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?

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?

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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?

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

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

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.

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

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.

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?

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?

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.

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

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?

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.

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

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?

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.

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

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

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...

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 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. . . .

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?

A few extra challenges set by some young NRICH members.

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

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

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

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.

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?

Given the products of adjacent cells, can you complete this Sudoku?

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

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?

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

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?

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

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?

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

Use the differences to find the solution to this Sudoku.

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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

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

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

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?"