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.

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

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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?

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

In this article, the NRICH team describe the process of selecting solutions for publication on the site.

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.

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

This challenge extends the Plants investigation so now four or more children are involved.

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

A Sudoku with clues given as sums of entries.

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

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

Find out about Magic Squares in this article written for students. Why are they magic?!

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

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

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.

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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

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?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

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

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

How many different triangles can you make on a circular pegboard that has nine pegs?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

How many possible necklaces can you find? And how do you know you've found them all?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

These practical challenges are all about making a 'tray' and covering it with paper.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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?