A | B--C--D / \ E F \ G H \ / I--J--K | L

Consider each face to be numbered 1 through 12.

Each vertex is the intersection of 3 faces. The vertex sum is therefore the sum of the values of those three faces.

The faces making up the vertices in the diagram above are:

1.ABC2.ABI3.ACD4.ADL5.AIL6.BCE7.BEG8.BGI9.CDF10.CEF11.DKF12.DKL13.EFH14.EGH15.FHK16.GHJ17.GIJ18.HJK19.IJL20.JKL.

What is the global vertex sum (20 vertices) and therefore the mean vertex sum?

How best can the faces be labeled so that the 20 vertices are as close as possible to the mean vertex sum?

*
“Close as possible” means that:
the sum of differences above (or below) the mean is at the optimum*

**or**the most vertex sums land on or have the best proximity to the mean as possible. The prior condition also applies; ie, any deviance is minimal.