Kyle and his teacher disagree about his test score - who is right?

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

A huge wheel is rolling past your window. What do you see?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Can you explain why a sequence of operations always gives you perfect squares?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

When is it impossible to make number sandwiches?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Here are some examples of 'cons', and see if you can figure out where the trick is.

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Four jewellers share their stock. Can you work out the relative values of their gems?

Can you make sense of the three methods to work out the area of the kite in the square?

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

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?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

What is the largest number of intersection points that a triangle and a quadrilateral can have?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Prove Pythagoras' Theorem using enlargements and scale factors.

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.