Explaining, convincing and proving

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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?

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

Can you work out the number of chairs at a cafe from the number of legs?

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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