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

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

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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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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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Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

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

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

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

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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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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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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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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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Surprise your friends with this magic square trick.

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

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

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

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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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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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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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

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

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

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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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Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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

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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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One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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

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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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Got It game for an adult and child. How can you play so that you know you will always win?