Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube

We all know what rubik cubes. For example, here is a 2×2 rubik cubes

2×2-rubik

and here is a 3×3 rubik cubes.

3x3-rubik Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube

3×3-rubik

The number of squares of a side for a NxN Rubik cubes can be computed by $tex_cd5a61e07d1e32517227f9371497c692 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ (6 side and each has $tex_abffeca062b81bc63ebecfb9a918e80d Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ squares). This applies when n >= 0.

How about the cubes? A corner cube is shared by three sides and a edge cube is shared by 2 faces so that should be taken into consideration.

If we consider a more general case for a cube that has x, y and z dimensions. We will have all the cubes that is x * y * z. If we deduct the inner cubes from the total number, we have the number of boundary cubes.

$tex_e5211462e797c6dff474859578e98c86 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$

So, for n-rubik cubes, the boundary cubes is $tex_53449623d835d1e9af702d9273c696cf Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ and apparently this applies for $tex_87bede9b72d66f60a9af03df02660a30 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ If n = 0, then apparent 0, if n = 1 then apparently this should be equal to $tex_1e9387788a4cd1e39a391e36d27c39b3 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ (considering 2 dimension square).

We can think of the other way to solve this. First we sum up the top and bottom boundary cubes.

which is $tex_342599be67a217d9fbbd4fa0ec55baad Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$

Then, we sum up the middle layers.

$tex_c757dee8bd0358c68ebc1a7ccd67eb7a Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$

The total equals $tex_936bff70b4537f80316a37835f348f3d Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ . For a NxN rubic cubes, it can be simplified to $tex_42b4ae65588bbaf92e97b57d31bea913 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ which is $tex_f5a9ebdbdedded9ef53575fa345b2efe Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$

This, again works for n >= 2.

If we subtract $tex_936bff70b4537f80316a37835f348f3d Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$ with $tex_a43c96978daf5c17d818d26e8a62d461 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$

$tex_c09e3ec71a8d4d84f3c50e97c1b5c7e8 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$
$tex_c5846672bc35aee312e25ec513d3886d Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$
$tex_852f74316f8190378b8079b801de9d22 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$
$tex_1d8a25e1c122d542d736a0a6da753e66 Calculate the Number of Boundary Cubes and Boundary Squares of a NxN Rubik Cube$