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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

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

The second segment will stretch from (1,1,1) to (2,2,2), the third from(2,2,2) to (3,3,3) and so on. So the tenth segment will have its ends at (9,9,9) and (10,10,10).

Establishing a rule for what is happening to each of the coordinates in turn is best achieved by thinking of the effect at seeing just a 2-D image.

In the last section there are a number of options to consider.