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

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.

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.

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?

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.

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

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

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?

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.

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?

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.

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.

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

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

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

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

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

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

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?

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.

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

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

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.

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

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

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

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?

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?

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

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.

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

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 to do up all four buttons on my coat? How about if I had five buttons? Six ...?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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

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

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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

What happens when you round these numbers to the nearest whole number?

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

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

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?

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?