Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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?

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

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?

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.

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?

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

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

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

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

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.

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.

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

Can you explain the strategy for winning this game with any target?

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.

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. . . .

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

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?

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

Given the products of adjacent cells, can you complete this Sudoku?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Can you find any two-digit numbers that satisfy all of these statements?

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?

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. . . .

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

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

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

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.

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.

The items in the shopping basket add and multiply to give the same amount. What could their prices be?

A game that tests your understanding of remainders.

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

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

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

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

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

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.

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?

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.

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

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

What is the smallest number of answers you need to reveal in order to work out the missing headers?

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

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?