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.

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

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

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.

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

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

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.

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?

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?

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

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?

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.

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

Can you beat the computer in the challenging strategy game?

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches.

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?

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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.

Can you number the vertices, edges and faces of a tetrahedron so that the number on each edge is the mean of the numbers on the adjacent vertices and the mean of the numbers on the adjacent faces?

Imagine picking up a bow and some arrows and attempting to hit the target a few times. Can you work out the settings for the sight that give you the best chance of gaining a high score?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Carry out some time trials and gather some data to help you decide on the best training regime for your rowing crew.

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

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

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

This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

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?

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.

Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

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

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

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

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

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?

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.

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

Can you guess the colours of the 10 marbles in the bag? Can you develop an effective strategy for reaching 1000 points in the least number of rounds?

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?

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

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?

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.

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 3x3 cube with these shapes made from small cubes?