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

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?

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?

Investigate the number of faces you can see when you arrange three cubes in different ways.

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

Here are many ideas for you to investigate - all linked with the number 2000.

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?

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

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?

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

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

What is the largest cuboid you can wrap in an A3 sheet of paper?

I cut this square into two different shapes. What can you say about the relationship between them?

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?

These pictures show squares split into halves. Can you find other ways?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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.

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

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

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

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

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?

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.

A follow-up activity to Tiles in the Garden.

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 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?

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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?

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

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

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?

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?

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

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

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

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?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

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?

Can you find ways of joining cubes together so that 28 faces are visible?

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

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

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