How many centimetres of rope will I need to make another mat just like the one I have here?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

This task follows on from Build it Up and takes the ideas into three dimensions!

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

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Here are two kinds of spirals for you to explore. What do you notice?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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

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

This challenge is about finding the difference between numbers which have the same tens digit.

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?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Use two dice to generate two numbers with one decimal place. 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?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

This activity involves rounding four-digit numbers to the nearest thousand.

Stop the Clock game for an adult and child. How can you make sure you always win this game?

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

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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.

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?