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# Gift of Gems

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Age 14 to 16

Challenge Level

Four jewellers had respectively 8 rubies, 10 sapphires, 100 pearls and 5 diamonds.

Each gave one gem from their collection to each of the rest.

Afterwards they noticed that they all had collections of gems of precisely equal value.

Can you work out the relative values of each gem?

Can you then work out how much each jeweller gained or lost?

This problem comes from *Lilavati* , a standard work on Hindu mathematics written by Bhasakaracharyya, who lived in the twelfth century of the Christian era. The book is written as instruction for a young and beautiful woman called Lilavati and it is thought that she was Basakaracharyya's
daughter.

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.

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?