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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:

A standard calibration for a spectrophotometric assay for lysozyme was created by diluting some initial concentrations of the reagents Tris HCl buffer, NaCl, Lysozyme and a Micrococcus lysodeikticus suspension.The final volume was 5ml.

The numbers in the calculation have become mixed up. Can you work out how consistently to fill in the table? There will be two numbers left over. What will these represent?

Here is a printable version of the table and cards that you can cut out.


Initial conc.

Final conc.

Dilution factor

Volume added































171 mM


106 /ml


500 pmol/ml


0.050 ml




0.250 ml


68 nmol/ml


50 mM


0.037 ml


0.292 ml


108 /ml


0.629 ml


4.371 ml


1000 mM