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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?

Mixed up Mixture

Age 14 to 16 Challenge Level:

Why do this problem?

This problem gives practice in converting between a variety of measures. Don't be put off if your students are not aware of the meaning of the various reagents: the table can be reconstructed using knowledge of units alone and a little common sense. Engaging with a problem containing some unfamiliar elements is a good general aspect of mathemaical training.

Possible approach

The most intimidating part of this question is the scientific names of the reagents. Encourage students to see past this to the mathematical structure of the problem. If it helps, the names of the reagents could be replaced by letters.

Key questions

  • What is the question asking mathematically?
  • Is it possible easily to rule out certain cards from certain positions? What possibilities remain?

Possible extension

You might naturally try Real-life equations next.

Possible support

Change the names of the reagents to letters or something simpler.