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Can you find a way of counting the spheres in these arrangements?

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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.

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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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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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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Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

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

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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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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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

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Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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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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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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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?”

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

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Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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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?

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

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

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

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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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Watch this animation. What do you see? Can you explain why this happens?

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

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

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

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

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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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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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Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

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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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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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Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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

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

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