A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

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

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

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

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

The Zargoes use almost the same alphabet as English. What does this birthday message say?

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?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

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

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

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

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

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.

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?

In how many ways can you stack these rods, following the rules?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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?

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.

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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

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?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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?

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

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Investigate the different ways you could split up these rooms so that you have double the number.

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

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

How many possible necklaces can you find? And how do you know you've found them all?