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.

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.

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?

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?

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.

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?

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

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?

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.

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

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

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?

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.

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

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

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?

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

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

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

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?

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

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

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

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?

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

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?

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?

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!

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

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

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?

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

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

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?

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.

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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