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 three-digit numbers to the nearest 100?

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

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

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?

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

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.

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

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

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

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

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

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

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.

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?

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

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.

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

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?

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?

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?

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

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

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

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 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?

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

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks.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

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

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

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?

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....?

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.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

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

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

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

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

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?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?