A cube has 8 vertices. If each vertex is cut off to form a triangle, the new solid will have 3 x 8 = 24 vertices. If each of these vertices is then connected directly to each of the others via a straight line segment, how many of these segments will go through the body of the solid, rather than along its surface?

I get 216 body penetrating connections.

Each vertex has a surface connection to its two siblings (sharing the same parent corner). In addition, each vertex has surface connections to its first cousins (sharing the same grandparent cube face).

Total family tree connections = 24 x 23 / 2 = 276

-Sibling connections = 8x3 = 24

-First cousin connections = 6x6 = 36

Note: Geometric hygiene requires that vertices limit their body penetrating connections to those with whom they are further removed than first cousins!

Posted by FrankM on 2008-02-14 18:05:10