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

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

How many models can you find which obey these rules?

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?

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.

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

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?

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

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?

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

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

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

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

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

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

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

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

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?

Investigate these hexagons drawn from different sized equilateral 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?

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

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?

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?

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

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

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?

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.

Investigate the different ways you could split up these rooms so that you have double the number.

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

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?

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?

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

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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?

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.

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

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

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?

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?

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?

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.

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.