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

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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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Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

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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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Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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

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

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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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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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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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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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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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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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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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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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Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

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

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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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Ben has five coins in his pocket. How much money might he have?

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

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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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This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

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48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

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What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

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We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.