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?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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 two grids?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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?

How many different rhythms can you make by putting two drums on the wheel?

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?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

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.

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

This activity investigates how you might make squares and pentominoes from Polydron.

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

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.

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?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

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 jugs.

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

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.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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!

Can you make square numbers by adding two prime numbers together?

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?

What could the half time scores have been in these Olympic hockey matches?