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

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?

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?

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?

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?

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

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

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?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

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?

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.

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?

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

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

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

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?

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

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

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

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

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

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?

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?

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?

What is the greatest number of squares you can make by overlapping three squares?

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?

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?

How many models can you find which obey these rules?

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?

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

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

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.

Can you visualise what shape this piece of paper will make when it is folded?

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.

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?

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

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

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?

Can you make the birds from the egg tangram?

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?

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?

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