Sums are for heights of skyscrapers visible from the edges. Only those higher than the previous one(s) count toward the sums.
On the odd-numbered rows and columns, which are normal ones containing 1-9, each sum must include 9, the height of the highest skyscraper, which can be seen from each edge. However this doesn’t necessarily apply for the even-numbered rows and columns where repeated numbers are allowed; for these rows and columns the height of the highest skyscraper may be less than 9.
Since this an ORC puzzle, I’ve stated placements in odd rows/columns.
x not 0, y not 7, w is unknown. There is a total on the diagonal.
It has been specified that all totals in the even-numbered rows and columns must contain at least one of 7,8,9; also that they contain at least one of 1,2,3 although that was said not to be particularly helpful.
Three killer cages have the same total, the other killer cage has twice that total. Because the killer cages cannot be FNC or NC while R5C9 + R6C8 also cannot be AK, the three equal ones must total 4, 5, 6, 7 or 8 (cannot be 2 because three of the four killer cages cannot be [11]) and the other one must total 8, 10, 12, 14 or 16
1. Lower total in C8 = 35 can be only made up as [9.8.7.6.5/9.8.7.641/9.8.75.42] (cannot be [9.8.7.63.2/9.8.753.2.1/9.86.5.4.3/9.86.5.42.1/97.6.5.4.31/8.7.6.5.4.3.2] which require more than 9 cells to provide gaps between consecutive digits) -> R1C8 = 9, placed for R1, R3C8 = 8, placed for R3, R5C8 = 7, placed for R5, R9C8 = {125}, max R6C8 = 5, R7C8 = {126}, max R8C8 = 5, no 8 in R1C7, no 8 in R6C9
[Note. Since these are long permutations, I worked them out by looking at groups of the 2, 3 or 4 missing digits which total 10.]
2a. Skyscrapers starting with 8 or 9 must total 8 (possible for even rows/columns), 9 or 17. Since no totals are given as 17 and xy or 2w cannot be 8, 9 or 17, rows/columns starting/finishing with 8,9 must be ones with unspecified totals.
2b. Unspecified upper totals only in C467, R1C8 = 9, no 8 in R1C7 -> R1C4 = 8, placed for R1
2c. Unspecified left-hand totals only in R24 -> R24C1 = {89}, locked for C1
2d. Unspecified lower totals only in C26 -> R9C26 = {89}, locked for R9
2e. Unspecified right-hand totals only in R68 -> R68C9 = [98], placed for C9
Clean-ups after steps 1 and 2:
R1C4 = 8 -> no 7 in R1C35, no 7,8,9 in R2C35, no 7,9 in R2C4
R1C8 = 9 -> no 8,9 in R2C7, no 8 in R2C8
R3C8 = 8 -> no 7 in R2C79 + R34C9, no 7,9 in R24C8 + R3C7, no 7,8,9 in R4C7
R4C1 = {89} -> no 8,9 in R5C2
R5C8 = 7 -> no 6 in R4C79 + R5C9, no 6,8 in R4C8 + R5C7, no 6,7,8 in R6C7
R8C9 = 8 -> no 7 in R79C9
R9C2 = {89} -> no 8,9 in R8C3
R9C6 = {89} -> no 8,9 in R8C57
Naked pair {89} in R24C1 -> no 7 in R3C1, no 7,9 in R3C2
3. Lower total in C8 = 35 can only be made up as [9.8.7.6.5/9.8.7.641] (cannot be [9.8.7.5.42] which would make R7C89 = {12}, not allowed by NC) -> R7C8 = 6, placed for R7, R9C8 = {15}
3a. R89C7 = [41/.5] -> no 4,5 in R9C9
Clean-up:
R7C8 = 6 -> no 5 in R6C78 + R7C9 + R8C8, no 5,7 in R7C7, no 5,6,7 in R8C7
4a. R1C9 = 7 (hidden single in C9), placed for R1
4b. R7C7 = 8 (hidden single in C7), placed for R7 and D\, no 7,9 in R6C6, no 9 in R6C7
4c. R5C7 = 9 (hidden single in C7), placed for R5, no 8 in R5C6
4d. R9C7 = 7 (hidden single in C7), placed for R6, no 8 in R9C6
4e. R9C26 = [89], no 7 in R8C3
4f. 8 in R5 only in R5C34 -> no 7,8,9 in R46C3
4g. R5C3 = 8 (hidden single in C3), placed for R5, no 7,9 in R4C4
4h. R37C3 = {79} (hidden pair in C3)
4i. R2C2 + R3C3 = {79} (hidden pair on D\), no 8 in R2C1
4j. R24C1 = [98]
Clean-ups:
R1C9 = 7 -> no 6 in R2C89
R4C1 = 8 -> no 7,9 in R4C2
R5C3 = 8 -> no 8 in R4C2, no 7,8,9 in R6C24
R5C7 = 9 -> no 8,9 in R4C6
R7C7 = 8 -> no 7,9 in R7C6, no 7,8,9 in R8C6
R9C2 = 8 -> no 7 in R8C1, no 7,9 in R8C2
R9C7 = 7 -> no 6 in R8C6
R3C3 = {79} -> no 8 in R2C4
R7C3 = {79} -> no 8 in R8C24
Naked pair {79} in R2C2 + R3C3 -> no 6 in R2C3 + R3C2
5. Lower total on diagonal = 20 must contain 8,9 -> R8C8 + R9C9 must total 3 (cannot be [21], NC) -> R9C9 = 3, placed for R9, C9 and D\, no 2,4 in R8C8
5a. R8C8 = 1, placed for D\, no 1,2 in R7C9
5b. R7C9 = 4, placed for R7 and C9
5c. Lower total in C8 = 35 (step 3) = [9.8.7.6.5] (only remaining permutation) -> R9C8 = 5, placed for R9
5d. R3C9 = 6 (hidden single in C9), placed for R3, no 5 in R24C9
5e. R5C9 = 5 (hidden single in C9), placed for R5
Clean-ups:
R3C9 = 6 -> no 5 in R24C8
R5C9 = 5 -> no 4 in R46C8
R7C9 = 4 -> no 3 in R6C8
R8C8 = 1 -> no 2 in R8C7
R9C8 = 5 -> no 4 in R8C7
R8C7 = {13} -> no 2 in R78C6
6a. 7 in C1 only in R67C1 -> no 6 in R6C12, no 7 in R7C2
6b. 6 in C7 only in R12C7 -> no 5,6 in R1C6, no 5 in R12C7, no 5,7 in R2C6
6c. 5 in C7 only in R34C7 -> no 4,5 in R3C6, no 4 in R34C7
7. 7,9 in R3 only in R3C3456, R3C6 = {79} -> R3C456 must contain one of 7,9 -> no 8 in R4C5
7a. R6C5 = 8 (hidden single in C5) -> no 7,9 in R7C45
7b. 9 in C5 only in R34C5 -> no 9 in R3C46
7c. 7 in R7 only in R7C13 -> no 6 in R8C2
[Now it’s time to use the killer cages; don’t think I can make any further progress without using them. See note at the start about possible killer cage totals before any eliminations.]
8a. R4C4 + R5C5 total 6, 7, 8 or 10, R5C9 + R6C8 total 6 or 7 so they must be two of the cages with the same total
8b. R23C6 is the only killer cage which can have twice the total so must total 12 (cannot total 14 because 6,8 only in R2C6 and no 7 in R2C6) -> R23C6 = [93], 3 placed for R3, no 9 in R3C5, no 2 in R3C7
8c. The three equal killer cages must total 6
8d. R1C67 = 6 = {24}, locked for R1
8e. R4C4 + R5C5 = 6 = {24}, locked for D\
8f. R5C9 + R6C8 = 6 = [51]
8g. R2C7 = 6 (hidden single in C7) -> no 5 in R3C7
8h. R3C7 = 1, placed for R3 and C7
8i. R8C7 = 3, placed for C7
8j. R4C7 = 5 (hidden single in C7)
8k. R3C3 = 9 (hidden single in R3), placed for C3 and D\
8l. R2C2 = 7 -> no 6 in R1C123
8m. R1C1 = 5, placed for R1, C1 and D\
8n. R1C5 = 6 (hidden single in R1), placed for C5
8o. R6C6 = 6 -> no 5 in R7C56
8p. R7C2 = 9 (hidden single in R7)
8q. R7C3 = 7, placed for R7
8r. R6C1 = 7 (hidden single in C1)
8s. R7C4 = 5 (hidden single in R7)
Clean-ups:
R1C5 = 6 -> no 5,6 in R2C4, no 5 in R2C5
R3C6 = 3 -> no 2,3,4 in R24C5, no 2,4 in R3C5 +R4C6
R3C7 = 1 -> no 1,2 in R24C8, no 1 in R4C6
R4C7 = 5 -> no 6 in R4C6, no 4,6 in R5C6
R4C8 = 3 -> no 2 in R4C9
R6C1 = 7 -> no 6 in R5C12
R6C8 = 1 -> no 2 in R6C7
R6C7 = 4 -> no 3 in R57C6
R7C3 = 7 -> no 6 in R6C34 + R8C3, no 6,7 in R8C4
R7C4 = 5 -> no 4,5 in R68C3, no 4 in R68C4
Naked pair {13} in R1C23 -> no 1,2,3,4 in R2C3
Naked pair {24} in R4C4 + R5C5, no 1,5 in R4C5, no 1,2,3,4 in R5C4
9a. R2C3 = 5, placed for C3, no 4,5 in R3C24
9b. R2C5 = 1, placed for C5
9c. R3C1245 = [4275], 4 placed for C1, 5 placed for C5
9d. R4C9 = 1, placed for C9 -> R2C9 = 2, no 3 in R2C8
9e. R2C8 = 4 -> no 4 in R1C7
9f. R1C67 = [42]
9g. R3C4 = 7 -> no 6 in R4C3, no 7 in R3C5
9h. R7C6 = 1, placed for R7, no 2 in R7C5
9i. R7C5 = 3, placed for R7 and C5
9j. R7C1 = 2, placed for C1, no 1,3 in R8C1
9k. R8C1 = 6, R9C1 = 1, placed for R9 and C1
9l. R5C1 = 3, placed for R5, no 2,4 in R5C2
9m. R5C2 = 1, placed for R5
9n. R5C6 = 2, placed for R5
9o. R5C5 = 4, placed for C5 and D\
9p. R9C5 = 2, placed for R9 and C5
9q. R9C3 = 6 (hidden single in C3), placed for R9
9r. R4C3 = 4 (hidden single in C3)
Clean-ups:
R2C3 = 5 -> no 4 in R2C4
R2C5 = 1 -> no 2 in R2C4
R3C1 = 4 -> no 3,4,5 in R4C2
R3C2 = 2 -> no 1 in R4C2
R5C1 = 3 -> no 2 in R4C2, no 2,3,4 in R6C2
R5C2 = 1 -> no 1,2 in R6C3
R5C5 = 4 -> no 3,5 in R4C6 + R6C4
R7C1 = 2 -> no 1 in R6C2, no 1,2,3 in R8C2
R7C5 = 3 -> no 2 in R6C4, no 2,3 in R8C4, no 3,4 in R8C6
R8C1 = 6 -> no 5 in R8C2
R9C3 = 6 -> no 5 in R8C4
R9C5 = 2 -> no 1 in R8C46
10a. R6C3 = 3, placed for C3
10b. R1C3 = 1, placed for R1 and C3, no 1 in R2C4