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!

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

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

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

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

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?

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?

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

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

How many different sets of numbers with at least four members can you find in the numbers in this box?

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?

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

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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

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?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

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?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

Here are many ideas for you to investigate - all linked with the number 2000.

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?

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

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

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 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?

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

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

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

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?

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.

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?

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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

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

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?

A follow-up activity to Tiles in the Garden.

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?

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?

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

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?