An investigation that gives you the opportunity to make and justify
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?
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
A Sudoku with clues given as sums of entries.
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Find out about Magic Squares in this article written for students. Why are they magic?!
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Can you draw a square in which the perimeter is numerically equal
to the area?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
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?
Zumf makes spectacles for the residents of the planet Zargon, who
have either 3 eyes or 4 eyes. How many lenses will Zumf need to
make all the different orders for 9 families?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's
there is one digit, between the two 2's there are two digits, and
between the two 3's there are three digits.
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
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 investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did he do it?
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
Find out what a "fault-free" rectangle is and try to make some of
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
These rectangles have been torn. How many squares did each one have
inside it before it was ripped?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
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.
Two children made up a game as they walked along the garden paths.
Can you find out their scores? Can you find some paths of your own?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
What happens when you round these three-digit numbers to the nearest 100?
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
What happens when you round these numbers to the nearest whole number?
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