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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?

Why 24?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
Hint for Final Challenge:
Take any prime number greater than 3, square it and subtract one. Repeat several times and record your results.
What is the highest common factor of the numbers you get? How is this related to the observations you made when working on the building blocks?