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.

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?

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?

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.

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

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

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?

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.

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

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 game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

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

A few extra challenges set by some young NRICH members.

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

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

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?

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

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

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

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.

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

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

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.

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

A challenging activity focusing on finding all possible ways of stacking rods.

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

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 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?

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?

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

Find the values of the nine letters in the sum: FOOT + BALL = GAME

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

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?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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

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. . . .

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?