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

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

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?

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

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?

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

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

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?

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?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

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. . . .

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

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?

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?

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

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?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

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

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 greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?

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?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

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?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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?

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

Can you fit the tangram pieces into the outline of the rocket?

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

Can you fit the tangram pieces into the outline of this junk?

Can you fit the tangram pieces into the outline of this plaque design?

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

Here are shadows of some 3D shapes. What shapes could have made them?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Which of these dice are right-handed and which are left-handed?

Can you fit the tangram pieces into the outline of Wai Ping, Wah Ming and Chi Wing?

Can you fit the tangram pieces into the outline of the telescope and microscope?

Can you fit the tangram pieces into the outline of these convex shapes?

Can you fit the tangram pieces into the outline of these rabbits?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

Can you fit the tangram pieces into the outlines of the workmen?

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

Here's a simple way to make a Tangram without any measuring or ruling lines.

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.