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 Post subject: SSS ORC UK 21D 24 25
PostPosted: Mon Apr 15, 2019 6:57 am 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 692
Location: Saudi Arabia
SSS ORC UK 21D 24 25

There are only a limited number of ORC solutions; hence I was beginning to run out of puzzles, so I though of using a killer approach to solving the Sky Scraper slums. In this sequence of Sky Scraper Sum Odd Row and Column puzzles: the killer cage totals are unknown - however they are all the same (or mathematically related). I call them unknown killers (UK), well I am British {Until Dublin gives me a passport}.

In earlier ones which I have not published the killer cages were just used to clear the slums, here they are part of the solution. In order of increasing difficulty, I think these go 24, 21D then 25. I expect Andrew to like them {clue}.

SSS: the clues look into the row or column and are the sum of the numbers seen with large numbers hiding smaller and equal ones.

ORC - so it is:
ORC: odd rows and columns are 1-9 no repeat; even ones are not (i.e. they can repeat).
NN: no nonets

For all cells:
AK: Anti-King - diagonally adjacent are not equal
FNC: Ferz Non-consecutive - diagonally adjacent are not consecutive
NC: adjacent cells are not consecutive

At least one of 7,8,9 in even rows and columns (likewise {but not very useful} there must be at least one of 1,2,3 in even rows and columns).

X not 0, y not 7 and w is unknown


SSS ORC UK D 21
Image

Note the 20 is the SkyScraper Sum along the diagonal

SSS ORC UK 24
Image

SSS ORC UK 25
Image


Last edited by HATMAN on Sun May 19, 2019 10:42 am, edited 1 time in total.

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 Post subject: Re: SSS ORC UK 21D 24 25
PostPosted: Thu May 16, 2019 5:24 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Thanks HATMAN for this latest batch of puzzles!

I'm also British but also a Canadian citizen so have both UK and Canadian passports.

HATMAN is probably right that 24 is the easiest of these three; 21D definitely seems harder after an easy start (but see my post for that puzzle), I haven't yet tried 25. 24 felt about the same level as other easier Mean ORC and SSS ORC puzzles. The final killer cage was definitely a key part of this puzzle, leading to the solution.

Here is my walkthrough for SSS ORC UK 24:
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.
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.
The four killer cages have the same total.

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 cannot be 8, 9 or 17, rows/columns starting/finishing with 8,9 must be ones with unspecified totals.

1a. Upper total in C1 = 11 must contain 9 -> R12C1 = [29] (cannot be [219], NC), placed for C1, 2 placed for R1, no 8 in R3C1 (NC)
1b. Unspecified lower totals only in C38, right-hand total in R9 = 14 must contain 9 -> R9C89 = [95], placed for R9, 5 placed for C9, all four killer cages must total 14
1c. Other unspecified lower total in C3 -> R9C3 = 8, placed for R9 and C3
1d. Left-hand total in R9 = 21 must include 8,9 with 14(3) cage at R9C1 -> R9C12 = [42], placed for R9, 4 placed for C1
1e. R34C1 = 14(2) cage = [68], placed for C1, 6 placed for R3
1f. Unspecified upper totals only in C57 -> R1C57 = {89}, locked for R1
1g. Unspecified right-hand totals only in R26 -> R26C9 = {89}, locked for C9
Clean-ups:
R1C1 = 2 -> no 1,3 in R1C2, no 1,2,3 in R2C2
R2C1 = 9 -> no 8 in R2C2, no 8,9 in R3C2
R3C1 = 6 -> no 5,6,7 in R24C2, no 5,7 in R3C2
R4C1 = 8 -> no 9 in R4C2, no 7 in R5C1, no 7,8,9 in R5C2
R9C1 = 4 -> no 3,5 in R8C1, no 3,4,5 in R8C2
R9C2 = 2 -> no 1 in R8C12, no 1,2,3 in R8C3
R9C3 = 8 -> no 7,8,9 in R8C24, no 7,9 in R8C3, no 7 in R9C4
R9C8 = 9 -> no 8,9 in R8C7, no 8 in R8C8
R9C9 = 5 -> no 4,5,6 in R8C8, no 4,6 in R8C9
R1C5 = {89} -> no 8,9 in R2C456
R1C7 = {89} -> no 8,9 in R2C78
R2C9 = {89} -> no 8,9 in R3C8
R6C9 = {89} -> no 8,9 in R57C8
R8C3 = {456} -> no 5 in R7C234
Naked pair {89} in R1C57 -> no 7 in R12C6

2a. R8C1 = 7, placed for C1, no 6 in R8C2
2b. R8C2 = 2 -> no 1,3 in R7C1
2c. R7C1 = 5, placed for R7 and C1
Clean-ups:
R7C1 = 5 -> no 4,5,6 in R6C2, no 4,6 in R7C2
R7C2 = 2 -> no 1,3 in R7C2, no 1,2,3 in R7C3
R8C1 = 7 -> no 7,8 in R7C2
R5C1 = {13} -> no 2 in R456C2
R6C1 = {13} -> no 2 in R7C2
Naked pair {13} in R56C1 -> no 1,3,4 in R5C2

3a. R7C2 = 9, placed for R7
3b. Left-hand total for R5 = 15 must contain 9 -> R5C12 = [15], placed for R5, 1 placed for C1
3c. Right-hand total for R5 = 30 must contain 9 and 8 for R5 so can only be [98742/9876] (because of NC, cannot be [98643] because no space for 7 in R5) -> R5C3 = 9, R5C5 = 8. R5C7 = 7, all placed for R5, 9 placed for C3, 8 placed for C5, 7 placed for C7, R5C9 = {26}
3d. R1C5 = 9, placed for R1 and C5
3e. R1C7 = 8, placed for C7
Clean-ups:
R1C7 = 8 -> no 7 in R12C8
R5C1 = 1 -> no 1 in R46C2
R5C2 = 5 -> no 4 in R4C2, no 4,5,6 in R46C3
R5C3 = 9 -> no 8 in R4C2, no 8,9 in R4C4 + R6C24
R4C2 = 3 -> no 2,4 in R3C2, no 2,3,4 in R3C3, no 2 in R4C3
R5C5 = 8 -> no 7 in R46C45, no 7,8,9 in R46C6
R5C7 = 7 -> no 6 in R46C67 + R5C68, no 6,7,8 in R46C8
R3C2 = {13} -> no 2 in R2C3
R5C6 = {234} -> no 3 in R46C57
R5C8 = {234} -> no 3 in R4C9
7 in R7 only in R9C56 -> no 6,7 in R8C5, no 6 in R9C56

4a. R6C3 = 2 (hidden single in C3) -> no 2,3 in R5C4
4b. R5C4 = {46} -> no 5 in R4C45
4b. 14(3) cage at R4C3 = [716] (cannot be 7{34}/[761], NC), 7 placed for C3, 6 placed for C5, no 1 in R3C3, no 6 in R5C4
4c. R3C3 = 5, place for R3 and C3
4d. Naked pair {46} in R78C3, locked for C3
4e. R5C4 = 4, placed for R5
4f. R5C9 = 6 (hidden single in R5), locked for C9
Clean-ups:
R3C3 = 5 -> no 4 in R2C2 + R3C4, no 4,5,6 in R2C4
R4C3 = 7 -> no 7,8 in R3C4
R4C4 = 1 -> no 2 in R3C4, no 1,2 in R3C5
R4C5 = 6 -> no 7 in R3C56, no 5 in R4C6
R5C4 = 4 -> no 3,5 in R6C4, no 4,5 in R6C5
R5C9 = 6 -> no 5 in R46C8, no 7 in R4C9
R6C3 = 2 -> no 3 in R6C2, no 1 in R6C4, no 1,2,3 in R7C4
R6C5 = {12} -> no 2 in R5C6, no 1,2 in R7C56

5a. R5C6 = 3, placed for R5 -> R5C8 = 2, no 2 in R6C5
5b. R6C2 = 7 -> no 6 in R7C3
5c. R7C3 = 4, placed for R7 and C3
5d. R6C5 = 1, placed for C5
5e. R8C3 = 6 -> no 6,7 in R7C4, no 6 in R9C4
5f. R7C4 = 8, placed for R7, no 7 in R7C5
5g. R7C5 = 3, placed for R7 and C5, no 2,4 in R8C5
5h. R3C5 = 4, placed for R3 and C5
5i. R89C5 = [57], placed for C5, 7 placed for R9
5j. R9C7 = 6 (hidden single in R9), placed for C7
Clean-ups:
R2C5 = 2 -> no 1,3 in R123C46
R3C5 = 4 -> no 4,5 in R2C6, no 3,4 in R4C6
R5C6 = 3 -> no 2 in R4C67, no 2,4 in R6C67
R4C6 = 1 -> no 2 in R3C6, no 1,2 in R3C7
R5C8 = 2 -> no 1 in R46C7, no 1,3 in R46C8, no 1,2 in R4C9
R4C9 = 4 -> no 3 in R3C89
R6C5 = 1 -> no 2 in R6C4
R7C3 = 4 -> no 4 in R6C4, no 3,4,5 in R8C4
R7C5 = 3 -> no 3 in R6C6, no 2 in R8C4, no 2,3,4 in R8C6
R8C5 = 5 -> no 6 in R7C6 + R8C46
R9C5 = 7 -> no 7,8 in R8C6
R9C7 = 6 -> no 5 in R8C67, no 7 in R8C8

6a. R3C4 = 9, placed for R3
6b. R3C7 = 3, placed for R3 and C7 -> R3C2 = 1, placed for R3, no 1 in R2C3, no 4 in R4C7
6c. R2C3 = 3, placed for C3 -> R1C3 = 1, placed for R1
6d. R3C6 = 8 -> no 9 in R4C7
6e. R4C7 = 5, placed for C7
6f. R4C9 = 4, placed for C9
Clean-ups:
R2C3 = 3 -> no 4 in R1C24, no 2 in R2C4
R2C4 = 7 -> no 6 in R1C4
R3C7 = 3 -> no 2 in R2C6 + R3C8, no 2,4 in R2C7 + R4C8, no 2,3,4 in R2C8
R2C6 = 6 -> no 5 in R1C6
R3C8 = 7 -> no 6 in R2C8, no 8 in R2C9

7a. R278C7 = [124], 2 placed for R7
7b. R3C89 = [72], 2 placed for C9
7c. R7C689 = [761], 1 placed for C9
Clean-ups:
R3C9 = 2 -> no 1 in R2C8
R7C7 = 2 -> no 1 in R68C6, no 2 in R6C8, no 1,2,3 in R8C8
R7C8 = 6 -> no 7 in R8C9
R8C7 = 4 -> no 3 in R9C6

8a. R18C9 = [73], 7 placed for R1
8b. R26C9 = [98]
Clean-up:
R6C9 = 8 -> no 9 in R6C8

and the rest is naked singles.

Solution:
2 6 1 5 9 4 8 3 7
9 9 3 7 2 6 1 5 9
6 1 5 9 4 8 3 7 2
8 3 7 1 6 1 5 9 4
1 5 9 4 8 3 7 2 6
3 7 2 6 1 5 9 4 8
5 9 4 8 3 7 2 6 1
7 2 6 1 5 9 4 9 3
4 2 8 3 7 1 6 9 5


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 Post subject: Re: SSS ORC UK 21D 24 25
PostPosted: Tue May 21, 2019 1:07 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Thanks HATMAN for adding the note about 20. I was struggling with this puzzle after an easy start, then checked my worksheet diagram again and found that I hadn't included the 20, so I checked with HATMAN what it is. After that I reworked my solving path from the start of step 5.

Here is my walkthrough for SSS ORC UK 21D:
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

Solution:
5 3 1 8 6 4 2 9 7
9 7 5 3 1 9 6 4 2
4 2 9 7 5 3 1 8 6
8 6 4 2 9 7 5 3 1
3 1 8 6 4 2 9 7 5
7 5 3 1 8 6 4 1 9
2 9 7 5 3 1 8 6 4
6 4 2 9 7 5 3 1 8
1 8 6 4 2 9 7 5 3

Interesting to note that the digits of the solution total 405. I think that's the first time I've seen that for an ORC puzzle.


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 Post subject: Re: SSS ORC UK 21D 24 25
PostPosted: Thu May 23, 2019 4:35 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
I sometimes find that Skyscraper sum puzzles require more care than usual in avoiding making invalid assumptions. This was one such puzzle. Step 2 and the start of step 3 took some thought; after that it was at about the same level as the other two in this batch.

Here is my walkthrough for SSS ORC UK 25:
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.
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.
The six killer cages have the same total.

1a. 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 cannot be 8, 9 or 17, rows/columns starting/finishing with 8,9 must be ones with unspecified totals.
1b. Unspecified left-hand totals only in R13 -> R13C1 = {89}, locked for C1
1c. Unspecified right-hand totals only in R57 -> R57C9 = {89}, locked for C9
1d. Unspecified upper totals only in C169, no 8,9 in R1C9 -> R1C16 = {89}, locked for R1
1e. Unspecified lower totals only in C48 -> R9C48 = {89}, locked for R9
Clean-ups:
R1C6 = {89} -> no 8,9 in R2C57
R3C1 = {89} -> no 8,9 in R234C2
R5C9 = {89} -> no 8,9 in R456C8
R7C9 = {89} -> no 8,9 in R78C8
R9C4 = {89} -> no 8,9 in R8C35
R9C8 = {89} -> no 8,9 in R8C7
Naked pair {89} in R13C1 -> no 7 in R2C12
Naked pair {89} in R57C9 -> no 7 in R6C89

2a. Upper total in C5 = 15 must contain 9 = [159/249/6.9] with possibly additional lower numbers hidden after the 4, 5 or 6 (cannot be [1239], NC) -> R1C5 = {126}
2b. Lower total 30 in C5 = 30 must contain 8,9 with 7 either included in the total or hidden after the 9 = [9.8.7.6/9.8.7 with two smaller numbers] (cannot be 97.8 with three smaller numbers totalling 13 because these could only be 6.52 or 64.3 which would require 8 cells for this total) -> R3C5 = 9, placed for R3 and C5, R5C5 = 8, placed for R5 and C5, R7C5 = 7, placed for R7 and C5
2c. R3C1 = 8, placed for R3 and C1 -> R1C1 = 9, placed for R1
2d. R5C9 = 9, placed for R5 and C9
2e. R7C9 = 8, placed for R7
2f. R123C5 = [159/249/6.9], no 6 in R2C5
Clean-ups:
R1C6 = 8 -> no 7 in R12C7, no 7,9 in R2C6
R3C1 = 8 -> no 7 in R3C2 + R4C12
R3C5 = 9 -> no 8,9 in R2C4 + R4C46, no 8 in R2C6
R5C5 = 8 -> no 7 in R45C46, no 7,8,9 in R6C46
R7C5 = 7 -> no 6 in R6C456 + R7C46 + R8C5, no 6,7,8 in R8C46
R7C9 = 8 -> no 7 in R8C89

3a. Killer cage at R1C5 totals at least 15 -> all six killer cages must total at least 15
3b. Killer cage at R7C5 must total at least 15 with visible numbers totalling 13 to make lower total of 30 for C5 -> R9C5 = 6, placed for R9 and C5
3c. R123C5 = [159/249] -> killer cage at R1C5 totals 15 -> all killer cages must total 15
3d. R79C5 = [76] -> R8C5 = 2 (cage total), 2 placed for C5
3e. R123C5 = [159], 1,5 placed for C5, 1 placed for R1
Clean-ups:
R1C5 = 1 -> no 2 in R1C4, no 1,2 in R2C46
R2C5 = 5 -> no 4,5,6 in R1C4 + R3C46, no 4,6 in R2C46
R8C5 = 2 -> no 1,2,3 in R7C46 + R9C6, no 1,3 in R8C46
R9C5 = 6 -> no 5 in R8C46, no 5,7 in R9C6
R2C6 = {35} -> no 4 in R123C7
R4C5 = {34} -> no 3 in R3C46, no 3,4 in R5C46
R6C5 = {34} -> no 4 in R7C46
Naked pair {34} in R46C5 -> no 2,5 in R5C46
Naked pair {59} in R7C46 -> no 4 in R6C5

4a. R6C5 = 3, placed for C5
4b. R9C6 = 4, placed for R9
4c. Naked pair {16} in R5C46, locked for R5
4d. Naked pair {59} in R7C46, locked for R7
Clean-ups:
R4C5 = 4 -> no 3,5 in R4C46
R6C5 = 3 -> no 2,4 in R6C46
R9C6 = 4 -> no 3,4,5 in R8C7, no 3,5 in R9C7

5a. R46C7 = {89} (hidden pair in C7)
5b. Killer cage at R4C5 = 15, R4C5 = 4 -> R4C67 = 11 = [29], 9 placed for C7, no 1 in R5C6
5c. R6C7 = 8, no 9 in R7C6
5d. R5C46 = [16]
5e. R7C46 = [95]
Clean-ups:
R4C6 = 2 -> no 1 in R3C6, no 1,2,3 in R3C7, no 2,3 in R5C7
R5C4 = 1 -> no 1,2 in R46C3, no 2 in R4C4 + R5C3
R5C6 = 6 -> no 5,7 in R5C7, no 5 in R6C6
R6C7 = 8 -> no 7 in R5C8
R7C4 = 9 -> no 8,9 in R6C3
R7C6 = 5 -> no 4,6 in R7C7, no 4 in R8C6, no 6 in R8C7

6a. R5C7 = 4, placed for R5, no 3,5 in R5C8
6b. R5C8 = 2, placed for R5
Clean-ups:
R5C7 = 4 -> no 3,4,5 in R46C8, no 3 in R6C6
R5C8 = 2 -> no 1 in R46C8, no 1,2,3 in R46C9
Naked triple {357} in R5C123 -> no 4,6 in R46C2

7a. R6C6 = 1 -> no 1,2 in R7C7
7b. R7C7 = 3, placed for R7 and C7, no 2 in R6C8
7c. R6C8 = 6, no 5 in R6C9
7d. Right-hand total in R6 = 2w, max R6C789 = [864] = 18 (R6C789 = [866] would only contribute 14 toward skyscraper sum) -> total must also contain 9 -> R6C2 = 9
Clean-up:
R7C7 = 3 -> no 2,4 in R7C8, no 2 in R8C67, no 2,3,4 in R8C8

8a. 6 in C7 only in R123C7 -> no 5 in R2C67
8b. 5 in C7 only in R13C7 -> no 6 in R2C7
8c. R2C6 = 3 -> no 2 in R12C7 + R3C6
8d. R3C6 = 7, placed for R3, no 6 in R3C7
8e. R3C7 = 5, placed for R3 and C7
8f. R1C7 = 6, placed for R1
8g. R2C7 = 1, placed for C7
8h. R8C7 = 7, placed for C7, no 6 in R7C8, no 8 in R9C8
8i. R9C7 = 2, placed for R9
8j. R9C8 = 9, placed for R9
8k. R7C8 = 1, placed for R7
Clean-ups:
R1C7 = 6 -> no 5,7 in R1C8, no 5,6,7 in R2C8
R2C7 = 1 -> no 2 in R12C8, no 1,2 in R3C8
R3C7 = 5 -> no 4 in R2C8, no 4,6 in R3C8
R7C8 = 1 -> no 1,2 in R8C9
R8C7 = 7 -> no 6 in R8C8
R9C4 = 8 -> no 7 in R89C3, no 9 in R8C4
R9C7 = 2 -> no 1 in R8C8
R8C8 = 5 -> no 4,6 in R8C9, no 5 in R9C9

9a. R3C8 = 3, placed for R3, no 2,4 in R3C9, no 2 in R4C8
9b. R4C8 = {67} -> no 6 in R3C9
9c. R3C9 = 1, placed for R3 and C9
9d. R3C4 = 2, placed for R3, no 3 in R2C4
9e. R2C4 = {57} -> no 6 in R3C3
9f. R3C3 = 4, placed for R3 and C3
Clean-ups:
R3C2 = 6 -> no 5,6 in R24C1, no 5 in R24C2, no 5,6,7 in R24C3
R3C3 = 4 -> no 3,4 in R2C2, no 3 in R2C3 + R4C23, no 5 in R2C4, no 4 in R4C4
R2C4 = 7 -> no 7 in R1C3, no 8 in R2C3
R3C4 = 2 -> no 1,2 in R2C3, no 1 in R4C4
R3C8 = 3 -> no 2,3,4 in R2C9, no 4 in R4C9
R4C4 = 6 -> no 5,7 in R5C3

10a. R2C3 = 9, placed for C3
10b. R4C3 = 8, no 7 in R5C2
10c. R5C3 = 3, placed for R5 and C3
10d. R5C2 = 5, placed for R5, no 5,6 in R6C3
10e. R5C1 = 7, placed for C1
10f. R6C3 = 7, no 6 in R7C23
10g. R7C3 = 2, placed for R7 and C3
10h. R1C3 = 5, placed for R1 and C3
10i. R7C2 = 4, placed for R7
10j. R7C1 = 6, placed for C1
10k. R9C3 = 1, placed for R9 and C3
10l. R8C3 = 6, no 5,7 in R9C2
10m. R9C2 = 3, placed for R9
10n. R9C1 = 5, placed for C1
10o. R9C9 = 7, placed for C9
10p. Naked pair {56} in R24C9, locked for C9
10q. R6C9 = 4, placed for C9
10r. R8C9 = 3, placed for C9
10s. R1C9 = 2, placed for R1, no 3 in R1C8
10t. R1C8 = 4, placed for R1, no 5 in R2C9
10u. R2C9 = 6, placed for C9
Clean-ups:
R4C9 = 5 -> no 6 in R4C8
R5C2 = 5 -> no 4 in R46C1
R5C3 = 3 -> no 2 in R4C2, no 3 in R6C4
R4C2 = 1 -> no 2 in R4C1
R7C2 = 4 -> no 3 in R6C1, no 3,4 in R8C1
R7C3 = 2 -> no 1 in R6C4, no 2 in R8C4
R9C2 = 3 -> no 2 in R8C1

10a. R2468C1 = [4321], no 3 in R1C2
10b. R1C24 = [73]
10c. Killer cage at R2C1 = 15, R2C13 = [49] = 13 -> R2C2 = 2
10d. Killer cage at R2C7 = 15, R2C79 = [16] = 7 -> R2C8 = 8
10e. Killer cage at R8C1 = 15, R8C13 = [16] = 7 -> R8C2 = 8

Solution:
9 7 5 3 1 8 6 4 2
4 2 9 7 5 3 1 8 6
8 6 4 2 9 7 5 3 1
3 1 8 6 4 2 9 7 5
7 5 3 1 8 6 4 2 9
2 9 7 5 3 1 8 6 4
6 4 2 9 7 5 3 1 8
1 8 6 4 2 9 7 5 3
5 3 1 8 6 4 2 9 7

I commented that the solution for 21D was unusual for an ORC puzzle in that the digits totalled 405.
This one is even more unusual; as well as totalling 405, all the even rows and columns contain 1-9. After I’d found that the totals for the killer cages were 15, I’d thought it quite likely that at least one of them might be [555].


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