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.

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?

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

Four small numbers give the clue to the contents of the four surrounding cells.

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

This Sudoku, based on differences. Using the one clue number can you find the solution?

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

A pair of Sudoku puzzles that together lead to a complete solution.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

A man has 5 coins in his pocket. Given the clues, can you work out what the coins are?

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

Use the clues about the shaded areas to help solve this sudoku

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

You need to find the values of the stars before you can apply normal Sudoku rules.

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?

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Given the products of diagonally opposite cells - can you complete this Sudoku?

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

A Sudoku that uses transformations as supporting clues.

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

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

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

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

Two sudokus in one. Challenge yourself to make the necessary connections.

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

Each clue number in this sudoku is the product of the two numbers in adjacent cells.

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?