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

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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?

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?

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?

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

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

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

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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

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.

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

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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.

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.

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?

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

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. . . .

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

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

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

It starts quite simple but great opportunities for number discoveries and patterns!

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

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

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.

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.

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?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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

Can you find sets of sloping lines that enclose a square?

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.

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

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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

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

Can all unit fractions be written as the sum of two unit fractions?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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.

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

It would be nice to have a strategy for disentangling any tangled ropes...