### A Square Deal

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

# Cubes Within Cubes

## Cubes Within Cubes

We had interlocking cubes (all the same size) in $10$ different colours, up to $1000$ of each colour. We started with one yellow cube. This was covered all over with a single layer of red cubes:

This was then covered with a layer of blue cubes. Then came a layer of green, followed by black, brown, white, orange, pink and purple for as long as there were enough cubes of that colour to cover the layer that came before.

The unused cubes were put away. The many-layered cube was then broken up and each colour made into cubes. These were just of the one colour and the largest cubes possible made. For example, the red layer made three $2\times 2\times2$ cubes with two $1\times 1\times1$ cubes left over, whereas the larger layers made much larger cubes as well as smaller ones.

What colour was the largest cube that was made?

Which colour made into cubes had no $1\times 1\times1$ cubes?

Which colour was made into the most cubes including the $1\times 1\times1$ cubes

### 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.