Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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 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?
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?
Investigate the different ways you could split up these rooms so that you have double the number.
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.
Explore one of these five pictures.
In how many ways can you stack these rods, following the rules?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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.
Can you find ways of joining cubes together so that 28 faces are visible?
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?
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.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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?
Have a go at this 3D extension to the Pebbles problem.
This challenge extends the Plants investigation so now four or more children are involved.
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
How many models can you find which obey these rules?
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?
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 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.
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?
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?
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?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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.
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?
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?
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?