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?

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?

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

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

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

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?

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?

How many models can you find which obey these rules?

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?

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.

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?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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?

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?

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. . . .

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?

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 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?

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 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.

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

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

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

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?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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?

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Two sudokus in one. Challenge yourself to make the necessary connections.

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.

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