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?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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

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?

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

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

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.

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

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

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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

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

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.

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

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

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?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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.

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

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?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

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.

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.?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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?

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

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

The Zargoes use almost the same alphabet as English. What does this birthday message say?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

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

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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

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?