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A basic method might be to take a set of random samples, and work out the mean proportion of red squares that are covered by the shape. The area of the shape should be approximately this proportion of the area of the main square.

How many random samples need to be taken in order to get an accurate result? Are the areas of some shapes more easily estimated by this method than others? Can you think of a better method?

This is an open ended problem, and we encourage you to send in your different ideas and results.