This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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.

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

A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

What is the shape of wrapping paper that you would need to completely wrap this model?

Can you find a way of representing these arrangements of balls?

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Make a cube out of straws and have a go at this practical challenge.

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

Exploring and predicting folding, cutting and punching holes and making spirals.

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

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Can you fit the tangram pieces into the outline of the child walking home from school?

Can you fit the tangram pieces into the outline of Little Ming playing the board game?

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

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Exchange the positions of the two sets of counters in the least possible number of moves

Can you fit the tangram pieces into the outline of Little Fung at the table?

Can you fit the tangram pieces into the outline of this brazier for roasting chestnuts?

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

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

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 outlines of these people?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

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

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

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 rabbits?

Can you work out what is wrong with the cogs on a UK 2 pound coin?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

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?

Can you fit the tangram pieces into the outline of Granma T?

Can you fit the tangram pieces into the outline of Little Ming?

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

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

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?