What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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?

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

How many ways can you find of tiling the square patio, using square tiles of different sizes?

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

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

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

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

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

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.

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?

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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.

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

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

Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away?

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

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

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

Can you find out in which order the children are standing in this line?

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?

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

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

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?

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?

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

This activity investigates how you might make squares and pentominoes from Polydron.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

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?

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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?

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?

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.

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

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

Find out what a "fault-free" rectangle is and try to make some of your own.

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?