Richard Guy wrote [as slightly reformatted]: << The corresponding problem where sums of {\em triples} of elements of a set are given has been settled by Boman & Linusson. The exceptions are precisely 3, 6, 27, 486. For n=27 they give five examples of which the simplest is {-4,-1^{10},2^{16}} and its negative, where exponents denote repetitions. For n=486 they give {-7,-4^{56},-1^{231},2^{176},5^{22}} and its negative.
This is so cool! I love results like this, where a simply stated problem has a straightforward answer, except for some small and/or strange set. (It is surely no coincidence that these are 3, 2*3, 3^3, 2*3^5.) I am interested to hear of mathematical results that have a small and strange set of exceptions. A few I can think of are these: * Any differentiable structure on n-space is equivalent to any other for all n except 4. * The alternating group A_n is simple for all n except 4 (Hmmm, is there a connection here?) * The automorphism group Aut(S^n) is isomorphic to S_n for all n except 2, 6. * The ring of integers of the imaginary quadratic field Z(sqrt(-n)) has non-unique factorization except for n = 1, 2, 3, 7, 11, 19, 43, 67, 163. * The ring of integers of the cyclotomic field Z(exp(2pi i/p) has non-unique factorization for all primes p except 2,3,5,7,11,13,19. * There exists no real division algebra R^k for k = 2^n for all n except 0, 1, 2, 3. Other such examples are solicited. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele