Explaining, convincing and proving

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?

Find the maximum value of n to the power 1/n and prove that it is a maximum.

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?

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

What have Fibonacci numbers got to do with Pythagorean triples?