This was an optional problem given to me which I just don't know how to approach. The full question states: "An arrangement of numbers has 'different differences' when the differences between all the neighbouring numbers are different-
e.g. '1423' the differences between these neighbours (in order) are 3, 2, 1
Given numbers 1 - 6 and the 3rd number being 3 (can't use it again), what is the sum of the last three numbers?"
basically in the format: _ _ 3 _ _ _
I initially looked at the example and tried to find any clues, e.g descending differences (/maybe ascending differences) and relationships between the actual neighbouring number, quickly found that this wasn't the right way to do this. Then just decided to play around and try some numbers that I thought would work, didn't see a new perspective or similar and kinda just pondered on until the next problem. However, I still really want any insight/a hint or something to lead me in the right direction to be able to solve this. Any help is welcome, Thanks.
[This solution uses the fact, mentioned in a comment, that the sum of the last three numbers is one of $12, 13, 14, 15$ or $16$.]
The possible differences are $1,2,3,4,5$ and each must occur. The only way to produce a difference of $5$ is for $1$ and $6$ to be adjacent. The difference of $4$ requires either $1$ adjacent to $5$ or $2$ adjacent to $6$, neither of which is possible if $1$ and $6$ are placed before the $3$. Therefore they are after, and the maximum possible sum of the last three numbers is $1+6+5=12$. Since the other choices are even larger, $12$ must be correct.