This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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?

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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. . . .

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?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?

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?

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

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,. . . .

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

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

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 find ways of joining cubes together so that 28 faces are visible?

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?

A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.

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.

Watch this animation. What do you see? Can you explain why this happens?

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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.

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?

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

Can you find a way of counting the spheres in these arrangements?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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?

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

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

Which of the following cubes can be made from these nets?

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

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

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?

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

A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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

How many different symmetrical shapes can you make by shading triangles or squares?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

When dice land edge-up, we usually roll again. But what if we didn't...?