Okay, here's my "logical" approach to this puzzle.

We have the eight cells labeled to .

There are two "central cells", and Code:

*---*
| a |
*---*---*---*
| b | c | d |
*---*---*---*
| e | f | g |
*---*---*---*
| h |
*---*

There are eight digits to install:

. . two "end" numbers and six "middle" numbers.

. .

Place a middle number in a central cell, say, 4 in cell . Code:

*---*
| x |
*---*---*---*
| x | 4 | x |
*---*---*---*
| x | x | x |
*---*---*---*
| h |
*---*

We see that its neighbors (3 and 5) cannot be placed

. . in cells

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 and

Code:

*---*
| a |
*---*---*---*
| b | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| h |
*---*

2 cannot be in cells

7 cannot be in cells Code:

*---*
| 7 |
*---*---*---*
| b | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| 2 |
*---*

3 cannot be in or . . . 3 may be in or

Pick one: say, Code:

*---*
| 7 |
*---*---*---*
| 3 | 1 | d |
*---*---*---*
| e | 8 | g |
*---*---*---*
| 2 |
*---*

6 cannot be in

Also, 6 cannot be in cell

Else 4 and 5 will be in cells and (and be adjacent).

. . Hence: . Code:

*---*
| 7 |
*---*---*---*
| 3 | 1 | d |
*---*---*---*
| e | 8 | 6 |
*---*---*---*
| 2 |
*---*

4 cannot be in cell . . .

And finally: . Code:

*---*
| 7 |
*---*---*---*
| 3 | 1 | 4 |
*---*---*---*
| 5 | 8 | 6 |
*---*---*---*
| 2 |
*---*
Solution

Disregarding rotations and reflections, there is **one** solution.