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Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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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..

Cubic Net

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

It is important to try to visualise the net wrapped up into a cube, using the following ideas
  1. You might like to think of the yellow and purple faces as the 'Top' and 'Bottom' of the 3D cube.
  2. The Blue - White - Red - Green faces wrap around to make the sides of the cube
  3. Twisiting the 'Top' of the cube will cause the top layer of the Blue - White - Red and Green faces to twist -- Why not try this out
  4. Twisting the 'Bottom' of the cube will cause the bottom layer of the Blue - White - Red and Green faces to twise -- Why not try this out?
You might also like to try to investigate the net whilst holding a 3D cube.