Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Can you find what the last two digits of the number $4^{1999}$ are?

This problem is designed to help children to learn, and to use, the two and three times tables.

The number 10112359550561797752808988764044943820224719 is called a 'slippy number' because, when the last digit 9 is moved to the front, the new number produced is the slippy number multiplied by 9.

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

Find a great variety of ways of asking questions which make 8.

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?

Chandrika was practising a long distance run. Can you work out how long the race was from the information?

Grandma found her pie balanced on the scale with two weights and a quarter of a pie. So how heavy was each pie?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

Where can you draw a line on a clock face so that the numbers on both sides have the same total?

How would you count the number of fingers in these pictures?

Use the information to work out how many gifts there are in each pile.

All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

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.

Number problems at primary level that require careful consideration.

When I type a sequence of letters my calculator gives the product of all the numbers in the corresponding memories. What numbers should I store so that when I type 'ONE' it returns 1, and when I type. . . .

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have?

This number has 903 digits. What is the sum of all 903 digits?

Number problems at primary level that may require determination.

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

There are four equal weights on one side of the scale and an apple on the other side. What can you say that is true about the apple and the weights from the picture?

Can you score 100 by throwing rings on this board? Is there more than way to do it?

These pictures and answers leave the viewer with the problem "What is the Question". Can you give the question and how the answer follows?

Put operations signs between the numbers 3 4 5 6 to make the highest possible number and lowest possible number.

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?