35763

1. To get an odd total there must be at least one odd and at least one even ball.
2. There must be more than two balls because with two balls there is only one pair.
3. There must be more than three balls because with three balls there is an odd number of solutions.
4. Given 1. above there must be one odd and three evens or vice-versa or two odds and two evens.
5. Because odd plus even equals odd, two odds and two evens produces too many odd results.
6. Three odds and one even or vice-versa produces the same number of odd and even totals.
Thus with a total of four balls there must be three odds and one even or vice-versa.