The network represents a map of the edges and vertices of a hypercube or more precisely, a tesseract when imposed within 2D space.

The orange squares represent the faces of the internal cell, a cube in 3D space, and is defined by vertices A to H. The larger bolder letters, I to P are the vertices of the outer cell/cube. The diagonal lines are the edges which link the inner to the outer forming the faces that configure the cells which interface to the inner and outer cells. Let each hypercell have a value which is the vertex sum of each of its enclosing faces (or three times the sum of its 8 vertices).

The vertices are to be numbered uniquely from 1 to 16, and for consistency of reference, let A always be 1.

Vertex values: A B C D E F G H I J K L M N O P = 136

Cell 1: ABCDEFGH Cell 2: IJKLMNOP Cell 3: ADFGILMP Cell 4: ABEFIJMN Cell 5: ABCDIJKL Cell 6: BCEHJKNO Cell 7: CDGHKLOP Cell 8: EFGHMNOP

Find sets of values for A through P so that all the cells have the same value.

Note: while the on-line calculator may be useful, a spreadsheet should prove more valuable for those able to use one.

one that works is

a=1 b=6 c=14 d=15 e=12 f=10 g=2 h=3 i=6 j=4 k=8

L=9 m=13 n=11 o=5 and p=7

I am currently running a program that will count all the solutions (so far its at over 500,000 so obviously won't be submitting all the solutions but simply how many there are)

Posted by Daniel on 2009-02-03 00:25:29