Okay, here's my "logical" approach to this puzzle.
We have the eight cells labeled $\displaystyle a$ to $\displaystyle h$.
There are two "central cells", $\displaystyle c$ and $\displaystyle h.$ Code:
*---*
| a |
*---*---*---*
| b | c | d |
*---*---*---*
| e | f | g |
*---*---*---*
| h |
*---*
There are eight digits to install:
. . two "end" numbers and six "middle" numbers.
. . $\displaystyle \underbrace{1}_{\text{end}},\underbrace{2,3,4,5,6, 7}_{\text{midde}},\underbrace{8}_{\text{end}}$
Place a middle number in a central cell, say, 4 in cell $\displaystyle c$. Code:
*---*
| x |
*---*---*---*
| x | 4 | x |
*---*---*---*
| x | x | x |
*---*---*---*
| h |
*---*
We see that its neighbors (3 and 5) cannot be placed
. . in cells $\displaystyle a,b,d,e,f,g.$
So both 3 and 5 cannot be installed.
Conclusion: the central cells must not contain a middle number.
Hence, 1 and 8 must go in cells $\displaystyle c$ and $\displaystyle f.$
Code:
*---*
| a |
*---*---*---*
| b | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| h |
*---*
2 cannot be in cells $\displaystyle a,b,d,e,g \quad\Rightarrow\quad 2 \in h.$
7 cannot be in cells $\displaystyle b,d,e,g \quad\Rightarrow\quad 7 \in a.$ Code:
*---*
| 7 |
*---*---*---*
| b | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| 2 |
*---*
3 cannot be in $\displaystyle e$ or $\displaystyle g$ . . . 3 may be in $\displaystyle b$ or $\displaystyle d.$
Pick one: say, $\displaystyle 3 \in b$ Code:
*---*
| 7 |
*---*---*---*
| 3 | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| 2 |
*---*
6 cannot be in $\displaystyle d.$
Also, 6 cannot be in cell $\displaystyle e.$
Else 4 and 5 will be in cells $\displaystyle d$ and $\displaystyle g$ (and be adjacent).
. . Hence: .$\displaystyle 6 \in g.$ Code:
*---*
| 7 |
*---*---*---*
| 3 | 1 | d |
*---*---*---*
| e | 8 | 6 |
*---*---*---*
| 2 |
*---*
4 cannot be in cell $\displaystyle e$ . . . $\displaystyle 4 \in d$
And finally: .$\displaystyle 5 \in e$ Code:
*---*
| 7 |
*---*---*---*
| 3 | 1 | 4 |
*---*---*---*
| 5 | 8 | 6 |
*---*---*---*
| 2 |
*---*
Solution
Disregarding rotations and reflections, there is one solution.