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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

Neerajan and Eilidh both noticed that the number of L-triominoes needed to tile a size $n$ L-triominoes was equal to $n^2$.
Philip from Wilson's School explained his thinking:
To tile a size 4 triomino you would simply put 4 size 2 triominoes together the same way you put four size 1 triominoes together to make a size 2.

Therefore you would put 4 size 4 triominoes together in the same way to form a size 8 triomino. Then 4 size 8 triominoes to make a size 16, and so on.

The triominoes could therefore tile size 8, 16, 32... $2^n$. These can therefore be made up of size 1 triominoes.

As it takes $4^1$ size 1 triominoes to tile a size 2 triomino, and $4^2$ size 1 triominoes to tile a size 4 triomino, it would therefore take $4^3=64$ size-1 triominoes to tile a size 8 triomino.
It would then take $4^4=256$ size 1 triominoes to tile a size 16 triomino and $4^5=1024$ size 1 triominoes to tile a size 32 triomino.
Finally, it would take $4^x$ triominoes to tile a size $2^x$ triomino.

To tile the size 3 triomino you simply use the 2x3 rectangular blocks at the top and on the right and then the rest you use a size 2 triomino and the size 1 in the middle.

To tile a size 5 triomino you simply add another size 2 triomino in the bottom left and for the rest you can use the 2x3 rectangular boxes. It's the same with a size 7 triomino, add a size 2 and then use 2x3 boxes to tile the rest. This works with all odd numbers.

Therefore all size triominoes can be tiled. If the size number is even you use the first method and if the size number is odd you use the second.
Isaac from Hampton School also sent us his solution, which you can read here.