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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?
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?
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.
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?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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?
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Why does the tower look a different size in each of these pictures?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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.
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
How many different sets of numbers with at least four members can you find in the numbers in this box?
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?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
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!
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.
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?
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?
Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.
Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.
There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?
In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Investigate the different ways you could split up these rooms so that you have double the number.
In how many ways can you stack these rods, following the rules?
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.
Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Can you find ways of joining cubes together so that 28 faces are visible?
If the answer's 2010, what could the question be?
"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
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?
Explore ways of colouring this set of triangles. Can you make symmetrical patterns?
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?
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?
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?
Sort the houses in my street into different groups. Can you do it in any other ways?
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?
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?
Investigate what happens when you add house numbers along a street in different ways.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
I cut this square into two different shapes. What can you say about the relationship between them?