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.

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

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 order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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

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....?

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

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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.

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

Throughout these challenges, the touching faces of any adjacent dice must have the same number. Can you find a way of making the total on the top come to each number from 11 to 18 inclusive?

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

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

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.

Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?

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?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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.

Here are some rods that are different colours. How could I make a yellow rod using white and red rods?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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

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?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

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

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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.

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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?

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.

How many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?

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'.

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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?

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

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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?