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These are all equivalent.
Another way of thinking about the problem is:
First you put a square with side of $x$ on each face ($6x^2$ cubes), then a column of $x$ cubes on each edge ($12x$ cubes), and finally one cube on each corner ($8$ cubes).
This would give us our $6x^2 + 12x + 8$ expression, which is what you get when you simplify the other expressions.
Cubes Within Cubes Revisited
Age 11 to 14
Challenge Level 
Yanqing from Devonport High School for Girls sent us a complete solution to this problem:
We would need to use $26$ red cubes and $98$ blue cubes.
For the $3 \times 3 \times 3$ cube, there is one yellow cube,
so there are $27 - 1 = 26$ red cubes.
It is the same for the blue cubes: there are $5 \times 5
\times 5$ cubes in total, and $3 \times 3 \times 3$ of them are not
blue.
For the next size up, there would be $7 \times 7 \times 7$
cubes in total, and $5 \times 5 \times 5$ that aren't green.
So there would be $343 - 125 = 218$ green cubes.
The amount you need for each layer added onto a cube with
edges of $x$ will be $(x+2)^3 - x^3$.
$(x+2)^3$ is the cube you will get when you add the next
layer, and $x^3$ is the number of cubes already there. (This is
Martha's method)
With Emma's method, the top and bottom of the layer would both
contain $(x+2)^2$ cubes.
Each added face has $x^2$ cubes, and the four vertical columns
$4x$ cubes altogether.
So the expression for the total number of cubes needed would
be $2(x+2)^2 + 4x^2 + 4x$.
With Charlie's method, the top and bottom of the layer would
both contain $x^2$ cubes.
Each column has $x + 2$ cubes, and there would be $4(x + 1)$
columns.
So the expression for the total number of cubes needed would
be $2x^2 + 4(x+1)(x+2) = 6x^2 + 12x + 8$.
Martha's method: \begin{eqnarray} (x+2)^3 - x^3 &=&
x^3 + 6x^2 + 12x + 8 - x^3\\ &=&6x^2 + 12x + 8
\end{eqnarray} Emma's method: $$ 2(x+2)^2 + 4x^2 + 4x = 6x^2 + 12x
+ 8 $$ Charlie's method: $$ 2x^2 + 4(x+1)(x+2) = 6x^2 + 12x + 8
$$
These are all equivalent.
Another way of thinking about the problem is:
First you put a square with side of $x$ on each face ($6x^2$ cubes), then a column of $x$ cubes on each edge ($12x$ cubes), and finally one cube on each corner ($8$ cubes).
This would give us our $6x^2 + 12x + 8$ expression, which is what you get when you simplify the other expressions.
Well done Yanqing.
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