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?

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

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

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?

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?

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?

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

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?

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

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

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?

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?

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?

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

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?

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?

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

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

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

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?

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

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Try this matching game which will help you recognise different ways of saying the same time interval.

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

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

How long does it take to brush your teeth? Can you find the matching length of time?

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

My coat has three buttons. How many ways can you find to do up all the buttons?

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

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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.

Find all the numbers that can be made by adding the dots on two dice.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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?

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

If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.