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. . . .

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

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.

Can you work out how many of each kind of pencil this student bought?

How many six digit numbers are there which DO NOT contain a 5?

Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

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.

Ranging from kindergarten mathematics to the fringe of research this informal article paints the big picture of number in a non technical way suitable for primary teachers and older students.

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

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

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

What would you do if your teacher asked you add all the numbers from 1 to 100? Find out how Carl Gauss responded when he was asked to do just that.

A combination mechanism for a safe comprises thirty-two tumblers numbered from one to thirty-two in such a way that the numbers in each wheel total 132... Could you open the safe?

Can you make a hypothesis to explain these ancient numbers?

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

This investigation is about happy numbers in the World of the Octopus where all numbers are written in base 8 ... Find all the fixed points and cycles for the happy number sequences in base 8.

There are some water lilies in a lake. The area that they cover doubles in size every day. After 17 days the whole lake is covered. How long did it take them to cover half the lake?

In how many ways can a pound (value 100 pence) be changed into some combination of 1, 2, 5, 10, 20 and 50 pence coins?

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

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 article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?

Problem one was solved by 70% of the pupils. Problem 2 was solved by 60% of them. Every pupil solved at least one of the problems. Nine pupils solved both problems. How many pupils took the exam?

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. . . .

This challenge is to make up YOUR OWN alphanumeric. Each letter represents a digit and where the same letter appears more than once it must represent the same digit each time.

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

Whenever two chameleons of different colours meet they change colour to the third colour. Describe the shortest sequence of meetings in which all the chameleons change to green if you start with 12. . . .

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Can you find any perfect numbers? Read this article to find out more...

Is there an efficient way to work out how many factors a large number has?

This is a game in which your counters move in a spiral round the snail's shell. It is about understanding tens and units.

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

This task depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together.

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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" .

Guess the Dominoes for child and adult. Work out which domino your partner has chosen by asking good questions.

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

I am thinking of three sets of numbers less than 101. Can you find all the numbers in each set from these clues?

A case is found with a combination lock. There is one clue about the number needed to open the case. Can you find the number and open the case?