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.

i think this is not right, but if i switch the multiplicities of -1 and 2 , then it seems to work: for the multiset { -4, -1^16, 2^10 } , its 3-at-a-time-sums multiset is { -6^120, -3^720, 0^1245, 3^720, 6^120 } , which is symmetric about 0 .

For $n=486$ they give $\{-7,-4^{56},-1^{231},2^{176},5^{22}\}$ and its negative.

this one seems to be fine, with 3-at-a-time-sums multiset { -15^1540, -12^40656, -9^392161, -6^1800568, -3^4358893, 0^5826304, 3^4358893, 6^1800568, 9^392161, 12^40656, 15^1540} , which is symmetric about 0 . mike