Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

Can you find a way to identify times tables after they have been shifted up?

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.

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

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?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

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.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

A game that tests your understanding of remainders.

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

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

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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 clues for this Sudoku are the product of the numbers in adjacent squares.

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

Given the products of diagonally opposite cells - can you complete this Sudoku?

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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.

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

A game in which players take it in turns to choose a number. Can you block your opponent?

Have you seen this way of doing multiplication ?

A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?

Find the highest power of 11 that will divide into 1000! exactly.

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

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

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

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

Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

Find the frequency distribution for ordinary English, and use it to help you crack the code.

Substitution and Transposition all in one! How fiendish can these codes get?

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

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

Factors and Multiples game for an adult and child. How can you make sure you win this game?

Can you work out what size grid you need to read our secret message?

Follow this recipe for sieving numbers and see what interesting patterns emerge.