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?

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?

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

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.

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

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

Benâ€™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?

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

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

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?

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

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

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

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

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

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?

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.

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?

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

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

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

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

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

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?

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

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

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

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

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

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?

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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

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?

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

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?

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.

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?

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

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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?

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

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?

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.