Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

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

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks.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.

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?

Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.

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

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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

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.

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.

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

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?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

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?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

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?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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

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

What could the half time scores have been in these Olympic hockey matches?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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?

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.