There is a 2 inch grid cube made up of 8 wire-frame cubes, each with wire where their edges should be and space in their middle and faces.

The quickest way to get from one vertex to the opposite vertex is 6 inches. How many such paths are there?

Let's see if I can get this formated correctly--The numbers show the number of shortest paths to each node (vertex).  The back side of the cube is to the top left, the front is to the lower right.  We're moving from the top, left, back corner to the bottom, right, front corner:

1--1--1
|  |  |
1--2--3
|  |  |
1--3--6


 
        1---2---3
        |   }   }
        |   |   |
        2---6---12
        |   |   |
        |   |   |
        3---12--30
 
                    1-----3-----6
                    |     |     |
                    |     |     |
                    |     |     |
                    3-----12----30
                    |     |     |
                    |     |     |
                    |     |     |
                    6-----30----90

There are 90 shortest paths fron one vertex to the opposite vertex.

(The value for the center node was changed after it was pointed out I messed up in my haste to be the first one with the right answer... Doh!)

Edited on June 8, 2004, 9:04 am

Posted by Erik O. on 2004-06-08 08:22:02