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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?

Calculating with Cosines

Age 14 to 18
Challenge Level

Try to make sense of each method one line at a time. The following mathematical topics may be useful:

Pythagoras' theorem

Cosines of angles greater than $90^{\circ}$

Distance between two points on a coordinate grid

Trigonometry in right-angled triangles