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

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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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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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This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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

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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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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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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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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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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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How many models can you find which obey these rules?

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

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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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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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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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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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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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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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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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An activity making various patterns with 2 x 1 rectangular tiles.

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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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Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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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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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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Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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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 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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It starts quite simple but great opportunities for number discoveries and patterns!

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

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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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A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

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

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What is the largest cuboid you can wrap in an A3 sheet of paper?

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What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

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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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I cut this square into two different shapes. What can you say about the relationship between them?

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

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What do these two triangles have in common? How are they related?

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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.