Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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.

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.

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?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

How many different symmetrical shapes can you make by shading triangles or squares?

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

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

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.

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.

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.

The clues for this Sudoku are the product of the numbers in adjacent squares.

A few extra challenges set by some young NRICH members.

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?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Use the differences to find the solution to this Sudoku.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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

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.

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.

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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

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. . . .

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

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?

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

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

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

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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.

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

A Sudoku with clues given as sums of entries.

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

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.

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.

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.

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

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

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