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. . . .

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?

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?

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.

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

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

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?

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

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.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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.

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

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

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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?

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

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?

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

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 convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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.

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

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

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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

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

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. . . .

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 largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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?

Have you seen this way of doing multiplication ?

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

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

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

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