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?

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

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

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

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?

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.

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.?

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?

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?

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

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 trains can you make which are the same length as Matt's, using rods that are identical?

How many models can you find which obey these rules?

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

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?

Can you find all the different ways of lining up these Cuisenaire rods?

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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

How many different triangles can you make on a circular pegboard that has nine pegs?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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.

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?

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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.

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

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

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?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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?

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

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

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?

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

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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?

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