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 Post subject: Assassin 380
PostPosted: Mon Jul 15, 2019 8:00 am 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
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Assassin 380
Found a really nice way to get started with this puzzle which got me hooked. Takes persistence to get the final breakthrough with many key steps but no big combo steps. Resists for a long way in (37 steps for me). Perfect for a milestone! SudokuSolver gives it 1.90 JSudoku has a bug so can't finish it.

code:
3x3::k:2560:2560:3073:3073:0000:0000:5891:5891:5891:0000:0000:0000:3077:3077:3077:5891:5894:5891:0000:3847:3847:3847:6152:6152:5894:5894:1545:0000:4106:2315:6152:6152:7436:7436:5894:1545:4106:4106:2315:7436:7436:7436:2317:4366:4366:2831:9488:7436:7436:9488:9488:2317:4366:6673:2831:9488:9488:9488:9488:4882:4882:4882:6673:7955:9488:7955:5140:5140:5140:6673:6673:6673:7955:7955:7955:2069:2069:1538:1538:1796:1796:
solution:
Code:
+-------+-------+-------+
| 3 7 8 | 4 5 9 | 1 2 6 |
| 1 6 4 | 7 3 2 | 9 8 5 |
| 2 9 5 | 1 6 8 | 7 3 4 |
+-------+-------+-------+
| 8 3 7 | 9 1 6 | 4 5 2 |
| 9 4 2 | 5 8 3 | 6 1 7 |
| 6 5 1 | 2 4 7 | 3 9 8 |
+-------+-------+-------+
| 5 2 6 | 3 9 4 | 8 7 1 |
| 4 1 3 | 8 7 5 | 2 6 9 |
| 7 8 9 | 6 2 1 | 5 4 3 |
+-------+-------+-------+


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 Post subject: Re: Assassin 380
PostPosted: Thu Jul 25, 2019 5:51 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Thanks Ed for your latest Assassin. Don't know whether my way in was the same as yours; I enjoyed the early steps of my solving path.

Definitely a hard one. Hope I'm not giving anything away by saying that, after I got stuck, I revisited areas where I'd worked before and found that I could get more from them.

Here is my walkthrough for Assassin 380:
Prelims

a) R1C12 = {19/28/37/46}, no 5
b) R1C34 = {39/48/57}, no 1,2,6
c) R34C9 = {15/24}
d) R45C3 = {18/27/36/45}, no 9
e) R56C7 = {18/27/36/45}, no 9
f) R67C1 = {29/38/47/56}, no 1
g) R9C45 = {17/26/35}, no 4,8,9
h) R9C67 = {15/24}
i) R9C89 = {16/25/34}, no 7,8,9
j) 19(3) cage at R7C6 = {289/379/469/478/568}, no 1
k) 20(3) cage at R8C4 = {389/479/569/578}, no 1,2
l) 29(7) cage at R4C6 = {1234568}, no 7,9
m) 37(8) cage at R6C2 = {12345679}, no 8

1a. 45 rule on R9 3 innies R9C123 = 24 = {789}, locked for N7, 7 also locked for R9, clean-up: no 2,3,4 in R6C1, no 1 in R9C45
1b. Killer pair 2,5 in R9C45 and R9C67, locked for R9
1c. 20(3) cage at R8C4 = {389/479/578} (cannot be {569} which clashes with R9C45), no 6

2a. 45 rule on N3 2 outies R4C89 = 7 = [25/34/52/61] -> R4C8 = {2356}
2b. 45 rule on N36 2 innies R4C7 + R6C9 = 12 = [39/48/57/84] -> R4C7 = {3458}, R6C9 = {4789}
2c. 29(7) cage at R4C6 = {1234568}, CPE no 1,2,6 in R6C56
2d. 8 in N5 only in R45C456 + R6C4, CPE no 8 in R4C7, clean-up: no 4 in R6C9

3a. 45 rule on N9 3(1+2) outies R6C9 + R79C6 = 13
3b. Min R6C9 = 7 -> max R79C6 = 6 -> max R7C6 = 5, max R9C6 = 4 (because no 1 in R7C6), clean-up: no 1 in R9C7
3c. 19(3) cage at R7C6 = {289/379/469/478/568}
3d. R7C6 = {2345} -> no 2,3,4,5 in R7C78
3e. 8 in N8 only in 20(3) cage at R8C4, locked for R8
3f. 20(3) cage (step 1c) = {389/578}, no 4
3g. R9C45 = {26} (cannot be {35} which clashes with 20(3) cage), locked for R9 and N8, clean-up: no 1 in R9C89
3h. Naked pair {34} in R9C89, locked for R9 and N9 -> R9C78 = [15], clean-up: no 4 in R56C7, no 7 in R6C9 (step 2b)
3i. R9C6 = 1 -> R6C9 + R7C6 = 12 = [84/93], no 5 in R7C6
3j. 4 in N8 only in R7C456, locked for R7, clean-up: no 7 in R6C1

[The first key step, a wellbeback-style one.]
4a. 45 rule on C789 1 outie R7C6 = 1 remaining innie R4C7
4b. R4C7 ‘sees’ all of N5 except for R6C56 and indirectly ‘sees’ R6C6 because R4C7 = R7C6 -> R4C7 = R6C5 = {34}
4c. R6C3 ‘sees’ all of N5 except for R4C45 -> at least one of {1234568} must be in R4C45
4d. 7,9 in N5 can only be in R4C45 and R6C6, R4C45 cannot contain both of 7,9 -> R6C6 = {79}

[The next key step but, as Ed said, many more are needed.]
5. 45 rule on N8 3 remaining innies R7C456 = 16 contains 4 = {349/457}
5a. 19(3) cage at R7C6 = {469/478} (cannot be {379} = 3{79} which clashes with R7C456 = {49}3, CCC) -> R7C6 = 4
5b. R7C6 = 4 -> R4C7 = 4 (step 4a) -> R6C9 = 8 (step 2b), clean-up: no 2 in R3C9, no 3 in R4C8 (step 2a), no 5 in R5C3, no 1 in R56C7, no 3 in R7C1
5c. 8 in N9 only in 19(3) cage at R7C6 = 4{78}, 7 locked for R7 and N9
5d. 7 in N8 only in 20(3) cage at R8C4 (step 3h) = {578}, 5 locked for R8 and N8
5e. Naked pair {39} in R7C45, locked for R7 and 37(8) cage at R6C2 -> R6C56 = [47], clean-up: no 2 in R5C7
5f. 9 in N5 only in R4C45, locked for R4 and 24(4) cage at R3C5
5g. 9 in C6 only in R12C6, locked for N2, clean-up: no 3 in R1C3
5h. 29(7) cage at R4C6 = {1234568}, 8 locked for N5
5i. R8C13 = {34} (hidden pair in N7)
5j. 1 in N7 only in R7C23 + R8C2, locked for 37(8) cage at R6C2, no 1 in R6C2
5k. Killer pair 2,6 in R4C78 and R56C7, locked for N6
5l. 7 in N6 only in R5C789, locked for R5, clean-up: no 2 in R4C3

6a. 45 rule on N78 R6C56 = [47] = 11 -> 2 remaining outies R6C12 = 11 = [56/65/92]
6b. 45 rule on N4 2 other innies R4C1 + R6C3 = 9 = {36}/[72/81] -> R4C1 = {3678}, no 5 in R6C3
6c. 29(7) cage at R4C6 = {1234568}, 5 locked for N5
6d. 16(3) cage at R4C2 = {178/349/358/367/457} (cannot be {169/259/268} which clash with R6C12), no 2
6e. 1 in N4 only in R4C1 + R6C3 = [81] or R45C3 = {18} or 16(3) cage = {178} -> 16(3) cage = {178/349/367/457} (cannot be {358}, locking-out cages)
6f. 7 of {178/367/457} must be in R4C2, 3 of {349} must be in R4C2 -> R4C2 = {37}

7a. 45 rule on R56789 3 remaining outies R4C236 = 16 = {178/358/367} (cannot be {268} which clashes with R4C89), no 2
[At this stage I saw
{358} can only be [385] (cannot be [358] because R45C3 = [54] clashes with 16(3) cage at R4C2 = {349/358} while 7 of 16(3) cage at R4C2 = {367} must be in R4C2) -> no 5 in R4C3, clean-up: no 4 in R5C3
However I then saw the more powerful short forcing chain.]
7b. Consider combinations for R4C89 (step 2a) = {25}/[61]
R4C89 = {25}, 5 locked for R4 => R4C236 = {178/367}
or R4C89 = [61], R6C7 = 2 (hidden single in N6), no 7 in R4C1 (step 6b) => 7 in R4 only in R4C236 = {178/367}
-> R4C236 = {178/367}, no 5, 7 locked for R4, clean-up: no 4 in R5C3, no 2 in R6C3 (step 6b)
7c. 1 of {178} must be in R4C3 -> no 8 in R4C3, clean-up: no 1 in R5C3
7d. 4 in N4 only in 16(3) cage at R4C2 = {349/457} -> R5C12 = {45/49}
7e. 5 in R4 only in R4C89 (step 2b) = {25}, locked for N6, 2 also locked for R4, clean-up: no 5 in R3C9, no 7 in R5C7
7f. Naked pair {36} in R56C7, locked for C7, 3 also locked for N6
7g. 1 in C1 only in R123C1, locked for N1, clean-up: no 9 in R1C1
7h. 1 in N4 only in R46C3, locked for C3

8. 45 rule for R6789 4 innies R6C3478 = 15 = {1239/1356}
8a. 2,5 only in R6C4 -> R6C4 = {25}
8b. Consider combinations for R6C3478
R6C3478 = {1239}, 2,9 locked for R6 => R5C3 = 2 (hidden single in N4), R4C3 = 7
or R6C3478 = {1356} => R6C8 = 1, R4C3 = 1 (hidden single in N4), R5C3 = 8
-> R45C3 = [18/72], no 3,6
8c. R4C236 (step 7b) = {178/367}
8d. 6,8 only in R4C6 -> R4C6 = {68}

9. 24(4) cage at R3C5 contains 9 = {1689/2679/3579} (cannot be {2589} because 2,5,8 only in R3C56)
9a. 7 of {2679/3579} must be in R3C5 -> no 2,3,5 in R3C5
9b. 5 of {3579} must be in R3C6 -> no 3 in R3C6
9c. Consider combinations for R4C236 (step 7b) = {178/367}
R4C236 = {178} => R8C6 = 5 => 24(4) cage = {1689/2679}
or R4C236 = {367}, 3 locked for R4 => 24(4) cage = {1689/2679}
-> 24(4) cage = {1689/2679}, no 3,5
9d. 3 in R4 only in R4C12, locked for N4, clean-up: no 6 in R4C1 (step 6b)
9e. R6C7 = 3 (hidden single in R6) -> R5C7 = 6
9f. 6 in R4 only in R4C456, CPE no 6 in R3C6
9g. 24(4) cage = {1689/2679}
9h. R3C6 = {28} -> no 8 in R3C5
9i. 24(4) cage = {1689} must be [68]{19} (cannot be [18]{69} which clashes with R34C6 = [86]) -> no 1 in R3C5
9j. Consider combinations for 24(4) cage = {1689/2679}
24(4) cage = {1689} => R3C6 = 8
or 24(4) cage = {2679} => R3C56 = [72], R4C45 = {69}, locked for R4 => R4C6 = 8
-> 8 in R34C6, locked for C6 -> R8C6 = 5

10. Consider placements for R4C1 = {38}
R4C1 = 3 => R4C6 = 8 (hidden single in R4) => R35C6 = [23]
or R4C1 = 8, R6C3 = 1 (step 6b), R4C3 = 7 => R5C3 = 2 => R5C6 = 3
-> R5C6 = 3
10a. 2 in C6 only in R123C6, locked for N2
10b. 12(3) cage at R2C4 = {156/237/246} (cannot be {129} because 2,9 only in R2C6, cannot be {138/147/345} because R2C6 only contains 2,6,9), no 8,9
[Now it’s getting a lot easier. Ed thought this comment should have been after step 11; the reason for it here was that it was consistent with the comment posted before my walkthrough, I had to go back and find a way to make progress after step 9.]
10c. R1C6 = 9 (hidden single in C6), clean-up: no 1 in R1C1, no 3 in R1C4
10d. 24(4) cage at R3C5 (step 9c) = {1689/2679} -> R3C56 = [68/72]
10e. 12(3) cage at R2C4 = {156/237} (cannot be {246} which clash with R3C56), no 4
10f. 6 of {156} must be in R2C6 -> no 6 in R2C45
10g. Killer pair 6,7 in 12(3) cage and R3C56, locked for N2, clean-up: no 5 in R1C3

11a. 12(3) cage at R2C4 (step 10e) = {156/237}, R3C56 (step 10d) = [68/72] -> combined cage 12(3) + R3C56 = {156}[72]/{237}[68]
11b. R1C34 = [48/75/84], R1C5 = {1358} -> combined cage R1C345 = [75/84]1/[48/75/84]3/[84]5 (cannot be [48]1/[48/57]5/[75]8) which clash with combined cage 12(3) + R3C56) -> no 8 in R1C5
11c. Combined cage R1C345 = [75/84]1/[75]3/[84]5 (cannot be [48/84]3 which clash with R1C12) -> R1C34 = [75/84]
11d. Killer pair 7,8 in R1C3 and R45C3, locked for C3 -> R9C3 = 9
11e. 8 in N2 only in R3C46, locked for R3

12. 15(3) cage at R3C2 = {159/249/348/357/456} (cannot be {168} because 1,8 only in R3C4, cannot be {258} which clashes with R3C6, cannot be {267} which clashes with R3C5)
12a. 12(3) cage at R2C4 (step 10e) = {156/237}
12b. Consider combinations for R1C12 = {28/37/46}
R1C12 = {28}, locked for R1 => R1C34 = [75], 12(3) cage = {237}, 2 locked for N2 => R3C6 = 8
or R1C12 = {37/46} => 15(3) cage cannot be {348} = {34}8 which clashes with R1C12
-> no 8 in R3C4
[Cracked, at last.]
12c. R3C6 = 8 (hidden single in N2), R4C6 = 6, R4C45 = {19}, 1 locked for R4 and N5, R3C5 = 6 (cage sum), R6C3 = 1 (hidden single in N4), R6C8 = 9, clean-up: no 2 in R7C1
12d. R2C6 = 2 -> 12(3) cage = {237}, locked for R2, 3 locked for N2
12e. R4C3 = 7 -> R5C3 = 2, R1C3 = 8 -> R1C4 = 4, clean-up: no 2,6 in R1C12
12f. Naked pair {37} in R1C12, locked for R1 and N1
12g. 15(3) cage = {159} (only remaining combination, cannot be {249} because 2,9 only in R3C2) -> R3C234 = [951]
12h. R3C9 = 4 -> R4C9 = 2, R3C1 = 2, R3C78 = [73], R4C8 = 5 -> R2C8 = 8 (cage sum)

and the rest is naked singles.

Rating Comment:
Maybe my steps are 1.5, but I'll rate my walkthrough for A380 at Hard 1.5 because I found it difficult to spot how to finish it. I used 6 forcing chains, that's more than usual.
Thanks Ed for your comments and a minor correction.

Loved your step 4! See walkthrough posted below.


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 Post subject: Re: Assassin 380
PostPosted: Fri Jul 26, 2019 8:27 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Loved that start Andrew. wellbeback will be proud! I think my different start is wellbebackish too. Andrew and I worked in all the same areas in a similar order but came at things differently. Perhaps it is a very narrow solving path. Thanks to Andrew for some corrections and simplifications.
A380 WT:
Preliminaries courtesy of SudokuSolver
Cage 6(2) n89 - cells only uses 1245
Cage 6(2) n36 - cells only uses 1245
Cage 7(2) n9 - cells do not use 789
Cage 8(2) n8 - cells do not use 489
Cage 12(2) n12 - cells do not use 126
Cage 9(2) n6 - cells do not use 9
Cage 9(2) n4 - cells do not use 9
Cage 10(2) n1 - cells do not use 5
Cage 11(2) n47 - cells do not use 1
Cage 20(3) n8 - cells do not use 12
Cage 19(3) n89 - cells do not use 1
Cage 29(7) n456 - cells ={1234568}
Cage 37(8) n4578 - cells ={12345679}

Note: no clean-up done unless stated.
1. "45" on r9: 3 innies r9c123 = 24 = {789} only: all locked for r9 and n7

2. 8 in n5 in the 29(7)r4c6 or in r4c45. r4c7 sees all those -> no 8 in r4c7 (Common Peer Elimination CPE)

3. "45" on n36: 2 innies r4c7 + r6c9 = 12 = [39/48/57]

4. r6c9 sees all 7,8,9 in n9 apart from r7c78 so must repeat there (clone)
4a. 37(8)r6c2 = {12345679}: must have 7 & 9 which are only in r67 -> 7,9 cannot repeat in r6c9 + r7c78
4b. -> r6c9 = 8, r4c7 = 4 (h12(2))
4c. no 2 in r9c6

5. "45" on n9: 2 remaining outies r79c6 = 5 = [41] only permutation
5a. r9c7 = 5
5b. r7c6 = 4 -> r7c78 = 15 and must have 8 for n9 = {78} only: 7 locked for n9 & r7

6. "45" on n8: 2 remaining innies r7c45 = 12 = {39} only: both locked for r7, n8 and for 37(8) cage

7. 8(2)n8 = {26}: both locked for r9, 6 for n8
7a. 7(2)n9 = {34}: both locked for n9
7b. 20(3)n8 = {578}: 5 locked for r8

8. hidden single 4 in n5 -> r6c5 = 4

9. hidden pair 3,4 in n7 -> r8c13 = {34} only

10. 29(7)r4c6 = {123568}[4]. r6c6 sees all of 2,5,6 -> no 2,5,6 in r6c6 (CPE)
10a. r6c6 = 7

11. 9(2)n6 = [72]/{36}(no 1) = 2 or 6

12. 17(3)n6: {269} blocked by 9(2)n6
12a. 17(3) = {179/359}(no 2,6)

13. 2 in r5 in 29(7)r4c6 or in n4: r6c3 sees all those -> no 2 in r6c3 (Common Peer Elimination CPE)

14. 2 & 8 must be in 29(7)r4c6: both locked for n5

15. "45" on n478: 2 remaining innies r4c1 + r6c3 = 9 = [81]/{36}(no 2,5,7,9)

16. 9 in n5 only in r4c45: 9 locked for r4 and 24(4)r3c5

17. "45" on n4578: 3 remaining innies r4c145 = 18 and must have 9 for r4 = {189/369}(no 5) = 6 or 8, 8 or 3


18. "45" on r56789: 3 remaining outies r4c236 = 16
18a. but {268} blocked by h18(3)r4 = 6 or 8
18b. but {358} blocked by h18(3)r4 = 8 or 3
18c. = {178/367}(no 2,5)
18d. no 4 in r5c3

19. hidden pair 2,5 in r4 -> r4c89 = {25}: both locked for n6 (could also do 2 outies n3 = 7)
19a. r3c9 = (14)
19b. 9(2)n6 = {36}: both locked for c7 and 3 for n6

20. "45" on n78: 2 remaining outies r6c12 = 11 = [92]/{56}(no 1,3; no 2 in r6c1) = 2 or 5 but not both

21. hidden killer pair 2,5 in r6 -> r6c4 = (25)

22. 4 in n4 only in 16(3) = {349/457}(no 1,2,6,8)
22a. can't have both 3,7 which must be in r4c2 -> no 3,7 in r5c12

This is a key step for my solution
23. if 1 in r4 is in h16(3)r4c236 = [718] -> 5 in r8c6
23a. or 1 in r4 in r4c45 in 24(4)r3c5 = {68}{19}
23b. both positions of 1 -> no 5 in r3c6

24. 24(4)r3c5: {3579} blocked by no 5,7 for r3c6
24a. {2589} blocked by no 2,5,8 in r4c56
24b. = {1689/2679}(no 3,5)

25. 3 in n5 only in 29(7) -> no 3 in r6c3
25a. no 6 in r4c1 (h9(2))
25b. hidden single 3 in r6 -> r56c7 = [63]
25c. no 3 in r4c3

26. 6 in n5 in r4c456: 6 locked for r4 and no 6 in r3c6 (CPE)

27. "45" on n4578: 2 outies r3c56 - 6 = 1 innie r4c1 = [72/18][3]/[68][8](no 2,8 in r3c5)
27a. note: 8 in r4c1 -> 8 in r3c6

28. 8 in r4 in r4c1 -> 8 in r3c6 or 8 in r4 in h16(3)r4c238 = [718] only
28a. 8 locked for c6 in r34c6
28b. r8c6 = 5

29. h16(3)r4c236 = [718/376] (no 8 in r4c3, no 3 in r4c6)
29a. r4c6 = (68): if r4c6 = 6 -> 7 in r4c3 -> 2 in r5c3 -> no 2 in r5c6
29b. or r4c6 = 8 -> r45c6 <>[82] because r3c6 = (28) (Note: much easier to just go r34c6 = [28]. Thanks Andrew!)
29c. -> no 2 in r5c6
29d. r5c6 = 3

30. 2 & 9 in c6 only in n2: both locked for n2

31. only combo with 9 in 12(3)n2 = {129}: but blocked since 2,9 only in r2c6
31a. -> no 9 in r2c6
31b. -> hidden single 9 in n2 -> r1c6 = 9

32. 24(4)r3c5 = {1689/2679}
32a. 1 blocked from r3c5 by r4c6 = (68)

33. 12(3)n2 must have 2 or 6 for r2c6 = {156/237/246}(no 8) = 6 or 7
33a. -> killer pair 6,7 with r3c5: both locked for n2

34. if 6 in c6 in r2c6 in 12(3) = {15}[6]
34a. or 6 in c6 in r4c6 in h16(3) = [376]
34b. note: has 5 in n2 or 7 in r4c3
34c. -> [75] blocked from 12(2)r1c34

35. 12(2)r1c3 = {48} only: both locked for r1

36. 10(2)n1 = {37} only: both locked for r1 and n1

37. 12(3)n2: {156} blocked by r1c5 = (15)
37a. {246} blocked by r1c4 + r3c6 = {248}
37b. = {237} only: all locked for n2 and r2

cracked. Finally! Much easier now.
Cheers
Ed


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 Post subject: Re: Assassin 380
PostPosted: Sun Aug 04, 2019 3:24 am 
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Grand Master
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Joined: Tue Jun 16, 2009 9:31 pm
Posts: 280
Location: California, out of London
I started the same way as Ed (pretty much) with the first key moves identical for a very nice start.
I got bogged down after that and eventually had to use one small contradiction chain and one longer one.
Here's how I did it. We all seemed to follow a similar path through the maze.
Congratulations on a fine puzzle Ed! I'm happy my style is proving a useful tool for you both.
Assassin 380 WT:
1. Innies r9 = r9c123 = +24(3) = {789}

2. 29(7) = {1234568} (No 7,9)
Whatever goes in r4c7 goes in n5 in r6c56.
Whatever goes in r6c3 goes in n5 in r4c45.
-> One each of (79) in r4c45 and r6c56

3! Innies n36 = r4c7,r6c9 = +12(2)
37(8) has no 8.
Since whatever goes in r4c7 goes in n5 in r6c56 -> r4c7 not 8.
Also r4c7 not 7 or 9.
-> [r4c7,r6c9] from [39], [48], [57]
Whichever of (789) is in r6c9 can only go in n9 in r7c78
-> whichever it is - is nowhere else in r67
But 37(8) must contain (79) somewhere in r67
-> [r4c7,r6c9] = [48]

4. -> 8 in n9 in r7c78
-> Remaining Innies n9 = +12(2) - can only be [75]
-> 19(3)r7 = [4{78}] and 6(2)r9 = [15]
-> 7(2)r9 = {34}
-> 8(2)r9 = {26}
(78) in n8 only in r8
-> 20(3)n8 = {578}
-> r7c45 = {39}
-> r6c56 = [47]
-> 9 in r4c45
(34) in n7 only in r8
-> Since r8c13 = +7(2) -> r8c13 = {34}

5. Remaining cells in n9 and in 37(8) are {1256}
-> r6c2 = r7c1
-> 11(2)c1 and r6c12 are both [92] or {56}

6. Remaining Innies n4 = r4c1,r6c3 = +9(2)
Whatever goes in r6c3 goes in n5 in r4c45 and therefore in n6 in r5.
-> Given previous placements r6c3 only from (1236)

7. Remaining innies n6 = r4c89 = [61] or {25}
But the former puts 9(2)n6 = [72] which leaves no value for r6c3
(Cannot be 3 since that puts another 6 in r4 at r4c1)
-> r4c89 = {25}
-> 17(3)n6 = {179} with 7 in r5
-> 9(2)n6 = {36}
Also 2 not in r6c3 -> r6c3 from (136)

8! Given r4c1,r6c3 = +9(2) and 9(2) cage in n4 ...
... -> there cannot be two numbers summing to 9 in r4c123.
-> At least one of (18) in r4 in n5 in r4c456
Similarly -> at least one of (36) in r4c456

1 cannot go in r4c6 and 8 cannot go in r4c45 (since it cannot also go in r6c3)

-> Either r4c45 = {19} and r4c6 = 3 or 6
or r4c6 = 8 and r4c45 = {39} or {69}

Trying the latter puts r8c6 = 5 and since 24(4) cannot contain all of (369) -> r3c6 = (NS)2
-> r4c45 cannot be {39}

9. -> r4c45 from {19} or {69}
The former puts r6c3 = 1 -> r6c8 = 9 -> r6c12 = {56}
The latter puts r6c3 = 6
Either way 6 in r6c123 -> 9(2)n6 = [63]
-> r4c456 from [{19}6] or [{69}8]

10. (Ugh) Trying r4c456 = [{69}8]
puts r8c6 = 5
puts r3c56 = [72]
puts r12c6 = [96]
puts r2c45 = {15}
puts 12(2)r1 = {48} and r1c5 = 3
leaves no solution for 10(2)r1
-> r4c45 = {19} and r4c6 = 6

11. -> r6c3 = 1
-> 17(3)n6 = [{17}9]
-> r6c12 = {56}
-> r6c4 = 2
Also r4c1 = 8
Also 16(3)n4 = [3{49}]
-> 9(2)n4 = [72]

12. Also r3c56 = [68]
-> 8(2)n8 = [62]
-> r12c6 = [92]
-> r2c45 = {37}
Also r8c6 = 5
-> r5c6 = 3
Also 12(2)r1 = [84]
-> 10(2)r1 = [37]
-> 31(5)n7 = [43789]
Also r3c78 = [73]
-> r24c8 = [85]
-> 6(2)c9 = [42]
Also r7c78 = [87]
-> r5c89 = [17]
Also r78c2 = {12}
Also remaining innie r3 -> r3c1 = 2
-> 15(3)r3 = [951]
etc.


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