A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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?

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

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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?

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?

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?

How many models can you find which obey these rules?

Why does the tower look a different size in each of these pictures?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

A follow-up activity to Tiles in the Garden.

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

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?

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?

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.

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

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?

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.

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.

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?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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?

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?

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

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?

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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?

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?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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

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

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?

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

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

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?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?