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?

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

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

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

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

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

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

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

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.?

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.

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

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?

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.

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.

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?

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.

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?

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

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?

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

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!

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

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

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

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

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

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

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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!

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

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

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?

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.

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?

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

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

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.

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.

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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?

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.

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

In this matching game, you have to decide how long different events take.

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?