Explaining, convincing and proving

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?

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Can you produce convincing arguments that a selection of statements about numbers are true?

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Can you find a rule which relates triangular numbers to square numbers?

Jasmine buys three different types of plant. How many triffids did she buy?

What fractions can you divide the diagonal of a square into by simple folding?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?