Explaining, convincing and proving

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?

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?

Generalise this inequality involving integrals.

In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?

Pizza, Indian or Chinese takeaway? If everyone liked at least one, how many only liked Indian?