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Rotating Triangle

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

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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!]

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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.


Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem begins with very simple ideas about tiling and enlargement but can lead learners to appreciate how proofs can be built up in stages, through breaking an idea down into special cases. It also introduces the intriguing mathematical notion of reptiles - shapes that can be tiled to make enlargements of themselves.

Possible approach

Start by introducing the smallest L-triomino and challenge learners to find a way to tile a size 2, 3 and 4 L-triomino with it, on squared paper. They may work in quite a haphazard way to start off with, and may not even find ways of tiling them all. There is the opportunity to discuss the number of tiles needed, and to make links with work on enlargements and similar shapes.
Once everyone has had a go at tiling the first few L-triominoes, and successful attempts are collected on the board for all to see, follow up with: "I wonder whether all sizes of L-triomino can be tiled"
Suggest the need for a systematic approach, gradually building up knowledge of how different sizes of L-triomino can be tiled.
Ask learners to start by considering how the tiling of the size 2 L-triomino helps in tiling the size 4 L-triomino. Can they develop their ideas further and suggest how they would convince someone that all size $2^n$ L-triominoes can be tiled? Bring the class together to share their insights.
Next, introduce the odd numbered sizes of L-triominoes. Share these diagrams with the class:
 size 1 to size 3from size 3 to size 5
Challenge learners to find simple ways of extending their tilings from one odd number to the next, in a way that will lead to a convincing argument that all odd sized L-triominoes can be tiled.
Finally, learners need to consider how the two arguments can be combined to prove that all L-triominoes can be tiled - this is a good opportunity to discuss the formal steps in writing down a mathematical argument.  

Key questions

How can I use my knowledge of tiling a size 2 L-triomino to tile a size 4?
How can I use my knowledge of tiling a size 3 L-triomino to tile a size 5?
How can I use my knowledge of tiling odd and $2^n$ sized L-triominoes to tile ANY size L-triomino?

Possible extension

Investigate tilings with these other reptiles:
 trapezium and L-tetromino
Come up with similar proofs that all sizes can be tiled.

Possible support

A nice way of showing how the $2^n$ sized L-triominoes can be built up is for each learner to create a size 2 L-triomino, and then to stick four of these together in their group of four to make a size 4, and then for four groups to get together to make a size 8...