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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

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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?

Tilting Triangles

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

As the triangle rotates compare what part of the square is not covered by it anymore and what part of the square is newly covered.

Does the triangle continue to cover the whole of the portion of the square it moves over as its size reduces. What is special when the triangles side is of length $ \sqrt 2 $ units?

Is there a point where the whole of the triangle is always in the square and what effect does that have?