Assume 14 < n < 30. Make boxes labelled 16n to 30n. Let S_{i} be the set of the first i elements. Let s_{i} be the sum of the elements of S_{i}. If s_{i} is less than or equal to 30n, put S_{i} in the box labelled s_{i}. Otherwise, put it in the box labelled s_{i}n. There are 15 boxes, and 16 subsets, so at least one box has two subsets. Clearly, a box with two subsets must have one, say S_{i}, with s_{i} = the box label, and one subset S_{j} with s_{j}n = the box label, with S_{j} containing S_{i}, obviously. So S_{j}  S_{i} is a subset whose elements have sum n. The complement of this set with respect to the full set of 16 elements has sum 30n.
You can probably express the above in 4 lines if you're especially terse.
