my question is the following. On an cube are numbers. The numbers are v, l, r, o, u and h. The twelve absolute amounts of the differences of these numbers are the numbers from 1 to 12. The differences are from the two sides which are next to each other. So the differences are
|v - l|; |v - r|; |v - o|; |v - u|; |u - l|; |u - r|; |u - h|; |h - l|; |h - r|; |h - o|; |o - l|; |o - r|
Now I have to find out one example of this distribution. And then I have to show that the absolute amount of the difference from two opposite sides of the cube with such a distribution is not larger than 17.
My problem is that I don't have an idea to analyse this distribution, so I just can try a long time. Thank you.
An example is: $v = 0,h=17,l=5,r=6,o=7,u=9$
For the second part of the question: suppose there are two opposite sides of the cube with a difference of 18. Consider the 4 paths from one of these sides to the other. Each of these paths have 2 differences that have to add up to 18 or have a difference of 18. We know the difference cant be 18 because that would imply that one of the differences of the path is larger than 12, and that is not allowed. So we need 4 times 2 differences that add op to 18 the possibilities are: $0+18,1+17,2+16,3+15,...,9+9$
But differences larger than 13 are not allowed, and each difference can only occur once, so the only possibilities are $10+8,11+7,12+6$
There are only 3 possibilities but we need 4, so it is impossible to have 2 opposing sides with a difference of 18. You can see that the same holds for any number larger than 18 too.