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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?

Pythagoras’ Comma

Age 14 to 16
Challenge Level

The topic of music can make a good connection between science and mathematics

The nature of sound and the working of the ear are rich areas of applied mathematics.

The ratio emphasis follows from harmonics or overtones and rests on ideas like lowest common multiple.

One teacher has offered the following comment : This Stage 4 work makes practice in multiplying and dividing fractions purposeful. And in school invites collaboration between the music and mathematics departments.