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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?


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?

Eight Ratios

Age 14 to 16 Challenge Level:

Oliver, from Nottingham High School, and Jonathan, both sent in very good solutions to this problem.

Here is Jonathan's solution:


Oliver's solution is below:


Well done to both Jonathan and Oliver for their excellent solutions!