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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

A Scale for the Solar System

Age 14 to 16
Challenge Level

This accessible problem will hopefully lead students into the whole area of solar and astronomical scales. The internet is full of sites offering information. Google anything and you'll start a trail of enquiry.

The methods often take some thinking about and nearly all of them require the imagination of different viewpoints. Grasping the methods and especially the required adjustment of perspective will not be immediate for most students and serves as a useful reminder that mathematical understanding isn't an "instant" or "not-at-all" experience.