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?

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

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

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

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?

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

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.

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

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

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

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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.

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?

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

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

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

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 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?

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

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?

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?

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.

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

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?

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

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

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!

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.

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.

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?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

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

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?

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

A description of some experiments in which you can make discoveries about triangles.

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

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?