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.
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.
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?
A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
A Sudoku with clues given as sums of entries.
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?
60 pieces and a challenge. What can you make and how many of the
pieces can you use creating skeleton polyhedra?
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
There are to be 6 homes built on a new development site. They could
be semi-detached, detached or terraced houses. How many different
combinations of these can you find?
In this town, houses are built with one room for each person. There
are some families of seven people living in the town. In how many
different ways can they build their houses?
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?
What happens when you try and fit the triomino pieces into these
Can you cover the camel with these pieces?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
My cube has inky marks on each face. Can you find the route it has
taken? What does each face look like?
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
This activity investigates how you might make squares and pentominoes from Polydron.
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Tom and Ben visited Numberland. Use the maps to work out the number
of points each of their routes scores.
Ben and his mum are planting garlic. Use the interactivity to help
you find out how many cloves of garlic they might have had.
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?
These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
How many different ways can you find to join three equilateral
triangles together? Can you convince us that you have found them
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?
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
Make a pair of cubes that can be moved to show all the days of the
month from the 1st to the 31st.
How many different rhythms can you make by putting two drums on the
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?
Using the cards 2, 4, 6, 8, +, - and =, what number statements can
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
What could the half time scores have been in these Olympic hockey
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
In this article, the NRICH team describe the process of selecting solutions for publication on the site.
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?
This problem is based on a code using two different prime numbers
less than 10. You'll need to multiply them together and shift the
alphabet forwards by the result. Can you decipher the code?
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
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!
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?