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.

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?

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.

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.

How many models can you find which obey these rules?

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?

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?

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

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

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

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?

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?

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!

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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 ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

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.

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?

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

Can you draw a square in which the perimeter is numerically equal to the area?

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!

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

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?

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.

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.

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

What is the best way to shunt these carriages so that each train can continue its journey?

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

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?

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.

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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?

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.

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?

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

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

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?

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

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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?