I have forgotten the number of the combination of the lock on my
briefcase. I did have a method for remembering it...
Complete the magic square using the numbers 1 to 25 once each. Each
row, column and diagonal adds up to 65.
This problem was very well answered by Martha from Tattingstone
We worked out this problem by finding out how many cubes would
cover 1 (yellow) cube, which was $3\times 3\times3$
(27). Then we found that a $5\times 5\times5$ cube
would be the next one and so on. The trouble was that the $3\times
3\times3$ cube was a totally different cube than the $1\times
1\times1$ cube. So with the 27 we had to take away 1 as the
$3\times 3\times3$ was only the skin. So in order to get the skin
each time it would be $5\times 5\times5$ - $3\times 3\times3$ 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:
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
The colour of the largest cube that was made was white.
The colour with no $1\times 1\times1$ cubes in it was black.
The colour with the most cubes made out of it including the
$1\times 1\times1$ cubes was blue.
Bronya, also from Tattingstone sent in a good solution too -