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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

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.

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?

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

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?

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?

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

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?

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

This task follows on from Build it Up and takes the ideas into three dimensions!

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 all the different ways of lining up these Cuisenaire rods?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same 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.

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

How many trapeziums, of various sizes, are hidden in this picture?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.