What is the largest cuboid you can wrap in an A3 sheet of paper?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
An investigation that gives you the opportunity to make and justify predictions.
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
Ben has five coins in his pocket. How much money might he have?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
Bernard Bagnall describes how to get more out of some favourite NRICH investigations.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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.
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.
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
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?
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?
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?
Investigate these hexagons drawn from different sized equilateral triangles.
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?
Why does the tower look a different size in each of these pictures?
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?
When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?
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?
In how many ways can you stack these rods, following the rules?
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!
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
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?
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?
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
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?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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?
How many models can you find which obey these rules?
Investigate what happens when you add house numbers along a street in different ways.
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.
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?
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?
I cut this square into two different shapes. What can you say about the relationship between them?
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
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?
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?
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.