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. . . .

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

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?

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. . . .

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.

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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. . . .

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.

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

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. . . .

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

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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?

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

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 NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Try out this number trick. What happens with different starting numbers? What do you notice?

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

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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

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?

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

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

Find the sum of all three-digit numbers each of whose digits is odd.

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. . . .

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?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

This task follows on from Build it Up and takes the ideas into three dimensions!

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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