### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Doodles

A 'doodle' is a closed intersecting curve drawn without taking pencil from paper. Only two lines cross at each intersection or vertex (never 3), that is the vertex points must be 'double points' not 'triple points'. Number the vertex points in any order. Starting at any point on the doodle, trace it until you get back to where you started. Write down the numbers of the vertices as you pass through them. So you have a [not necessarily unique] list of numbers for each doodle. Prove that 1)each vertex number in a list occurs twice. [easy!] 2)between each pair of vertex numbers in a list there are an even number of other numbers [hard!]

### Russian Cubes

How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted in order 'blue then red then blue then red then blue then red'. Having finished one cube, they begin to paint the next one. Prove that the girl can choose the faces she paints so as to make the second cube the same as the first.

# Why 24?

### Why do this problem?

This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights. It is well suited for students who are working on the difference of two squares.

Initially working on the building blocks then gives students the opportunity to work on harder mathematical challenges than they might otherwise attempt.

The problem is structured in a way that makes it ideal for students to work on in small groups.

### Possible approach

This task might ideally be completed in groups of three or four.
Hand out a set of building block cards (Word, PDF) to each group. (The final challenge will need to be removed to be handed out later.) Within groups, there are several ways of structuring the task, depending on how experienced the students are at working together.

Each student, or pair of students, could be given their own building block to work on. After they have had an opportunity to make progress on their question, encourage them to share their findings with each other and work together on each other's tasks.

Alternatively, the whole group could work together on all the building blocks, ensuring that the group doesn't move on until everyone understands.

When everyone in the group is satisfied that they have explored in detail the challenges in the building blocks, hand out the final challenge.

The teacher's role is to challenge groups to explain and justify their mathematical thinking, so that all members of the group are in a position to contribute to the solution of the challenge.

It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.

### Key questions

What important mathematical insights does my building block give me?
How can these insights help the group tackle the final challenge?

### Possible extension

Of course, students could be offered the Final Challenge without seeing any of the building blocks.

### Possible support

Encourage groups not to move on until everyone in the group understands. The building blocks could be distributed within groups in a way that plays to the strengths of particular students.
The final challenge could also be done as a proof sorter activity using this set of cards (Word, PDF).