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We have received a very elegant strategy for solving this problem from Tom and Adam, of Alcester Grammar School in England:
We have found a solution to this puzzle. Our solution works, because we selected all the nine colours for the bottom face. Their position did not matter as long as there was one of each on the base.
For each colour, we then moved up one level, right one space (if they were on the far right already, they moved to the left row), and forward once (if they were at the back already, they moved to the front row).
We repeated the same procedure to fill the top level.
This ensured that there was one of each colour in each row, column, and on each level. Because of this, each of the faces could only contain one of each colour, as when we consider a whole face, all of the colours will just shift one place, inwards, and move off the row and level that they were on before. Also, as every colour was on the initial face, there would have to be every single colour on all the other faces too.
Thank you Tom and Adam for such a neat solution.
Anybody who would like to check that the procedure works can return to the problem page and try the strategy out for themselves.