We can prove that the corner cells "u" and "v" cannot have an even and an odd number. This is because if this were so, then the sums "u" + "v" and "v" + "u*" will be odd, and since 0 is an even number, the sums "a" + "b" + "c" and "d" + "e" + "f" should be odd as well. The only way that the sum of three integers will result in an odd number is when 1) two of them are even and one is odd, or 2) when all three are odd. Since the corner cells are assumed to be odd and even, neither of these two statements are compatible with the fact that we only have 3 even and 3 odd bone numbers at our disposal. This proves that "u" and "v" cannot have different parity. This eliminates 16 possibilities.