Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches.

After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?

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

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.

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?

Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat. . . .

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. . . .

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

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.

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.

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

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

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?

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

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

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.

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?

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

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

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

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?

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

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?

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.

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.

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.

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. . . .

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

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

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

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

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

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

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

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.

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. . . .

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

What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them.

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.

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.

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

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

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

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.

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