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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?

Burning Down

Age 14 to 16
Challenge Level


Two different candles are lit. They burn at different rates and one is 3 cm longer than the other.

The longer one was lit at 5.30 pm and the shorter one at 7 pm.

At 9.30 pm they were both the same length.

The longer one, burned out at 11.30 pm and the shorter one burned out at 11 pm.

How long was each candle originally?