What is the length of the longest side?

Question

The side lengths of a nonagon are consecutive integers.

The perimeter is 2016.

What is the length of the longest side?

The answer is 228.

CONTEXT: I did the UK Junior Kangaroo Challenge in 2016 and was given a keychain with this question on it. I recently found it and solved it pretty quickly but not with a particularly elegant method. I am posting this as I am interested to see how others would solve it.

Answer 1

Let the shortest side be $a$ units. Then the side lengths are $a$, $a+1$, ..., $a+8$.

Their sum is $9a+(1+2+3+...+8)=9a+36$

So $9a+36=2016$. Subtract $36$ and divide by $9$ to get $a=220$ as the shortest side.

Then the longest side is $220+8=228$.

Answer 2

If the shortest side is $x$, then the next one is $x+1$ and so on, until $x+8$. The perimeter id $$P=x+(x+1)+...+(x+8)=9x+\frac{8\cdot 9}2=9x+36=2016$$ This yields $x=220$ and the longest side is $220+8=228$

Answer 3

The average side length is $2016/9=224$, which is an integer. Oh, nice. The other eight sides are the four previous and four next integers. So the longest side is $228$.