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?

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.

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.

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?

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

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?

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.

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?

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?

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

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

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?

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

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

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

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?

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.

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

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?

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?

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

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?

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?

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

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.

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?

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?

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?

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?

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?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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?

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

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!

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.

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