The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Four cards are shuffled and placed into two piles of two. Starting with the first pile of cards - turn a card over... You win if all your cards end up in the trays before you run out of cards in. . . .

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.