I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
How many different symmetrical shapes can you make by shading triangles or squares?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
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.
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?
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 thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
An investigation that gives you the opportunity to make and justify predictions.
Find out about Magic Squares in this article written for students. Why are they magic?!
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
This task challenges you to create symmetrical U shapes out of rods and find their areas.
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?
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?
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?
Can you draw a square in which the perimeter is numerically equal to the area?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
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?
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?
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.
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?
Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E
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 many solutions can you find to this sum? Each of the different letters stands for a different number.
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
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?
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.
How many models can you find which obey these rules?
This Sudoku, based on differences. Using the one clue number can you find the solution?
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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.
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?
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?
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Four small numbers give the clue to the contents of the four surrounding cells.