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?

A follow-up activity to Tiles in the Garden.

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?

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

Investigate these hexagons drawn from different sized equilateral triangles.

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

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.

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 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?

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

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

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?

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.

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

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?

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?

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

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?

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.

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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 number of faces you can see when you arrange 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?

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

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

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

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

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?

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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?

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

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

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

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

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

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.

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

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

How many models can you find which obey these rules?

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?

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

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.

"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 ...?

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

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?