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

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

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

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Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

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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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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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Here are two kinds of spirals for you to explore. 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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This activity involves rounding four-digit numbers to the nearest thousand.

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

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

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

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

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

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

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

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

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

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

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

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

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In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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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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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.

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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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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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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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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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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Surprise your friends with this magic square trick.

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How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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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?