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

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

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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?

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

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

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

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

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

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

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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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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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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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Make one big triangle so the numbers that touch on the small triangles add to 10.

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An activity centred around observations of dots and how we visualise number arrangement patterns.

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

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