Let's take into consideration a few cases:

N % 4 == 0

In this case, we can pair up complementary values (i and n-i+1) and give them the same sign. Because N is a multiple of 2², the number of pairs is an even number (N/2 is a multiple of 2¹). We have reduced the problem to an even number of identical values, which can be given alternate signs, in order to add up to 0. In this case, the answer is 15.

N % 4 == 2

In this situation, we cannot apply the method described above, because we get one extra pair. The best we can do is assigning alternate signs from the very beginning (-1 +2 -3 +4 -5 +6, and so on). Each pair of two consecutive values gives either a +1 or a -1. Because there is an extra pair, we can choose to either have a final result of 14 (which is not valid), or 16 (which is the best we can do, considering that the sequence contains an odd quantity of odd numbers, thus it cannot evaluate to zero - an even number). In this case, the answer is 16.

N % 4 == 3

In this situation, we can group the first three elements in a fixed manner: +1 +2 -3 (adding up to zero). This reduces the problem to the last N-3 values. Considering that N-3 is a multiple of 4, we can apply the strategy described in the first case. The answer is 15.

N % 4 == 1

The same idea goes for this case: we leave the first element by itself (+1) and reduce the problem to N-1 values (which is also a multiple of four, so a zero sum can be obtained). Just like the second case, which had a lower bound of 1 for the number of extra flair pieces, the answer is 16.

Python solution (Sergiu Puscas)

    1 print 15 + [0, 1, 1, 0][input()%4]