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

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. . . .

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

Can you find different ways of creating paths using these paving slabs?

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. . . .

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.

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.

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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?

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.

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?

From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .

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

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?

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?

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

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. . . .

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

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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

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

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

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

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

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

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

This article for primary teachers suggests ways in which we can help learners move from being novice reasoners to expert reasoners.

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

In this article for primary teachers we consider in depth when we might reason which helps us understand what reasoning 'looks like'.

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

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.

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. . . .

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. . . .

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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?

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 is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

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?

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. . . .

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.

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?

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

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

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

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.