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 $4 x$ cubes altogether.
So the expression for the total number of cubes needed would be $2 (x + 2)^{2} + 4 x^{2} + 4 x$ .

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 $2 x^{2} + 4 (x + 1) (x + 2) = 6 x^{2} + 12 x + 8$ .

Martha's method:

$\begin{matrix} (x + 2)^{3} - x^{3} & = & x^{3} + 6 x^{2} + 12 x + 8 - x^{3} \\ = & 6 x^{2} + 12 x + 8 \end{matrix}$

Emma's method:

$2 (x + 2)^{2} + 4 x^{2} + 4 x = 6 x^{2} + 12 x + 8$

Charlie's method:

$2 x^{2} + 4 (x + 1) (x + 2) = 6 x^{2} + 12 x + 8$

These are all equivalent.

Another way of thinking about the problem is:

First you put a square with side of $x$ on each face ( $6 x^{2}$ cubes), then a column of $x$ cubes on each edge ( $12 x$ cubes), and finally one cube on each corner ( $8$ cubes).

This would give us our $6 x^{2} + 12 x + 8$ expression, which is what you get when you simplify the other expressions.

Well done Yanqing.