Answer
Reasoning
The first observation is that all of the possible gaps must exist: 1, 2, 3, 4, 5, 6, 7, and let's call these Gap1, …, Gap7.
Looking at Gap7, this can only happen when:
[Gap7a: 1 is next to 8]
Looking at Gap6, this can happen when:
[Gap6a: 1 is next to 7]
this is not allowed because 1 and 7 are given squares that are not next to each other.
[Gap6b: 2 is next to 8]
Looking at Gap5, this can happen when:
[Gap5a: 1 is next to 6]
[Gap5b: 2 is next to 7]
this is not allowed because [Gap6b: 2 is next to 8], which would mean that we can't also have [Gap7a: 1 is next to 8].
[Gap5c: 3 is next to 8]
this is not allowed because we couldn't then also have: [Gap7a: 1 is next to 8] and [Gap6b: 2 is next to 8].
We can't have the 8 above the 1, as there is then no place for [Gap6b: 2 is next to 8].
So, we have two possible places for the 8, to the left of the 1, or to the right of the 1.
8 to the left of 1
What can we place to the right of 1?
3 – no, because the gap between the 1 and 3, and the gap between 3 and 5, are both 2.
4 – no, because the gap between 4 and 5 is 1, which we already have.
6 – no, because the gap between 5 and 6 is 1, which we already have.
So the 8 can't go to the left of 1.
8 to the right of 1
We know [Gap5a: 1 is next to 6].
Let's try 6 to the left of 1. No matter where we place the 3 and 4 we end up with duplicate gaps. So this doesn't work.
So, the 6 must go above 1, but the 4 can't go to the left of 1 because of duplicate gaps. Therefore, this doesn't work either.
Which means that the final answer is:
