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 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?

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

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

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.

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

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 of tiling the square patio, using square tiles of different sizes?

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.

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

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

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?

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.

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?

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!

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?

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.

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

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

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?

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.

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

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?

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?

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

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!

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?

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

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.

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

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?

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

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

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

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

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

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?

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.

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 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?

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.

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.

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

Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?

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?