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

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

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

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

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

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

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

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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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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?

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

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

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Make some loops out of regular hexagons. What rules can you discover?

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

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

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

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

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

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

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

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

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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?

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

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

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Surprise your friends with this magic square trick.

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

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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.

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.