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

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

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

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

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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.

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

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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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This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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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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These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

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

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

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

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Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

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Surprise your friends with this magic square trick.

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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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Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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Try out this number trick. What happens with different starting numbers? What do you notice?

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Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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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Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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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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Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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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?

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

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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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Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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Find the sum of all three-digit numbers each of whose digits is odd.