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!

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

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.

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

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

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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?

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

A follow-up activity to Tiles in the Garden.

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?

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

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

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?

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

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

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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?

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

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

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

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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!

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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.

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

Investigate the different ways you could split up these rooms so that you have double the number.

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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?

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

How many models can you find which obey these rules?

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

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?

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

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

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

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