Very roughly this is the net of a dodecahedron; each letter represents a pentagon.

       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. ABC  2. ABI  3. ACD  4. ADL  5. AIL
 6. BCE  7. BEG  8. BGI  9. CDF 10. CEF
11. DKF 12. DKL 13. EFH 14. EGH 15. FHK
16. GHJ 17. GIJ 18. HJK 19. IJL 20. 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.

The sum of the numbers from 1 to 12 is 6*13 = 78.  Each number then appears on five of the vertices, for a total of 390.  There are twenty vertices that share this, so each averages 390/20 = 19.5.

Posted by Charlie on 2007-06-01 14:56:11