A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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?

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

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.

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.

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

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?

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.

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.

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?

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

How many models can you find which obey these rules?

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?

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

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

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?

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

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

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

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?

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

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.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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

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

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 is the smallest number of coins needed to make up 12 dollars and 83 cents?

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?

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

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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

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?

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?

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?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

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?

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.

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?

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

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?

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

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!

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