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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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We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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How many models can you find which obey these rules?

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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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We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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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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In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

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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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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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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 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?

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

Challenge Level

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

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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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Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

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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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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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Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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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 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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Investigate these hexagons drawn from different sized equilateral triangles.

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

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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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This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

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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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How many faces can you see when you arrange these three cubes in different ways?

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Why does the tower look a different size in each of these pictures?

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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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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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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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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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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?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

Challenge Level

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

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In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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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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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 make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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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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A follow-up activity to Tiles in the Garden.

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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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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?