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?
A cube edge shares a common cube face with six other cube edges.
Therefore, a cube edge does not share a common cube face with five other cube edges.
Therefore, there are 30 (5*12/2) of these cube edge combinations.
A line segment through the body must have endpoints that lie on cube edges that do not share a common cube face.
There are four line segments per cube edge combination.
Therefore, there are 120 (4*30) line segments through the body.
Posted by Bractals
on 2008-02-14 22:00:42