This Penta Problem was very well answered by Martha from Tattingstone School:

We worked out this problem by finding out how many cubes would cover 1 (yellow) cube, which was 3x3x3 (27). Then we found that a 5x5x5 cube would be the next one and so on. The trouble was that the 3x3x3 cube was a totally different cube than the 1x1x1 cube. So with the 27 we had to take away 1 as the 3x3x3 was only the skin. So in order to get the skin each time it would be 5x5x5 - 3x3x3 etc. Here are my results:

To answer the second part of question, when this large cube is broken up, Martha drew another table which clearly shows how each colour was made into the largest possible cubes:

Martha continues:

The only problem with our results is that from the orange cubes downwards (in the table) they are all over 1000 and it said in the question "up to 1000 of each colour".

So, in answer to the three questions that were posed, Martha says:

The colour of the largest cube that was made was white.
The colour with no 1x1x1 cubes in it was black.
The colour with the most cubes made out of it including the 1x1x1 cubes was blue.

Bronya, also from Tattingstone sent in a good solution too - well done.