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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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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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Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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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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Is there an efficient way to work out how many factors a large number has?

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

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

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

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

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

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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

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

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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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Can you explain the strategy for winning this game with any target?

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

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

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

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

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

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

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

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

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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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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.

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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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Surprise your friends with this magic square trick.

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

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

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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?”

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

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

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Here are two kinds of spirals for you to explore. What do you notice?