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

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

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

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?

Investigate the number of faces you can see when you arrange three cubes in different ways.

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

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

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 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.

Investigate these hexagons drawn from different sized equilateral triangles.

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

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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.

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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.

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

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

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?

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?

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?

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.

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

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.

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?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

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?

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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

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?

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?

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 ...?

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.

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

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?

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?