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

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?

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?

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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.

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.

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

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?

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?

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?

Can you make the birds from the egg tangram?

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?

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.

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?

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

Can you put these shapes in order of size? Start with the smallest.

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

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

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?

How many models can you find which obey these rules?

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

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

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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?

If you count from 1 to 20 and clap more loudly on the numbers in the two times table, as well as saying those numbers loudly, which numbers will be loud?

Make a flower design using the same shape made out of different sizes of paper.

Here are some ideas to try in the classroom for using counters to investigate number patterns.

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?

Watch this "Notes on a Triangle" film. Can you recreate parts of the film using cut-out triangles?

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

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

Using a loop of string stretched around three of your fingers, what different triangles can you make? Draw them and sort them into groups.

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

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

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

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?

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

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

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?

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

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?

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?

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