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. . . .

Make some loops out of regular hexagons. What rules can you discover?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

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. . . .

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?

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. . . .

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

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. . . .

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.

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.

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.

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

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.

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?

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

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

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?

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

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.

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

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!

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

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 winner is the player to take the last counter.

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

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

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.

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. . . .

Explore the effect of combining enlargements.

Explore the effect of reflecting in two intersecting mirror lines.

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.

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

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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?

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?

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

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.

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

What's the largest volume of box you can make from a square of paper?

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

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

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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