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?
Why does the tower look a different size in each of these pictures?
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 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?
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
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?
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?
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.
Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.
How many tiles do we need to tile these patios?
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?
How many different sets of numbers with at least four members can you find in the numbers in this box?
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.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.
Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?
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?
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.
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?
How many models can you find which obey these rules?
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
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?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
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? What do you think is happening to the numbers?
In how many ways can you stack these rods, following the rules?
In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you 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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?
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.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
An investigation that gives you the opportunity to make and justify predictions.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What is the largest cuboid you can wrap in an A3 sheet of paper?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
If the answer's 2010, what could the question be?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?