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

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

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

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.

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?

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?

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.

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?

Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?

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?

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?

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?

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?

Here is a version of the game 'Happy Families' for you to make and play.

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?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

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?

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

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?

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?

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

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

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

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

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

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

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

How many models can you find which obey these rules?

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?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

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

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

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

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

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

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

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

Reasoning about the number of matches needed to build squares that share their sides.

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

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

Exploring and predicting folding, cutting and punching holes and making spirals.

Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

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

Can you make the birds from the egg tangram?