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

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

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

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

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

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

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

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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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Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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

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

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

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

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

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Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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

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

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

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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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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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Nim-7 game for an adult and child. Who will be the one to take the last counter?

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This activity involves rounding four-digit numbers to the nearest thousand.

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