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.

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

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

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

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?

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.

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

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

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

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

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.

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?

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?

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

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

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?

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

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?

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.

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?

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

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

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?

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

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

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?

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?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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

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

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.

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

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?

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

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?

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

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

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.

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

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.