In this article, the NRICH team describe the process of selecting solutions for publication on the site.

This article for primary teachers suggests ways in which to help children become better at working systematically.

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 work out some different ways to balance this equation?

What two-digit numbers can you make with these two dice? What can't you make?

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

Have a go at balancing this equation. Can you find different ways of doing it?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

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

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.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Can you replace the letters with numbers? Is there only one solution in each case?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

This challenge extends the Plants investigation so now four or more children are involved.

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

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

Can you find the chosen number from the grid using the clues?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

How many different triangles can you make on a circular pegboard that has nine pegs?

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

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

Find out about Magic Squares in this article written for students. Why are they magic?!

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

How many trains can you make which are the same length as Matt's, using rods that are identical?

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

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?