Suppose that for a given configuration, the sum of the vertexes and faces is a number $n$. If we change the number in a vertex from $a$ to $-a$, the total sum of the vertexes can vary by $\pm 2$ and the total sum of the faces by $\pm 6, 2$. This means that in total, $n$ can vary by $\pm 8, 4, 0$. Note that the new result is congruent modulo 4 to the old one. Now, since all results must be congruent modulo 4, and if we place ones in all the vertexes we obtain 14, the only attainable numbers are: $\pm 2, 6, 10, 14$ Doing some casework: -All ones, 14. -One minus one, 6. -Using two minus ones: *Opposite corners, -2 *In the same edge, 6 *Same face, different edges, 2 -We have checked the only three cases that could give 10, a higher number of minus ones will always result in a number smaller than 10, therefore 10 is not attainable. -Five minus ones, -6 -Two ones, in opposite corners, -10. -Finally, all minus ones, -2. This means that -14 is not attainable. All the possible values are: -10, -6, -2, 2, 6, 14.