Challenge Level

A *polite number* is a number which can be written as the sum of two or more consecutive **positive** integers.

For example, $21=10+11$ is polite as it is the sum of 2 consecutive positive integers, and $10=1+2+3+4$ is polite as it is the sum of four consecutive positive integers.

Here are some questions to think about:

- Is 63 a polite number?
- If you add up three consecutive integers, what sort of answers do you get?
- Are all multiples of 5 polite?

An impolite number is one that cannot be written as a sum of two or more consecutive **positive **integers.

- Can you find an impolite number?
- Can an impolite number be odd?

Can you find a rule for identifying impolite numbers?

You could try looking at the numbers between 1 and 20 and seeing which of these is a polite number to help you find a rule.

You could consider what happens if you add 2 consecutive numbers, 3 consecutive numbers, etc.

You could consider what happens if you add 2 consecutive numbers, 3 consecutive numbers, etc.

Can you explain why your rule works?

There are two stages to explaining why the rule works, which can be tackled in either order.

To show that any number which can be expressed as a power of 2 is impolite, can you show that the sum of a set of consecutive numbers always has an odd factor?

To show that all other numbers are polite, can you show that if a number has an odd factor, it can be split up into a sum of consecutive numbers?

To show that any number which can be expressed as a power of 2 is impolite, can you show that the sum of a set of consecutive numbers always has an odd factor?

To show that all other numbers are polite, can you show that if a number has an odd factor, it can be split up into a sum of consecutive numbers?

When you have explored this problem, you might like to take a look at the different proofs offered in the problem Impossible Sums.

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical 'content'.

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*