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

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?

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?

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

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.

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.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

These practical challenges are all about making a 'tray' and covering it with paper.

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

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

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?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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.

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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

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?

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

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?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

An activity making various patterns with 2 x 1 rectangular tiles.

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!

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

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

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

How many models can you find which obey these rules?

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

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.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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

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.

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

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

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