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?

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?

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?

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

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

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?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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?

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?

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

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?

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

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

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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?

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?

What happens when you try and fit the triomino pieces into these two grids?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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?

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

How many different rhythms can you make by putting two drums on the wheel?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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

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

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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

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

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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?

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?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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

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.

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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