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 found it helpful to think in terms of pencil marks placed a short distance from the vertices, instead of truncation.

Two pencil marks can be joined in the surface iff they are on the same face. There are 2x4=8 marks on each face, and each mark occurs on 2 faces.

Hence, a mark may be surface joined up with 7 distinct marks on one of its faces and 6 distinct marks on the other of its faces. (One of the partner marks also occurs on the same two faces). Since 2 marks participate in each joining, there are 13/2 surface connections per mark.

There are 24 marks, giving 24x23/2 connections in total. The number of body piercing connections is then

24x23/2 - 24x13/2 = 120.

Note: It's not easy to come up with a problem that's neither too hard nor too soft. Like the little bear said, this one was jussst right!.

 

Posted by FrankM on 2008-02-15 07:30:21