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?

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

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

A follow-up activity to Tiles in the Garden.

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

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 these hexagons drawn from different sized equilateral triangles.

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?

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?

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

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

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?

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

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.

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.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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?

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

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?

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

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?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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

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.

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

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

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?

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

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

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

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?

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

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?

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?

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

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

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

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

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.

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.

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?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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?

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

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