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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.


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.

Factorising with Multilink

Age 14 to 16
Challenge Level

Why do this problem?

When students first meet factorisation, they often don't make the connection between factorising an algebraic expression and breaking a number into factor pairs. This problem introduces factorisation using a visual representation that allows students to make that connection and discover for themselves the properties necessary for a quadratic expression to factorise.

This problem is a nice example of a Low Threshold High Ceiling task, as students can use the representation to factorise more and more complicated examples. You can read more about Low Threshold High Ceiling tasks in this article.

Possible approach

If multilink is available, students can create "squares", "sticks" and "units". Alternatively, these worksheets have sets of different bases that can be cut out and used: Base Three Base Four Base Five Base Six.
Ask the students to organise themselves into groups of three or four, and select which base each student will use. If using multilink, give them time to create some squares, sticks and units in their bases.
"I'd like you each to take 1 square, 7 sticks and 12 units and put them together to make a rectangle."
"In your groups, compare the rectangles you've made. Have you arranged them in the same way? The challenge is to find an arrangement that works in all bases."

Take a look at the second video to see how Becky and Charlie resolve their different rectangles into one that works for both of them.
"Now I'd like you to try to make a rectangle using 1 square, 5 sticks and 8 units."
Give groups some time to work on this. 
Bring the class together for discussion. "Did anyone manage to make a rectangle?"
(It is possible to make rectangles for bases 3, 4 and 8, but it's not possible to make the same arrangement for all bases.)
"Your challenge is to come up with a way of deciding quickly whether a combination of squares, sticks and units can be made into a rectangle that works in all bases."
"To help you approach this in a systematic way, here are three questions to have a go at."
Circulate, and listen out for interesting insights that will be worth sharing with the whole class.
Finally, bring the class together and invite groups to share their findings.

Key question

What is the connection between the number of sticks you need, the number of units you need, and the dimensions of the rectangle?

Possible support

Start students off on families of examples such as:

$x^2 + 3x + 2$
$x^2 + 4x + 3$
$x^2 + 5x + 4$


Possible extension

Think about the rectangles it's possible to make if you use 2, 3, 4... squares, some sticks and some units.

Finding Factors consolidates the ideas met here and has factorisations which require negative terms.