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?

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?

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?

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?

An activity making various patterns with 2 x 1 rectangular tiles.

This challenge invites you to create your own picture using just straight lines. Can you identify shapes with the same number of sides and decorate them in the same way?

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

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

How many models can you find which obey these rules?

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?

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?

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

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.

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?

Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!

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.

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

These pictures show squares split into halves. Can you find other ways?

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

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.

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

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?

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?

Explore the triangles that can be made with seven sticks of the same length.

This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.

Can you make a rectangle with just 2 dominoes? What about 3, 4, 5, 6, 7...?

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?

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?

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

What do these two triangles have in common? How are they related?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

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?

Can you split each of the shapes below in half so that the two parts are exactly the same?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Follow these instructions to make a five-pointed snowflake from a square of paper.

It's hard to make a snowflake with six perfect lines of symmetry, but it's fun to try!

Did you know mazes tell stories? Find out more about mazes and make one of your own.

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?

What shape is made when you fold using this crease pattern? Can you make a ring design?

Can you make five differently sized squares from the tangram pieces?

Follow the diagrams to make this patchwork piece, based on an octagon in a square.

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Have a go at drawing these stars which use six points drawn around a circle. Perhaps you can create your own designs?