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

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

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?

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

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?

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?

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?

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.

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

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 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?

Can you deduce the pattern that has been used to lay out these bottle tops?

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

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

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?

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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

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?

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

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?

Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

Can you make the birds from the egg tangram?

Use the tangram pieces to make our pictures, or to design some of your own!

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?

How many models can you find which obey these rules?

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

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?

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

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

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?

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?

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?

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

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

A game to make and play based on the number line.

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

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

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

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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

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