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

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?

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.

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!

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

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?

Here is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.

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

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

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

Investigate these hexagons drawn from different sized equilateral triangles.

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

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

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 follow-up activity to Tiles in the Garden.

This problem is intended to get children to look really hard at something they will see many times in the next few months.

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?

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

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

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?

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

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

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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.

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

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

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

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?

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.

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?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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.

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

How many faces can you see when you arrange these three cubes in different ways?

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?

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

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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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

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?

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

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?

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

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

"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 ...?