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.

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 three-digit numbers to the nearest 100?

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

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks.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

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

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?

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

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

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?

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.

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

Can you work out some different ways to balance this equation?

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

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

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

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

Using all ten cards from 0 to 9, rearrange them to make five prime numbers. Can you find any other ways of doing it?

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

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?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

Can you use the information to find out which cards I have used?

How many models can you find which obey these rules?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

An activity making various patterns with 2 x 1 rectangular tiles.

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?