This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens?

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

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

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

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

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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

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?

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

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

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

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?

Got It game for an adult and child. How can you play so that you know you will always win?

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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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

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

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?

Try out this number trick. What happens with different starting numbers? What do you notice?

This challenge asks you to imagine a snake coiling on itself.

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

Are these statements always true, sometimes true or never true?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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

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.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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.