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Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Kite in a Square

Age 14 to 18
Challenge Level

Kite in a Square printable sheet

ABCD is a square. M is the midpoint of the side AB.
By constructing the lines AC, MC, BD and MD, the blue shaded quadrilateral is formed:

What fraction of the total area is shaded?

Once you have had a think about this, you might like to take a look at these three different methods for finding the shaded area. In each case, all the statements have been muddled up. Can you rearrange them back into their original order?


Coordinates approach




Similar Figures approach




Pythagoras approach



Thanks to Jerome Foley for drawing our attention to this problem.


We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.