First, cover the 3x3x3 cube in red frosting, because you'll need that regardless.
Once you've taken the whole cake apart, what have we actually got?
Of all the twenty-seven cubes:
That's a total of fifty-four frosted faces. Since you need to rearrange the cubes to form a large cube that's either solid purple or green, you will at the very least need to colour in that many faces for both of them too. You also need to have them decorated similarly to the layout above, for symmetry's sake if nothing else.
Next, cover three of the faces of the centre-most cube purple and the other half green (but don't frost opposite sides of that cube the same colour). This will let it act as a corner piece for either a purple or green cube.
Then take two of the cubes with three red faces, and colour all remaining faces of one of them purple, the other green. That's two more corners for each coloured cube sorted.
The trick is to leave no wasted space, and to remember only three cubes can be missing one of the colours (i.e. you must have 26 cubes incorporating green, and the same goes for the other two colours). This is permissible since these pieces will double as the centre of the 3x3x3 cube when their missing colour is not being displayed.
Next, let's move on to the six cubes with one red face. Three of them needs to have two green faces and three purple faces, the other three need three green faces and two purple faces.
Why? Because so far we've only covered for two corners each of a green, red or purple cube, and this is how you're going to make the rest. We still need six corners of these three colours total, so we know at least eighteen of these twenty seven total cubes will need to have half their faces painted red, purple or green - six each.
Keeping symmetry in mind, that means the six cubes of three green faces will be divided into three cubes with two red faces, one purple - and three cubes of two purple faces and a red. The same goes for the three red-faced cubes and the three-purple faced cubes, for their corresponding colours.
That leaves only six more edge cubes in green and purple. And what do you know, that's also the EXACT amount of cubes you still haven't completely coloured in frosting! They each have four blank faces, so give them two green and two purple faces each (make sure same-colour faces are adjacent), and your cake is complete!
OR
THE EASY SOLUTION FOR ABSOLUTE CRETINS:
Combine the cubes into one and frost the outside red. Move the bottom layer to the top, then the front of the cube to the back, then the left side of the cube to the right. You now have a new cube uncoloured on the outside. Frost those sides purple.
De-assemble the cube and then paint all unfrosted sides green.