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

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

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

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

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

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

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

What two-digit numbers can you make with these two dice? What can't you make?

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

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

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

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?

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

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?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

Can you find the chosen number from the grid using the clues?

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

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?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

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.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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.

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!

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.

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!

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

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

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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?

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.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

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

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?