am remembering correctly): It is almost certainly true that for all n sufficiently large, you can arrange the integers from 1 to n in a circle in such a fashion that any two consecutive numbers add up to a perfect square. But no one can prove it. I just learned about another nice example from Barry Cipra. His Bricklayer's Challenge ( http://www.pavelspuzzles.com/2012/11/the_bricklayers_challenge.html) appears to be solvable for every n, but nobody knows how to prove it for infinitely many n. I hereby rule out answers to my question in which the "needle-ness" of the needle takes a long time to verify. This is the case for some Ramsey theory problems that are shown to be solvable by the probabilistic method. It is also the case for codes that come close to the Shannon bound (which again are proved to exist by the probabilistic method). Jim Propp