L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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

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

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. . . .

Can you make sense of these three proofs of Pythagoras' Theorem?

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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

Prove Pythagoras' Theorem using enlargements and scale factors.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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

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

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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.

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

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. . . .

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?

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?

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. . . .

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Keep constructing triangles in the incircle of the previous triangle. What happens?

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.

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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.

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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

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. . . .

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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. . . .

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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...

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

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?