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.

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?

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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

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?

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?

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

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

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?

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

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.

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 the statements, can you work out how many of each type of rabbit there are in these pens?

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.

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 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?

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

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?

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

There were 22 legs creeping across the web. How many flies? How many spiders?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

Place the digits 1 to 9 into the circles so that each side of the triangle adds to the same total.

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?

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.

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?

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

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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?

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 your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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

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.

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?

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

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?

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

Fill in the missing numbers so that adding each pair of corner numbers gives you the number between them (in the box).

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 1871 a mathematician called Augustus De Morgan died. De Morgan made a puzzling statement about his age. Can you discover which year De Morgan was born in?

There are a number of coins on a table. One quarter of the coins show heads. If I turn over 2 coins, then one third show heads. How many coins are there altogether?

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

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?

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?

56 406 is the product of two consecutive numbers. What are these two numbers?

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).

In this problem you have to place four by four magic squares on the faces of a cube so that along each edge of the cube the numbers match.