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Question 1134379: A 4x4x4 cube is ruled on all six of its faces with 16 congruent squares. How many paths are there along the faces of the cube from A to B, travelling along the lines and always moving closer to point B? (Remember, : the paths are drawn on all six faces!)
My work: I think it would be 12!/4!x4!x4! x 6 faces
What are your thoughts?
Sorry forgot to add diagram on the other one! http://tinypic.com/r/166m8si/9''
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
I think I answered this a few days ago with the wrong answer; I overlooked the fact that the paths have to stay on the surface of the cube. If the paths don't have to stay on the surface of the cube, the number of paths is
The problem where the path has to stay on the surface of the cube is more complicated; and it is more complicated than your attempted solution.
Consider all the separate cases where you move on one face from A to reach a point which is on some plane with point B. There are 5 cases: to get from A to another point on a plane containing A to a point on a plane containing B, you need to move 4 units in one direction; and you can move either 0, 1, 2, 3, or 4 units in the other direction.
(1) Moving 4 units in one direction and 0 units in the other direction from A.
There are 3 directions in which you can do this; in each direction the number of different ways to do it is C(4,4). Each of them puts you at another corner of the cube. To get to B from there, you have to move 4 units in one direction and 4 units in the other direction; the number of ways to do that is C(8,4).
Total number of paths for this case: 
(2) Moving 4 units in one direction and 1 unit in the other direction from A.
On each of the three faces containing A, there are 2 ways to do this; the number of ways to do it is C(5,4). Each of them puts you at a point on a plane containing B. To get to B from there, you have to move 4 units in one direction and 3 units in the other direction; the number of ways to do that is C(7,4).
Total number of paths for this case: 
(3) Moving 4 units in one direction and 2 units in the other direction from A.
Again on each of the three faces containing A, there are 2 ways to do this; the number of ways to do it is C(6,4). Each of them puts you at a point on a plane containing B. To get to B from there, you have to move 4 units in one direction and 2 units in the other direction; the number of ways to do that is C(6,4).
Total number of paths for this case: 
(4) Moving 4 units in one direction and 3 units in the other direction from A.
This case will be exactly the same as case (2); it's just that now you are moving 4 units in one direction and 3 units on the first face and 4 units in one direction and 1 unit in the other direction on the second face, instead of the other way around.
Total number of paths for this case: 
(5) Moving 4 units in one direction and 4 units in the other direction from A.
And this case will be exactly the same as case (1); it's just that now you are moving 4 units in one direction and 4 units on the first face and 4 units in one direction and 0 units in the other direction on the second face, instead of the other way around.
Total number of paths for this case: 
ANSWER: The number of paths from A to B, staying on the surface of the cube, is
210+1050+1350+1050+210 = 3870
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