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

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

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. . . .

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?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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. . . .

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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

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?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Which set of numbers that add to 10 have the largest product?

Choose any three by three square of dates on a calendar page...

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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

Are these statements always true, sometimes true or never true?

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

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

A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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.

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

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.

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?

Are these statements always true, sometimes true or never true?

When is it impossible to make number sandwiches?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

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.

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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

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

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

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

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

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?

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. . . .

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

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.