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

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

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

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.

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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.

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. . . .

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

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. . . .

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

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal. . . .

When is it impossible to make number sandwiches?

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. . . .

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

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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.

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

Can you find the areas of the trapezia in this sequence?

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

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

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.

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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 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?

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

Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

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

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

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

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

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

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

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

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

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.

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. . . .