Challenge Level

This problem provides an opportunity to discuss sums of consecutive numbers and to use algebra to prove facts about them. This problem introduces some ideas from number theory, such as divisibility and odd/even cases, and polite numbers.

There is also the opportunity to discuss "if and only if", and whether implications work both ways. In this case it is true that all numbers with an odd factor are polite, and that all polite numbers have an odd factor but students may find it difficult to see the difference between these two statements.

Students can consider both algebraic and pictorial representations of these numbers (Polite numbers are also sometimes called "staircase numbers" as the Young Diagram representation looks like a staircase).

Ask students if they can find any more polite numbers and show that they are polite. Can they find any numbers which are not polite?

Can students find some polite numbers which can be expressed as a sum of consecutive numbers in more than one way?

Students can consider the numbers 1 to 20 and which of these are polite. They can also be encouraged to try and find numbers which are polite in more than one way, and to try and link these to the factors of the number.

Once an initial investigation into polite numbers has given the students a feel for the problem, they can start to consider sums of consecutive numbers. They might find that:

- If you add up two consecutive numbers, you get an odd number. (Can they also argue that any odd number can be written as the sum of two consecutive numbers?)
- If you add three consecutive numbers you always get a multiple of three.
- If you add four consecutive numbers you don't get a multiple of four.

A possible conjecture is that if you add up an odd number of consecutive numbers then you get a multiple of that odd particular number. Students should also consider if the converse is true, that if a number has an odd factor, it can be split up into a sum of consecutive numbers, where the number of consecutive numbers is equal to the odd factor.

This second conjecture seems to fall down when considering numbers such as $14$. We have $14 = 7 \times 2$ so it should be possible to write $14$ as a sum of $7$ consecutive numbers, centred around $2$. However if we try this we get:

$$-1+0+1+2+3+4+5=14$$

The numbers $-1$ and $0$ are not allowed to be in our sum as they are not positive, but we can cancel the $-1$ and $1$ to get $$2+3+4+5=14$$

As a slightly more tricky example, $82=2\times 41$ can be expressed as a sum of $41$ numbers centred around $2$, where quite a lot cancel to leave a sum of $4$ consecutive numbers.

Some students may find spreadsheets helpful when investigating polite numbers.

After this students could be challenged to prove that all numbers which are powers of 2 are impolite, and that all numbers with an odd factor are polite. There are various methods they can use to show that these are true and there are some hints in the Getting Started section.

What happens if you add up two consecutive numbers? What does that tell you about odd numbers?

What happens if you add up three, or five consecutive numbers?

Which numbers are impolite? What is common about them?

Does your implication work "both ways"? I.e. is it true that all polite numbers have an odd factor, and that all numbers with an odd factor are polite?

Can you find an expression for the sum of $k$ consecutive numbers starting at $a$? Can you show that the sum of a set of consecutive numbers always has an odd factor?

Can you show that if a number has a factor of the form 2k+1, then it can be split up into the sum of 2k+1 consecutive numbers?

Can use these ideas to write 44 as the sum of 11 consecutive numbers (where some will be negative)? Can you use this to write 44 as the sum of consecutive **positive** numbers?

Students could start by investigating Summing Consecutive Numbers.

Since the difference between two triangular numbers is equal to a sum of consecutive numbers, students could be challenged to use the formula for triangular numbers to prove that polite numbers must have at least one odd factor.

Which numbers can be written as the sum of an arithmetical progression with common difference $2$?