User talk:Dushyant1978

Count of cubes on facets of Rubik's Cube
Mathematical Formula: 2(NxN) + (N-2) [(NxN) – (N-2)(N-2)] Where N is dimension of Cube.

Dry run of Mathematical Formula:

For N=3;

2(3x3) + (3-2)[(3x3) – (3-2)(3-2)] = 26

For N=4;

2(4x4) + (4-2)[(4x4) – (4-2)(4-2)] = 56

For N=5;

2(5x5) + (5-2)[(5x5) – (5-2)(5-2)] = 98

How to deduce the formula:

1.	Dissect the cube along any face of the cube so that it gives you NxNx1 (sort of wall of smaller cubes). This will give you N such walls. So if you have 3x3 cube you will get 3 walls each of NxNx1.

2.	Count the number of cubes of two of the wall (obtained in step #1). These will correspond to opposite faces of the Rubik’s cube. This will give 2(NxN) cubes.

3.	Now we have to count the cubes on faces of remaining (N-2) walls.


 * a.	We need to count the number of cubes on the periphery of the walls.


 * Total Number of cubes in a Wall = NxN


 * Number of cubes which are not on faces = (N-2) (N-2)


 * Number of cubes on periphery = (NxN) – (N-2)(N-2)


 * b.	Total cubes on periphery of (N-2) walls = (N-2) [(NxN) – (N-2)(N-2)]

4.	Total cubes on faces of Complete cube = Count of Step#2 + count of Step #3(b) = 2(NxN) + (N-2) [(NxN) – (N-2)(N-2)]

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