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

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

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

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

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

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.

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?

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?

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?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

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

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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?

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. . . .

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

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?

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?

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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. . . .

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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.

An investigation that gives you the opportunity to make and justify predictions.

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

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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

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. . . .

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

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.

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

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

What happens when you round these three-digit numbers to the nearest 100?

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

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?

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

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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

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.

Are these statements always true, sometimes true or never true?