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. . . .

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. . . .

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.

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

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?

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. . . .

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?

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?

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?

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

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

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?

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

This challenge asks you to imagine a snake coiling on itself.

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?

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?

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. . . .

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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. . . .

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

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?

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

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

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

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

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.

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?

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?

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.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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

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

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

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

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?

Here are two kinds of spirals for you to explore. What do you notice?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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

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

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

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

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?