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?

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

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

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

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"?

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

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

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

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.

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

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?

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

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?

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

This challenge extends the Plants investigation so now four or more children are involved.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

This article for teachers suggests ideas for activities built around 10 and 2010.

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

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

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?

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?

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

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.

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?

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.

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?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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 letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

A follow-up activity to Tiles in the Garden.

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.

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

A challenging activity focusing on finding all possible ways of stacking rods.

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

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?

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?

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?

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?

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

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?

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

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?

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

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?

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?