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

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. . . .

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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

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.

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

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?

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.

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

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

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

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. . . .

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

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?

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

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

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

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?

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?

An account of some magic squares and their properties and and how to construct them for yourself.

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. . . .

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

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.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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 see how to build a harmonic triangle? Can you work out the next two rows?

What is the total number of squares that can be made on a 5 by 5 geoboard?

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?

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?

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

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

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.

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.

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do 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.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

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

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?

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

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