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

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

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

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.

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?

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.

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?

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

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?

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

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

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

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

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?

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.

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

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

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

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.

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

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?

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?

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?

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

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

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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!

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 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?

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

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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

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!

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?

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?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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

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

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

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

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.

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

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

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