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.

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.

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

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

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.

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?

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

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

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?

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.

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

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

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

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

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

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?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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 to have an ordered approach.

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?

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?

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

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?

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

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?

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?

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?

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?

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

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

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?

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

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?

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

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.

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.

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

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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

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.?

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.

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?