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Congratulations to Josie for indeed solving this one (please refer to her post for the solution). For anyone still wondering how to start working this problem themselves, here are some starter tips:
The cell immediately adjacent to any outer 1, as ed indicated, will of course be a 9. And row 7 is obviously 1 through 9 in order from left to right. Those values and the 3 in the middle cell can then be eliminated as remaining possibilities for their respective row, column and box in typical Sudoku fashion as a 'first cut'.
Next, we know column 4 has 8 buildings visible from the bottom up, so one building must be 'out of order' somewhere in this column (i.e. it'll be hidden by a taller building). Since we already know from the above that the 4 for this column resides just three rows up from the bottom (on row 7), the 'out of order' building could then really only be one of 1, 2 or 3. As well, the 5 and 6 for this column could only reside in the central block while the 8 could only reside in the top block, so they too can be eliminated from the balance of their respective blocks.
Finally, since an outer 1 is adjacent to a 9, it's a fair bet too that the closest any other 9 can be to an outer row, column or corner will be the indicated outer 'Skyscraper' number of spaces into the grid; that is, if '4' buildings are indicated to be visible, the very closest the 9 could be must be at least 4 cells into the grid. Similarly, the closest 8 could only be the indicated number minus 1 cells in, the closest 7 would be the number minus 2 cells in, the closest 6 would be the number minus 3 cells in, etc. Work this pattern around all four sides, and you'll have numerous more eliminations to make, further reducing the remaining cell possibilities.
From there, it's really just a matter of determining through brute force what remaining available cell numbers still satisfy the indicated outer 'visible' Skyscraper numbers, eliminating possibilites in Sudoku fashion as the remaining building locations become known/fixed.
And yes, the diagonal numbers, while perhaps not critical, were simply provided to be of assistance, as Josie mentioned did prove helpful.
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