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.