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.
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 is your chance to investigate the number 28 using shapes, cubes ... in fact anything at all.
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 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 these hexagons drawn from different sized equilateral triangles.
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 and explain the patterns that you see from recording just the units digits of numbers in the times tables.
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?
Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
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?
How many tiles do we need to tile these patios?
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?
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?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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"?
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
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.
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?
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.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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.
Investigate what happens when you add house numbers along a street in different ways.
How many different sets of numbers with at least four members can you find in the numbers in this box?
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
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?
Investigate the different ways you could split up these rooms so that you have double the number.
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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?
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?
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
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.
Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?
How many models can you find which obey these rules?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
In how many ways can you stack these rods, following the rules?
If the answer's 2010, what could the question be?
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?
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?
This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.
These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?