This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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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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In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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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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This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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

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

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This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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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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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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How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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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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We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

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What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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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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A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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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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Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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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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A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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

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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

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Can you describe this route to infinity? Where will the arrows take you next?

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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

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How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

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These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

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Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?