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

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?

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

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

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 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?

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?

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?

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?

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

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

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?

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?

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

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

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

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

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

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?

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

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?

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?

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?

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.

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.

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

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

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

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?

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?

This problem is intended to get children to look really hard at something they will see many times in the next few months.

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?

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

A follow-up activity to Tiles in the Garden.

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.

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!

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

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 challenge encourages you to explore dividing a three-digit number by a single-digit number.

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

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?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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?

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

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?

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?