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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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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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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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Find your way through the grid starting at 2 and following these operations. What number do you end on?

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

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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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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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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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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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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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Can you describe a piece of paper clearly enough for your partner to know which piece it is?

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

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Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.

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

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A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

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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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Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

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

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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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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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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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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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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 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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Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

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In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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How many pieces of string have been used in these patterns? Can you describe how you know?

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How many loops of string have been used to make these patterns?

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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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If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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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.

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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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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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Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

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Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

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

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