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

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

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

How many two-digit primes are there between 10 and 99 which are also prime when reversed?

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

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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

Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?

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...

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

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

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?

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

Can you arrange the red and blue cards so that the rules are all followed?

Can you describe this route to infinity? Where will the arrows take you next?

We are given two factors of a number with eight factors. Can you work out the other factors of the number?

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

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?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Can you find a fraction with the following properties?

Can you find the next time that the 29th of February will fall on a Monday?

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 you find the factors?

Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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

If you take a number and add its square to its cube, how often will you get a perfect square?