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

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

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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. . . .

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

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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?

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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. . . .

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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. . . .

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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?

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Can you find out which 3D shape your partner has chosen before they work out your shape?

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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.

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.

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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.

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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. . . .

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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?

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. . . .

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

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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. . . .

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

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

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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. . . .

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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?

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

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

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Use your knowledge of place value to try to win this game. How will you maximise your score?

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

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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

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

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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.

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

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

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

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

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

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

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Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

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

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

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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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. . . .

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I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

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I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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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?

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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. . . .

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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?

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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. . . .

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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.