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

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

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

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?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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

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

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

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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....?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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

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?

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

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 ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Investigate the different ways you could split up these rooms so that you have double the number.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

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

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.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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 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?

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

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

My coat has three buttons. How many ways can you find to do up all the buttons?

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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.

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

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

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 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?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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

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?