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 Post subject: LC Killer NC 4
PostPosted: Wed Oct 30, 2019 3:53 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 693
Location: Saudi Arabia
Low Clue Killer NC 4

It is NC which allows me to have bigger cages and cut out more clues.

The original puzzle from JSudoku was symmetrical however I have merged some cages to make it harder, so some cages are not symmetric.

This puzzle type is about cage boundaries (rather than about cages) hence when I discuss symmetry below, I am discussing symmetry of cage boundaries.

N5 is self symmetric.

The cell boundaries that are internal to nonets (i.e. thin cell boundary lines) are have symmetrical cage boundaries. This is not necessarily the case for cell boundaries that are on the edge of nonets. Be careful here as it is easy to make mistakes.

If a cell boundary is completely within a nonet e.g. the r1c78 boundary it either has no cage boundaries or both cage boundaries and the boundary at r9c23 is the same. The cage boundaries at r1c67 do not have to be the same as the boundaries at r9c34.



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 Post subject: Re: LC Killer NC 4
PostPosted: Wed Jan 01, 2020 11:37 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Here is how I now interpret the cage boundary conditions:
I was confused by the other comments about what must be or may not be symmetric so I’ve started again from the statement that the original puzzle had symmetrical cage boundaries, with at least two cells in each cage, and that merging cages has been done without splitting them so that merged cages must have at least four cells. That seems to be enough to define the cage boundaries.

My walkthrough for LC Killer NC 4:
The usual uncaged killer rules, the number is in the top left corner with top dominating left as usual. There are no single cages and no diagonally-connected cages.
“The original puzzle from JSudoku was symmetrical however I have merged some cages to make it harder, so some cages are not symmetric.
This puzzle type is about cage boundaries (rather than about cages) hence when I discuss symmetry below, I am discussing symmetry of cage boundaries.
N5 is self symmetric.”
Non-consecutive (NC) horizontally and vertically.

Prelims, based just on positions of totals.
a) 20(3) cage in R1C12 + R2C1 = {389/479/569/578}, no 1,2
b) 9(?) cage at R7C1 must include R7C2 -> 7(2) cage in R6C12 = {16/25} (cannot be {34}, NC)
c) Because of symmetry 42(?) cage at R4C4 must be 42(7) cage = {3456789}, no 1,2
d) The two cells in N5 missing from the 42(7) cage cannot be R46C5 because ?? in R7C5 blocks a cage starting at R6C5, cannot be R5C46 because ?? in R5C7 blocks a cage starting at R5C6 -> R4C6 + R6C4 = {12}, locked for N5
Now to use the symmetry of the cages.
e) There’s a 20(3) cage in R1C12 + R2C1 -> there must be an 11(3) cage in R8C9 + R9C89 (since a 3-cell cage merged with a 2-cell cage cannot total 11) = {128/137/146/236/245}, no 9
f) R6C4 must be part of 21(4) cage at R6C3 since 21(3) cage cannot contain one of 1,2 -> 21(4) cage in R67C34
g) R4C6 must be part of 21(?) cage at R3C6 must be at least 21(4) cage since 21(3) cage cannot contain one of 1,2 -> contains at least R34C67
h) Because of the non-total cage in R4C89 corresponding with R6C12, there must be a ??(4) cage in R34C89 -> there must be a 9(2) cage in R7C12 = {18/27/36} (cannot be {45}, NC), no 4,5,9
i) There must be a ?(2) cage in R3C45, no 9
j) 8(?) cage at R1C7 must contain at least R2C7, no 8,9 in R12C7
k) 12(?) cage at R8C3 must contain at least R9C3

1a. R2C9 must be part of 18(?) cage at R1C8
1b. 45 rule on N3 R3C789 must total at least 19 -> no 1 in R3C789
1c. 45 rule on R12 2 innies R2C23 = 7 = {16/25} (cannot be {34}, NC)
1d. 20(3) cage in R1C12 + R2C1 = {389/479/578} (cannot be {569} which clashes with R2C23), no 6
1e. 3,5 of {389/578} must be in R1C1 because of NC -> no 3,5 in R1C2 + R2C1
1f. 16(?) cage at R3C1 cannot be {79} in R3C12 or R34C1 which would clash with 20(3) cage, using NC in the case of R34C1 -> 16(?) cage must contain at least 3 cells
1g. 11(3) cage in R8C9 + R9C89 = {128/137/146/236/245}
1h. 8 of {128} must be in R9C9 (NC) -> no 8 in R8C9 + R9C8
11. 2 of {245} must be in R9C9 (NC) -> no 5 in R9C9

2a. Killer pair 1,2 in R6C12 and R6C4, locked for R6
2b. 21(4) cage at R6C3 cannot contain both of 1,2, R6C4 = {12} -> no 1,2 in R7C34
2c. 45 rule on N7 1 innie R7C3 = any outies + 5 -> min R7C3 = 5
2d. 21(?) cage at R3C6 must be 21(4) because blocked by R34C89 and by total at R5C7
2e. 21(4) cage at R3C6 cannot contain both of 1,2, R4C6 = {12} -> no 1,2 in R3C67 + R4C7

3a. 8(?) cage at R1C7 cannot contain R2C6 because 12(?) cage at R8C3 cannot contain R8C4, 8(?) cage cannot contain R2C8 because 12(?) cage cannot contain R8C2 -> 8(?) cage must be 8(2) cage R12C7 = {17/26/35}, no 4
3b. Although R8C45 and R9C45 may have been separate cages in the original puzzle, 12(?) cage at R8C3 cannot be R89C3 + R9C45 cannot be [26]{13} (maximum value for R9C45 using step 2c, cannot be {12} NC) because 19(4) cage at R8C1 cannot be {3457}, NC) -> R89C3 must be 12(2) cage with R7C3 = 5 -> 12(2) cage = {39/48}, clean-up: no 2 in R2C2 (step 1c)
3c. 21(4) cage at R6C3 contains 1,2 in R6C4 = {1578} (cannot be {1569/2568}, NC) -> R6C3 + R7C4 = {78}, R6C4 = 1 -> R4C6 = 2, clean-up: no 6 in R6C12
3d. Naked pair {25} in R6C12, locked for R6 and N4
NC eliminations:
R4C6 = 2 -> no 3 in R35C6 + R4C57
R7C3 = 5 -> no 6 in R7C2, no 4 in R8C3, clean-up: no 3 in R7C1, no 8 in R9C3
R6C3 = {78} -> no 7,8 in R5C3
R7C4 = {78} -> no 7,8 in R7C5 + R8C4

4. Cages at R4C3 and R5C7 must correspond -> 16(4) cage R4C3 + R5C123, ??(4) cage R5C789 + R6C7 since there are no other totals in these cells
4a. 16(4) cage at R4C3 = {1348} (only remaining combination), locked for N4 -> R6C3 = 7, R7C4 = 8, clean-up: no 1 in R7C12
4b. Naked pair {69} in R4C12, locked for R4
4c. No total is specified for a R4C12 cage -> must be part of 16(?) cage at R3C1 = 16(3) cage R3C1 + R4C12 -> R3C1 = 1 (cage total), clean-up: no 6 in R2C23 (step 1c)
4d. R2C23 = [52] -> R6C12 = [52], clean-up: no 3,6 in R1C7, no 7 in R7C1
NC elimination:
R2C2 = 5 -> no 4 in R1C2 + R2C1, no 4,6 in R3C2
R2C3 = 2 -> no 3 in R13C3 + R2C4
R6C1 = 5 -> no 4 in R5C1, no 6 in R7C1
R6C2 = 2 -> no 1,3 in R5C2, no 3 in R7C2
4e. R7C12 = [27]
4f. 20(3) cage at R1C1 = {389/479}, 9 locked for N1
4g. 3,4 only in R1C1 -> R1C1 = {34}
4h. 18(?) cage at R2C2 cannot be 18(3) in R2C2 + R3C23 -> 18(4) cage in R23C23, R2C23 = [52] = 7 -> R3C23 = 11 = [38]
4i. R1C12 = [49], R2C1 = 7, R1C3 = 6, R4C12 = [96], clean-up: no 1 in R1C7, no 4 in R9C3 (step 3b)
NC eliminations:
R1C3 = 6 -> no 5,7 in R1C4
R7C1 = 2 -> no 3 in R8C1
R7C2 = 7 -> no 8 in R8C2
R7C4 = 8 -> no 9 in R7C5 + R8C4
R1C4 = {23} -> no 2,3 in R1C5

5a. Naked pair {39} in R89C3, 3 locked for C3 and N7
5b. Naked pair {14} in R45C3, locked for N4 -> R5C12 = [38]

6a. ?(2) cage at R3C2 must contain 2 (cannot be {45}, NC), locked for R3 and N2 -> R1C4 = 3
6b. 45 rule on N1 with R1C4 = 3, 1 remaining outie R2C4 = 9 (because R2C45 cannot total 9) -> 18(3) cage in R1C34 + R2C4
6c. 9 in N3 only in R3C789
6d. 45 rule on N3 3 innies R3C789 = 19 = {469}, 4,6 locked for R3 and N3, clean-up: no 2 in R1C7
6e. R2C56 = {46} (hidden pair in N2) = 10 -> R1C56 = 9 = {18}, locked for R1
6f. 21(4) cage at R3C6 = {2478} (cannot be {2469} because 6,9 only in R3C7, cannot {2568}, NC) -> R3C7 = 4, R3C6 = 7, R4C7 = 8
NC eliminations:
R3C6 = 7 -> no 6 in R2C6
R2C6 = 4 -> no 3 in R2C7, clean-up: no 5 in R1C7
R2C5 = 6 -> no 5 in R3C5
R1C7 = 7 -> no 8 in R1C6
R4C7 = 8 -> no 7 in R4C8, no 9 in R5C7
6g. R1C56 = [81]
6h. R2C7 = 1
6i. R3C45 = [52]

7a. R6C6 = 8 (hidden single in N5), no 9 in R57C6 + R6C57
7b. R5C5 = 9 (hidden single in N5)
7c. 3 in R4 only in R4C89, locked for N6 -> R6C7 = 6, no 5 in R5C7 (NC)
7d. R5C7 = 2
7e. Naked pair {49} in R6C89, 4 locked for R6 and N6 -> R6C5 = 3
7f. Naked triple {359} in R789C7, locked for N9
NC eliminations:
R5C7 = 2 -> no 1 in R5C8
R6C5 = 3 -> no 4 in R7C5
7g. 11(3) cage at R8C9 = {128} (cannot be {146} which clashes with R7C89) -> R9C9 = 8, R8C9 + R9C8 = {12}, locked for N9
7h. Naked pair {46} in R7C89, locked for N9, 6 locked for R7 -> R7C567 = [139], R89C1 = [86], R8C8 = 7
NC eliminations:
R8C8 7 -> no 6 in R7C8
6 in R8 only in R8C46 -> no 5 in R8C5
7i. R8C5 = 4, R89C2 = [14], R8C9 = 2, R9C8 = 1, R8C4 = 6
7j. Naked pair {47} in R45C4, 7 locked for C4 and N5 -> R4C5 = 5
NC eliminations:
R4C5 = 5 -> no 4 in R4C4
R8C5 = 4 -> no 5 in R8C6
The remaining naked singles
R1C89 = [25]
R2C89 = [83]
R4C3489 = [4731]
R5C34689 = [14657]
R7C89 = [46]
R3C89 = [69]
R6C89 = [94]
R8C367 = [395]
R9C34567 = [92753]

Finally defining the remaining cages, from symmetry
8a. 16(3) cage at R3C1 must correspond with R6C89 + R7C9 (R7C78 cannot also be part of this cage because of 4 in R6C9) -> 19(3) cage R6C89 + R7C9
8b. 18(3) cage at R1C3 must correspond with R89C6 + R9C7 -> 17(3) cage R89C6 + R9C7
8c. 19(4) cage at R1C5 must correspond with R89C45 (cannot be two separate cages because no total in R9C4) -> 19(4) cage R89C45
8d. ??(?) cage at R7C5 must total at least 10 so must also include R78C78 -> 29(6) cage R7C5678 + R8C78

Solution:
Attachment:
Low Clue Killer NC 4.jpg
Low Clue Killer NC 4.jpg [ 78.5 KiB | Viewed 4326 times ]


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