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

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

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.

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

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

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

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

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 find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

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

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

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

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?

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

This challenge extends the Plants investigation so now four or more children are involved.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

Can you draw a square in which the perimeter is numerically equal to the area?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?

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?

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Are all the possible combinations of two shapes included in this set of 27 cards? 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.

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

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.

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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

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

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

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

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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

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?

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

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

The pages of my calendar have got mixed up. Can you sort them out?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

A challenging activity focusing on finding all possible ways of stacking rods.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?