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

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

Can you draw a square in which the perimeter is numerically equal to the area?

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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.

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?

An investigation that gives you the opportunity to make and justify predictions.

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.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

This task challenges you to create symmetrical U shapes out of rods and find their areas.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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?

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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?

A Sudoku that uses transformations as supporting clues.

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

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

If we had 16 light bars which digital numbers could we make? How will you know you've found 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.

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.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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

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.

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

These practical challenges are all about making a 'tray' and covering it with paper.

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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

In this article, the NRICH team describe the process of selecting solutions for publication on the site.

How many models can you find which obey these rules?

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

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?

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?