Why do this problem?

This is a tough problem for learners who relish the challenge of working with large and difficult numbers. It would become more accessible if calculators were used.

Key questions

Have you found out how cubes are needed to cover the single cube?

Have you remembered that there is only "up to $1000$ of each colour"?

What is the cube root of $1000$?

Have you made list of the cubes up to $1000$?

Possible support

Suggest starting by finding how many cubes are needed to cover one (yellow) cube. This can be done practically with interlocking cubes.