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?

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.

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

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

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?

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

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

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.

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?

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?

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

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?

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

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

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 follow-up activity to Tiles in the Garden.

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

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?

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.

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

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?

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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.

Explore the different tunes you can make with these five gourds. What are the similarities and differences between the two tunes you are given?

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

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

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

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?

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?

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?

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

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?

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

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

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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.

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.

How many models can you find which obey these rules?

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

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?