Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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.

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Can you describe this route to infinity? Where will the arrows take you next?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Got It game for an adult and child. How can you play so that you know you will always win?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Delight your friends with this cunning trick! Can you explain how it works?

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.

Can you explain the strategy for winning this game with any target?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

To avoid losing think of another very well known game where the patterns of play are similar.

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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?