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Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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How many centimetres of rope will I need to make another mat just like the one I have here?

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An investigation that gives you the opportunity to make and justify predictions.

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Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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It starts quite simple but great opportunities for number discoveries and patterns!

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While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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What happens when you round these three-digit numbers to the nearest 100?

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Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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This task follows on from Build it Up and takes the ideas into three dimensions!

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Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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What happens when you round these numbers to the nearest whole number?

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In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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Can you find a way of counting the spheres in these arrangements?

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Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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Can you describe this route to infinity? Where will the arrows take you next?

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It would be nice to have a strategy for disentangling any tangled ropes...

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We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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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.

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A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.