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?

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. . . .

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?

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.

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?

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

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?

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.

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. . . .

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

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

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.

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?

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.

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

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?

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?

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

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

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

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

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

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

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. . . .

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

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

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.

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

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.

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. . . .

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?

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. . . .

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

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

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

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

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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?

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.

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?

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?

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