In my view the trick in this question is to think about adding up numbers not how you add them up.Since this was a programming question you don't need to know the formula you can add up the numbers in a loop. It wouldn't even be much slower since you need a loop of size N anyway for the purpose of adding up numbers in the input.

Xor solution and adding solutions are the same solution it's just a matter of what binary function you use for calculating sum. You can use any commutative, associative binary function which has inverse. You can use addition, xor or even multiplication.