### Two Cubes

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

### Common Divisor

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

##### Stage: 4 Challenge Level:

For this problem, you will either need multilink cubes or these sets of number base sheets to cut out: Base Three Base FourBase Five Base Six.

In the video below, Charlie and Becky show how you can make rectangles using sets of squares, sticks and units.

Can you make a rectangle to represent $x^2 + 7x + 12$?
Can you do it in more than one base?

Watch the video below to see how Charlie and Becky tackled this question:

Take 1 square and 12 units in your chosen base. Put them together with some sticks to make rectangles that will work in all bases.

Charlie and Becky made $x^2 + 7x + 12$ into a rectangle with length $x+4$ and width $x+3$.

How many different rectangles can you make?

Imagine you had 1 square, lots of sticks and 100 units. What can you say about the dimensions of the rectangles it is possible to make?

Now, take 1 square and 12 sticks in your chosen base. Put them together with some units to make rectangles that will work in all bases.

How many different rectangles can you make?

Imagine you had 1 square, 100 sticks and lots of units. What can you say about the dimensions of the rectangles it is possible to make?

If you had 1 square, $p$ sticks and $q$ units, what can you say about the dimensions of the rectangles it is possible to make?

Extension

Think about the rectangles it's possible to make if you use two, three, four... squares, some sticks and some units.

Many thanks to Kenneth Ruthven and Paul Andrews whose ideas inspired this problem.