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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

A triomino is a shape made from three squares. Here is an L-triomino:

Here is a size 2 L-triomino:
double sized L-triomino
It can be tiled with four size 1 L-triominoes:
size 2 triomino tiled

Can you work out how to use the tiling of a size 2 L-triomino to help you to tile a size 4 L-triomino? Click here for a hint.

Devise a convincing argument that you will be able to tile a size 8, 16, 32... $2^n$ L-triomino using size 1 L-triominoes.
How many size 1 L-triominoes would you need to tile a size 8... 16... 32... $2^n$ L-triomino? 
What about odd sized L-triominoes? The diagram below shows the region which needs to be tiled to turn a size 1 L-triomino into a size 3 L-triomino.
 Size 1 to size 3
Can you find a quick way of tiling the region, using combinations of the 'building blocks' below?
2 by 3 rectangle and size 2 triomino

In the same way, can you find a way of adding to your size 3 tiling to tile a size 5? Then a size 7, 9, 11...? Click here for a hint.
Devise a convincing argument that you will be able to tile any odd sized L-triomino using size 1 L-triominoes. 
Combine your ideas to produce a convincing argument that ANY size of L-triomino can be tiled.