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15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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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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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

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A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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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, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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Delight your friends with this cunning trick! Can you explain how it works?

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Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be. . . .

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Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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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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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?

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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How many moves does it take to swap over some red and blue frogs? Do you have a method?

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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Can all unit fractions be written as the sum of two unit fractions?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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

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

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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

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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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

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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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Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

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Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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Can you work out how to win this game of Nim? Does it matter if you go first or second?

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

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