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

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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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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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In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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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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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 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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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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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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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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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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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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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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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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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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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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This challenge extends the Plants investigation so now four or more children are involved.

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Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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A challenging activity focusing on finding all possible ways of stacking rods.

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

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Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

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

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A follow-up activity to Tiles in the Garden.

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These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

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Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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

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Investigate what happens when you add house numbers along a street in different ways.

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

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In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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Sort the houses in my street into different groups. Can you do it in any other ways?

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Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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