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In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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These practical challenges are all about making a 'tray' and covering it with paper.

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What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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An investigation that gives you the opportunity to make and justify predictions.

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A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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What could the half time scores have been in these Olympic hockey matches?

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This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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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 find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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How many ways can you find of tiling the square patio, using square tiles of different sizes?

This task challenges you to create symmetrical U shapes out of rods and find their areas.

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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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This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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Can you draw a square in which the perimeter is numerically equal to the area?

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Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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Investigate the different ways you could split up these rooms so that you have double the number.

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Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

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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?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

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You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

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Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

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How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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Using all ten cards from 0 to 9, rearrange them to make five prime numbers. Can you find any other ways of doing it?

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How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

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My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?

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Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

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There are lots of different methods to find out what the shapes are worth - how many can you find?

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These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

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Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

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Find out about the lottery that is played in a far-away land. What is the chance of winning?

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Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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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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This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?