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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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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Tracking Points

Age 14 to 16 Challenge Level:
By viewing the 3-D coordinates as if you could not see the third dimension you can extend what is happening in the 2D case and this is explored in the problem Coordinate Patterns first published in September 2004.