10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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?

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?

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?

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?

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

How many different triangles can you make on a circular pegboard that has nine pegs?

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?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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

Can you find all the different ways of lining up these Cuisenaire rods?

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.

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

Investigate the different ways you could split up these rooms so that you have double the number.

Can you draw a square in which the perimeter is numerically equal to the area?

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?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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 we had 16 light bars which digital numbers could we make? How will you know you've found them all?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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.

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 find all the different triangles on these peg boards, and find their angles?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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?

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.