A challenging activity focusing on finding all possible ways of stacking rods.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

How many different symmetrical shapes can you make by shading triangles or squares?

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?

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?

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?

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

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.

Use the clues about the symmetrical properties of these letters to place them on the grid.

This activity investigates how you might make squares and pentominoes from Polydron.

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

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

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

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 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?

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

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

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?

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.

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.

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

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?

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

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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

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?

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

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?

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?

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.

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

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

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

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?

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.

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

Can you find all the different triangles on these peg boards, and find their angles?

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?

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

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

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

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

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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?