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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?

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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Find the Fake

Age 14 to 16 Challenge Level:

There are 12 identical looking coins, one of which is a fake.

The counterfeit coin is of a different weight to the rest.

Using only a simple balance, what is the minimum number of weighings needed to locate the fake coin and how is this done?