Cube Paths
Problem
The diagram shows a skeletal view of a 2 by 2 by 2 cube with one route 'down' the cube.
How many routes are there on the surface of the cube from A to
B?
(No `backtracking' allowed, i.e. each move must be away
from A towards B.)
Student Solutions
Chin Siang from Tao Nan School, Singapore solved this problem using a Pascal's Triangle method.
You are only allowed to move on the surface of the cube and only in the directions of the arrows shown in the sketch. (You might like to mark out a cube and put arrows on all the line segments and mark the numbers of routes for yourself before reading on).
How many ways can you get to each `junction point'? Look at the arrows that go towards that point and consider where you might have come from to reach it. To get the number of routes to that point you add up the numbers at the other ends of the segments which have arrows going towards it.
The numbers on the sketch give the total number of different routes to each point. So there are 54 routes from A to B. Can you see the
symmetry?