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.

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.

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?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

A Sudoku with clues given as sums of entries.

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

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?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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?

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.

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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 make dice stairs using the rules stated? How do you know you have all the possible stairs?

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?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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?

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.

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 ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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?

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

What happens when you try and fit the triomino pieces into these two grids?

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

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?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

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.

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

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

Can you use the information to find out which cards I have used?

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

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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!

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.