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 Post subject: SSS ORC 11
PostPosted: Wed Sep 20, 2017 10:06 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 693
Location: Saudi Arabia
SSS ORC 11

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-concecutive - diagonally adjacent are not consecutive
NC: adjacent cells are not consecutive

There are six three-in-a-rows.

An easy start, but it gets very difficult.
I would be grateful for your views on difficulty.

At least one of 7,8,9 in even rows and columns
X not 0, y not 7


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 Post subject: Re: SSS ORC 11
PostPosted: Sun Mar 04, 2018 12:51 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 another interesting and challenging puzzle.

Here is my walkthrough for SSS ORC 11:
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.
Six of the even numbered rows/columns have three identical numbers in a row.
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.

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 upper totals only in C6 and C8, right-hand total in R1 = 23 with only one other digit -> R1C68 = [98], R1C9 = 6 (cage sum), all placed for R1, 6 also placed for C9, no 7 in R1C7, no 8,9 in R2C5, no 8 in R2C6, no 7,8,9 in R2C7, no 5,6,7,9 in R2C8, no 5,7,8,9 in R2C9 (AK, FNC, NC)
1c. Unspecified left-hand totals only in R4 and R7 -> R47C1 = {89}, locked for C1, no 8,9 in R35678C2 (AK,FNC, NC)
1d. Unspecified lower totals only in C6, C7 and C9, 8,9 cannot both be in R9C67 because of NC -> R9C9 = {89}, one of R9C67 must contain one of 8,9, no 8,9 in R8C789 (AK, FNC, NC), no 8,9 in R9C23458
1e. Unspecified right-hand totals only in R45689, R9C9 = {89}, no 8,9 in R8C9 -> one of R456C9 must contain one of 8,9, upper total in C9 = 23 with 6 in R1C1 must be {689} -> R9C9 = 9, placed for R9 and C9 -> 8 in one of R456C9, no 8 in R37C9
1f. 8 in C9 only in R456C9 -> no 7,8,9 in R5C8, no 7 in R5C9 (AK, FNC,NC)
1g. 8 in R9 only in R9C67 -> no 7,9 in R8C6, no 7 in R8C7 (FNC)
1h. 23 total = {689} cannot include 7 -> no 7 in R34C9
1i. 7 in C9 was only in R678C9 -> no 6,7,8 in R7C8 (FNC,NC)
1j. Left-hand total in R1 = 18 must contain 7,9 = {279} -> R1C12 = [27] (there cannot be a hidden digit in R1C2, NC), placed for R1, 2 also placed for C1, no 1,3,6,7 in R2C1, no 1,2,3,6,8 in R2C2, no 6,7,8 in R2C3 (AK,FNC, NC)
1k. Upper total in C1 = 21 containing 2 cannot contain both of 8,9 -> R1234C1 = [2469], 4,6,9 place for C1, 6 also placed for R3, R7C1 = 8, placed for R7, no 5,7 in R2C2, no 3,4,5,7 in R3C2, no 5,6,7,8 in R4C2, no 7 in R678C2 (AK,FNC, NC)
1l. R3C2 = {12} -> no 1,2 in R234C3 (AK,FNC, NC)

2a. Left-hand total in R5 = 27 must contain at least four digits -> no 7 in R5C1
2b. Upper total in C5 = 26 must contain 9 -> remaining digits must total 17 so must start with one of 1,3,4, maximum value for second digit = 6 -> no 5 in R1C5, no 7 in R2C5
2c. Lower total in C5 = 18 must contain 9 -> remaining digits must total 9 so must start with one of 1,2,3,4 and have least one more digit -> no 5,6,7 in R9C5, no 9 in R8C5
2d. Upper total in C4 = 10 must contain one of 7,8,9 = {19/37}, possibly with repetition -> R1C4 = {13}, R2C4 = {1379}, no 2 in R2C5 (FNC)
2e. R2C4 = {1379} -> either R2C4 = {13} when R3C4 = {1379} or R2C4 = {79} -> no 8 in R3C4 (NC)
2f. Right-hand total in R7 = 16 must contain 9 = {259/349/79} (cannot be {1249} because of NC) = [9n52/9413/9n7] -> R7C9 = {237}
2g. Left-hand total in R6 = 12 must contain one of 7,8,9 = {138/147/39} (cannot be {129} NC) -> R6C1 = {13}, R6C2 = {134}, no 5,6 in R6C3
2h. R6C1 = {13} -> no 2 in R57C2 (FNC)
2i. Left-hand total in R8 = 13 must start with one of 1,5 (cannot start with 3 because row must contain one of 7,8,9) -> no 3,7 in R8C1
2j. R9C1 = 7 (hidden single in C1), placed for R9, no 6 in R89C2 (FNC, NC)
2k. 3 in C1 only in R56C1 -> no 3,4 in R5C2, no 4 in R6C2 (AK,FNC, NC)
2l. R6C2 = {13} -> no 2 in R567C3 (FNC,NC)
2m. 2 in C3 only R89C3 -> no 1,3 in R8C24, no 1 in R89C3, no 1,2,3 in R9C24 (AK,FNC,NC)
2n. R9C2 = {45} -> no 5 in R8C1, no 4,5 in R89C3 (AK,FNC,NC)
2o. R568C1 = [531], 5 placed for R5, no 4 in R4C2, no 6 in R5C2, no 1,3,4 in R7C2, no 2 in R8C2 (AK,FNC,NC)
2p. R7C2 = {56} -> no 5,6 in R7C3 (NC)
2q. R8C2 = {45} -> no 4 in R7C3 (AK,FNC)
2r. Left-hand total in R5 = 27 must contain 9, R5C1 = 5 -> remaining visible digits = 22 = {679} with at least one hidden digit between the 6 and 7 -> no 7 in R5C2, no 7,8,9 in R5C34, no 8,9 in R5C5, no 8 in R5C6
2s. R5C2 = 1, placed for R5, no 2 in R4C2, no 1 in R6C3 (AK,NC)
2t. 9 in R5 only in R5C67 -> no 8 in R46C6 + R5C7, no 8,9 in R46C7 (AK,FNC,NC), also 6,7 must be visible in 27 total -> no 6,7 in R5C7, no 6 in R5C8
2u. R5C9 = 8 (hidden single in R5), placed for C9, no 7,8,9 in R46C8, no 7 in R6C9 (AK,FNC,NC)
2v. 7 in R5 only in R5C56 -> no 6,7,8 in R46C5, no 6 in R46C6 + R5C56 (AK,FNC,NC)
2w. 6 in R5 only in R5C34 -> no 5,6,7 in R5C3, no 5,7 in R46C4, no 7 in R6C3 (AK,FNC,NC)
2x. Left-hand total in R6 = {39} -> no 4,8 in R6C3, no 8 in R6C4 (NC or blocked by 9)
2y. Left-hand total in R8 = 13 = {148/157} -> no 6,9 in R8C3, no 9 in R8C4

[Looking in more detail at C5 …]
3a. R1C4 = {13}, R1C5 = {134}, R12C5 cannot be [13/14/31/41] which clash with R1C4 (AK,FNC,NC), cannot be [34/43] (NC) -> no 1,3,4 in R2C5
3b. R2C5 = {56} -> no 5 in R3C456 (AK, FNC,NC)
3c. R12C5 cannot be [15/16/35] because remaining visible digits cannot total 18,19,20 from 6,7,8,9, R12C5 cannot be [45] (NC) -> R1C5 = {34}, R2C5 = 6, placed for C5, no 7 in R2C4 + R3C456, no 5,7 in R2C6 (FNC, NC)
3d. R1C5 = {34} -> no 3 in R12C4, no 3,4 in R2C6 (AK,FNC,NC)
3e. R1C4 = 1, placed for R1
3f. R1C3 = {345} -> no 4 in R2C23 (AK,FNC)
3g. R1C7 = {345} -> no 4 in R2C78 (AK,FNC,NC)
3h. R2C2 = 9 -> no 8,9 in R3C3 (AK,FNC)
3i. Total 26 = [3689/4679] must contain at least one hidden digit -> no 9 in R34C5
3j. Lower total in C5 = 18, upper total 26 = [3689/4679] may include [368½79] -> lower total must contain 9 = [972/981/95½4] -> R9C5 = {124}, R8C5 = {1278}, no 2 in R8C4, no 1,2 in R89C6 (AK,FNC,NC)
3k. 9 in C5 only in R67C5 -> no 9 in R7C46 (AK)

4a. Right-hand total in R3 = 22 must include 9 and occupy at least four cells -> no 9 in R3C78
4b. 9 in R3 only in R3C46 -> no 8 in R3C5 (NC)
4c. 8 in R3 only in R3C678 -> no 7 in R34C7 (FNC,NC))
4d. 7 in C7 only in R67C7 -> no 6 in R6C78 + R7C7, no 6,7 in R7C6 (AK,FNC,NC)
4e. 22 total must contain 8,9 = {1489/589} (cannot be {2389) NC) -> R3C9 = {15}, no 7 in R3C8
4f. R3C3 = 7 (hidden single in R3), placed for C3, no 8 in R4C3, no 6,8 in R4C4 (FNC,NC)
4g. {1489} cannot have 5 in R3C78, 5 of {589} must be in R3C9 -> no 5 in R3C78
4h. R3C9 = 5 (hidden single in R3), placed for C9, no 4 in R24C9 + R3C8, no 4,5,6 in R4C8 (AK,FNC,NC)
4i. R8C5 = 8 (hidden single in C5) -> no 7 in R78C4, no 7,9 in R7C5, no 8 in R9C6 (AK,FNC,NC)
4j. R56C5 = [79] (hidden pair in C5), 7 placed for R5, no 7 in R46C6, no 6 in R56C4, no 9 in R5C6 (AK,FNC,NC)
4k. Lower table in C5 = 18, R68C5 = [98] -> R9C5 = 1, placed for R9 and C5
4l. Upper total in C5 = 26, R256C5 = [679] -> R1C4 = 4, placed for R1 and C5
4k. R5C37 = [69] (hidden pair in R5), 6 placed for C3, 9 placed in C7, no 9 in R46C6 (AK)
4l. R9C7 = 8 (hidden single in R9), placed for C7, no 8 in R8C6, no 7 in R8C8 (AK,FNC)
4m. R2C9 = {123} -> no 2 in R3C8 (AK,FNC)
4n. R5C468 = {234} -> no 3 in R4C3579 + R6C379 (AK,FNC)
4o. R4C9 = {12} -> no 1 in R3C8, no 2 in R5C8 (AK,FNC)
4p. R5C8 = {34} -> no 4 in R46C7 (AK,FNC)
4r. 2 in R5 only in R5C46 -> no 2 in R4C5 (AK,FNC)
4s. R4C5 = 5, placed for C5, no 4 in R345C46 (FNC,NC)
4t. R5C8 = 4 (hidden single in R5) -> no 5 in R46C7, no 3 in R4C8, no 3,5 in R6C8, no 4 in R6C9 (AK,FNC,NC)
4u. Naked pair {12} in R46C9, locked for C9
4v. R278C9 = [374], 7 laced for R7, no 3 in R3C8, no 3,4,5 in R79C8, no 3,5,6 in R8C8 (AK,FNC,NC)
4w. R3C8 = 8, placed for R3
4x. Right-hand total in R2 = 20 = {389} -> R2C8 = 8

5a. R4789C3 = [4182] (hidden quad in C3), 1 placed for R7, 2 placed for R9, no 3 in R345C4 + R4C2, no 1 in R6C2, no 1,2 in R6C4, no 2 in R7C4 (AK,FNC,NC)
5b. R5C4 = 2, placed for R5 -> R5C6 = 3, no 1 in R4C4, no 2 in R4C67 + R6C7, no 3 in R6C4, no 2,4 in R6C6 (FNC,NC)
5c. R8C13 = [18], left-hand total in R8 = 13 -> R8C2 = 4, no 5 in R79C2 (NC)
5d. R7C2 = 6, placed for R7
5e. R9C2468 = [4536], no 4,6 in R8C4, no 4 in R8C6, no 2,3,4,6 in R8C7 (AK,FNC,NC)
5f. R7C8 = 9 (hidden single in R7)
5g. R3C7 = 4 (hidden single in R3), placed for C7, no 3,5 in R2C7 + R4C6, no 3 in R3C6 (FNC,NC)
5h. R6C7 = 7 (hidden single in C7)
5i. R3C5 = 3 (hidden single in R3), placed for C5, no 2 in R2C6 + R3C46 + R4C4 (FNC,NC)
5j. R3C2 = 2 (hidden single in R3) -> no 1 in R4C2 (NC)
5k. R7C5 = 2, placed for R7, no 3 in R6C4 + R7C46 + R8C6, no 1,3 in R6C6 (FNC,NC)
5l. R7C7 = 3 (hidden single in R7), placed for C7 -> R1248C7 = [5261], 5 placed for R1, no 1,6 in R2C6, no 1 in R3C6, no 2,4 in R68C8, no 4 in R7C6 (FNC,NC)
5m. R2C3 = 5 (hidden single in C3)
5n. R7C46 = [45], no 5 in R8C4, no 6 in R8C6 (NC)
5o. R6C8 = 1 -> no 2 in R6C9 (NC)
5p. R46C9 = [21] -> no 1 in R4C8 (NC)
5q. R3C6 = 9, placed for R3

[Now for uniqueness, six of the even numbered rows/columns have three identical numbers in a row.]
6. There are already four groups of identical numbers
R123C6 = [999], R123C8 = [888], R678C6 = [555] and R8C345 = [888], two more are required so R123C4 = [111] and R6C345 = [999]

and the rest is naked singles.

When HATMAN wrote "but it gets very difficult":
I was expecting that I would need to use forcing chains, but I didn't. I think my technically hardest parts were at the start of step 3.
I had to make several restarts, including one where I'd overlooked a permutation in step 3j, hope I haven't missed any other valid permutations, and also because I'd omitted a few AK/FNC/NC eliminations, or occasionally deleted the wrong digit while doing those; I solve using Excel worksheets so make all my placements and eliminations manually.

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


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