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.

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.

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?

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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?

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.

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.

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

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

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?

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?

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

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.?

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?

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?

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

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

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

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

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 work out which cities Mohamed, Sheng, Tanya and Bharat live 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.

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?

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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?

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

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

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

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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.

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?

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

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.

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

This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.

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

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

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

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?

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

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

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?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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?

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