### 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.

# Finding Factors

### Why do this problem?

This problem could be used as an introduction to factorising quadratic expressions, or to develop fluency in this skill. Unlike standard factorisation exercises, this task is intended to engage students and motivate them to factorise lots of expressions and develop efficient strategies in the process.

Relating to this month's theme, the action of filling in the table given the headers is straightforward; probing the table and then working out the headers is the inverse or undoing action. It is challenging to look for the least amount of information required in general to enable the undoing action to be completed.

### Possible approach

Students will need to be able to expand pairs of brackets of the form $(x \pm a)(x \pm b)$ before embarking on this problem.

Introduce the class to the problem using the first interactivity.
Using a Level 1 challenge, reveal ten of the cells to the class (making sure that at least one cell in each row and column is revealed). Ask the students to work in pairs or threes, and give each group one of the revealed cells: "Can you find two expressions that multiply together to give the expression in your cell?"

After giving them some time to work on this, bring the class back together. Ask the first group to feed back the expressions they came up with.
"Can I fill this in on the grid?"
"What other information do I need?"
"Does anyone else have that information?"
Using the class's suggestions, fill in the rest of the grid and then reveal the headers.

If a computer room is available, students could work in pairs using the interactivity. Another option, if students have access to computers outside school, is to ask them to work on the different challenges for homework and then discuss their strategies in the next lesson.

If computers are not available, the task can be recreated by asking each student to create a multiplication grid of their own, and then draw a blank grid for their partner. As in the interactivity, the challenge is to ask for as few entries as possible from the grid in order to work out what the headers are.

Once students have had a chance to develop and share efficient strategies, they can move on to the second interactivity which introduces quadratics with a coefficient of $x^2$ greater than $1$.

### Key questions

What does the coefficient of $x$ tell you about the headers?
(For the second interactivity) What does the coefficient of $x^2$ tell you about the headers?