Exploring balance and centres of mass can be great fun. The resulting structures can seem impossible. Here are some images to encourage you to experiment with non-breakable objects of your own.

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

Help the bee to build a stack of blocks far enough to save his friend trapped in the tower.

Use these four dominoes to make a square that has the same number of dots on each side.

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

Use the information about Sally and her brother to find out how many children there are in the Brown family.

Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits.

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Can you find a path from a number at the top of this network to the bottom which goes through each number from 1 to 9 once and once only?

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.

Sam sets up displays of cat food in his shop in triangular stacks. If Felix buys some, then how can Sam arrange the remaining cans in triangular stacks?

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

Amy's mum had given her £2.50 to spend. She bought four times as many pens as pencils and was given 40p change. How many of each did she buy?

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

You have a set of the digits from 0 – 9. Can you arrange these in the 5 boxes to make two-digit numbers as close to the targets as possible?

Can you go from A to Z right through the alphabet in the hexagonal maze?

There are three buckets each of which holds a maximum of 5 litres. Use the clues to work out how much liquid there is in each bucket.

Use the information to work out how many gifts there are in each pile.

Can you make a 3x3 cube with these shapes made from small cubes?

Cassandra, David and Lachlan are brothers and sisters. They range in age between 1 year and 14 years. Can you figure out their exact ages from the clues?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Peter, Melanie, Amil and Jack received a total of 38 chocolate eggs. Use the information to work out how many eggs each person had.

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

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?

Using only six straight cuts, find a way to make as many pieces of pizza as possible. (The pieces can be different sizes and shapes).

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

I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to?

You have two sets of the digits 0 – 9. Can you arrange these in the five boxes to make four-digit numbers as close to the target numbers as possible?

Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.

A shunting puzzle for 1 person. Swop the positions of the counters at the top and bottom of the board.

Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

Can you use the information to find out which cards I have used?

Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off?

Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?

The discs for this game are kept in a flat square box with a square hole for each disc. Use the information to find out how many discs of each colour there are in the box.

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. Can you use the information given to find out how many eggs are in each basket?