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?

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?

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

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?

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

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?

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

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

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

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.

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?

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

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

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 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?

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.

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?

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.

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

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

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?

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

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.

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

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...

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

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

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?

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.

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?

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.

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

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?

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 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.

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?

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. . . .

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

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?

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?

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

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

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

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.

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?