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

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?

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

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?

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

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?

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

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.

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

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 each work out the number on your card? What do you notice? How could you sort the cards?

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?

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?

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?

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.

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

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?

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

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

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

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

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?

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

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?

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

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

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?

Can you make the birds from the egg tangram?

Have you ever noticed the patterns in car wheel trims? These questions will make you look at car wheels in a different way!

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

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?

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

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?

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

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

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?

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

Can you each work out what shape you have part of on your card? What will the rest of it look like?

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?

How many models can you find which obey these rules?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

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

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?