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

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?

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

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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?

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

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.

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?

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

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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.

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

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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?

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

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

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

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?

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

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?

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?

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Find the sum of all three-digit numbers each of whose digits is odd.

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

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

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

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

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?

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.

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

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

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

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

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

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?

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

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

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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

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?

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

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