Explaining, convincing and proving

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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.

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

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 find the value of this function involving algebraic fractions for x=2000?

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.

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