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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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Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

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Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Tower of Hanoi

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

Look at the sequence below:
$1, 2, 4, 8, 16...$

Can you describe how to get from one term to the next?

Can you describe the $n^{th}$ term of the sequence?

Now try adding together terms from the sequence:
$1 + 2$
$1 + 2 + 4$
$1 + 2 + 4 + 8$
Do you notice anything interesting?

Can you predict what $1 + 2 + 4 + ... + 64 + 128$ would be? Check to see if you are right.

How could you write the answer to $1 + 2 + 4 + ... + 2^n$?
Justify why your formula works.

Hint for building block B: Think about moving all the discs except the largest one onto the middle peg first.

Hint for Extension question: Current theory and observations suggest that between 13.5 and 14 billion years have elapsed since the Big Bang.