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

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

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

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?

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?

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

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

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 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?

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 happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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

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

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?

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?

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.

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

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

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

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

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

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?

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?

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?

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.

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?

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 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.

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.

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

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

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

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?

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!

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?

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

Investigate what happens when you add house numbers along a street in different ways.

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

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

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