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?

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

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?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

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

How many models can you find which obey these rules?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

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?

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

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

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?

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

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

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

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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

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. . . .

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.

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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

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

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

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

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

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?

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?

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?

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.

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

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

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?