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?

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

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

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.

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

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?

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.

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

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

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?

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?

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

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

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.

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?

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.

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?

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

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

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

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

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

How many models can you find which obey these rules?

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

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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?

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.

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

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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?

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

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?

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?

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

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.

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

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

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

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?

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?

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

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?