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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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How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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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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When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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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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If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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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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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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Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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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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This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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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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How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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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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Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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How many models can you find which obey these rules?

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Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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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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In how many ways can you stack these rods, following the rules?

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The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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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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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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While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

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There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

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Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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

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In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

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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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Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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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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Investigate these hexagons drawn from different sized equilateral triangles.