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?

Divide 24 vertices into 12-12 such each half present on the
same face of the cube.
Lines from points of same set to same set always go 
through face.
8 points can connect to 12 points of other set always passing
through the body.
The other 4 points has 8 points + 1 pt (diagonaaly opposite)

No. of lines that pass through the body
=8*12+4*9=96+36=132.

If these 8 points fall on vertices as well as the other, then
no. of lines through the body=8*8+4*5=84.

Edited on February 15, 2008, 4:11 am

Posted by Praneeth on 2008-02-15 04:04:48