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How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

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How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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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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A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

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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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What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

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Can you find ways of joining cubes together so that 28 faces are visible?

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One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

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What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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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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Design an arrangement of display boards in the school hall which fits the requirements of different people.

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In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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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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How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

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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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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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Make a cube out of straws and have a go at this practical challenge.

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How will you go about finding all the jigsaw pieces that have one peg and one hole?

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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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A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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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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Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

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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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Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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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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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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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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What is the best way to shunt these carriages so that each train can continue its journey?

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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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Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

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How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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How many different triangles can you make on a circular pegboard that has nine pegs?

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Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

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Can you make a 3x3 cube with these shapes made from small cubes?

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Can you cut up a square in the way shown and make the pieces into a triangle?

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Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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

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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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Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

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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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Exploring and predicting folding, cutting and punching holes and making spirals.