This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Can you work out where the blue-and-red brick roads end?

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Which of these triangular jigsaws are impossible to finish?

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

What fractions can you find between the square roots of 65 and 67?

Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.

This article sets some puzzles and describes how Euclid's algorithm and continued fractions are related.

An introduction to proof by contradiction, a powerful method of mathematical proof.

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

Using the interactivity, can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

If the yellow equilateral triangle is taken as the unit for area, what size is the hole ?

Ranging from kindergarten mathematics to the fringe of research this informal article paints the big picture of number in a non technical way suitable for primary teachers and older students.

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?