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Assassin 56

 
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CathyW
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PostPosted: Fri Jun 22, 2007 10:49 am    Post subject: Assassin 56 Reply with quote

Mad and Sad

Bother! Gone wrong somewhere - will have to start over later. Better get on with the day job in the meantime though.
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unkx80
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PostPosted: Fri Jun 22, 2007 4:16 pm    Post subject: Reply with quote

I guess, after applying naked subsets, locked candidates and some 45-tests, the key to solving this puzzle is:
Finding two large killer subsets, {6,7,8,9} in C3 followed by {1,2,3,4,5} in N4.
(I normally play killer sudokus using SumoCue with markups and configurations enabled.)
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Para
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PostPosted: Fri Jun 22, 2007 8:10 pm    Post subject: Reply with quote

Hi all

I think this walk-through typically shows how i tackle solving killers. Going through the basics of initial cage-combos and 45-test till i hit some conflicting combinations and run with it (and clean up properly), skipping the remaining 45-tests still in the grid. That is mostly the reason i sometimes don't use 45-tests to solve killers, just because there's something more interesting to work with (i like working cage combo's over 45-tests).

[edit]
Ok counted wrong on a 45-test. See that is why i rather not do 45 test. Crying or Very sad Feeling like a rookie again. I have to recheck my steps. Thanks Glyn.
It would be soooooo much handier when you make a mistake the puzzle actually turns up wrong instead off ending up with the proper solution.

[edit]
Here is the redraft with a little trick to replace the busted 45-test. This trick's pretty easy to see but even easier to overlook.

Walkthrough assassin 56

1. 23(3) in R1C2 = {689} -->> locked for N1

2. R1C456 = {123} -->> locked for R1 and N2

3. 19(3) in R1C7 and R6C7 = {289/379/469/478/568}: no 1

4. 22(3) in R4C1 = {589/679}: no 1,2,3,4; 9 locked for N4

5. 13(4) in R5C6 = {1237/1246/1345}: no 8,9

6. 26(4) in R8C4 = {2789/3689/4589/4679/5678}: no 1

7. 24(3) in R8C8 = {789} -->> locked for N9

8. 45 on R89: 2 innies: R8C19 = 3 = {12} -->> locked for R8

9. 45 on R123: 3 outies: R4C258 = 8 = {125/134}: no 6,7,8,9; 1 locked for R4

10. 45 on C12: 3 outies: R126C3 = 16 = {169/268}(only possible combinations) -->> R6C3 = {12}; R12C3 = {68/69} -->> 6 locked in R12C3 for C3

11. 45 on C1234: 2 innies: R18C4 = 10 = [19/28/37]: R8C4 = {789}

12. 45 on C89: 3 outies: R126C7 = 22 = {589/679}: no 2,3,4; 9 locked for C7
12a. 45 on C89: 1 innie and 1 outie: R6C7 = R1C8 + 3 -->> R6C7 = {789}; R1C8 = {456}
12b. R12C7 = {58/59/67/69} -->> 19(3) in R1C7 = {469/568}: no {478} clashes wih R12C7 -->> R12C7 = {58/69}: no 7; R1C8 = {46} -->> R6C7: no 8(step 12a); 6 locked in 19(3) for N3

13. 45 on C6789: 2 innies: R18C6 = 10 = [19/28/37] -->> R8C6 = {789}
13a. Naked Triple {789} in R8C468 -->> locked for R8

14. 45 on N1: 1 innie and 1 outie: R4C2 = R3C3 + 1 -->> R4C2 = {2345}; R3C3 = {1234}

15. 45 on N3: 1 innie and 1 outie; R3C7 = R4C8 + 1: R3C7: no 1,7,8; R4C8: no 5

16. 12(3) in R2C4 = {147/156/246/345}: no {129/138/237} needs 2 of {456789} in R23C4 -->> R3C3 = {123}(only place for {123}); R23C4 = {45/46/47/56}: no 8,9
16a. Clean up: R4C2: no 5(step 14); R4C5 and R4C8: no 2(step 9(When {125} R4C2 = 2(no {15}); R3C7: no 3(step 15)

17. 11(3) in R1C9 = {137}: {128} clashes with R8C9; {245} clashes with R3C7 -->> R1C9 = 7; R23C9 = {13} -->> locked for C9 and N3
17a. R8C9 = 2; R8C1 = 1

18. 11(3) in R1C1 = {245} -->> locked for C1 and N1
18a. Clean up: R4C2: no 3(step 14); R4C58: no 4(step 9(When {134} R4C2 = 4: no {13})); R3C7: no 5(step 15)

Little Trick on step 14, put it to replace the broken 45-test:
19. 45 on N1: “R4C2 = 2 -->> R3C3 = 1” blocked by R6C3 (R6C3 = {12}sees both cells, so they can’t contain both {12}) -->> R4C2: no 2; R3C3: no 1
19a. R4C2 = 4; R3C3 = 3; R23C9 = [31]; R23C2 = [17]
19b. Clean up: R23C4 = {45}(step 16) -->> locked for C4 and N2

Alternative step 19:
19. 12(3) in R6C1 = {38}[1]/[741]/[651]/{37}[2]: [642] blocked by R4C2 -->> R6C2: no 1,2,6
19a. 1 in N4 locked for C3
19b. R3C3 = 3; R4C2 = 4(step 14); R23C9 = [31]; R23C2 = [17]
19c. Clean up: R23C4 = {45}(step 16) -->> locked for C4 and N2

20. 45 on N4: 1 outie: R4C4 = 8
20a. 15(3) in R4C3 = 8{25}(last possible combination) -->> R45C3 = {25} -->> locked for C3 and N4
20b. R6C3 = 1; R8C3 = 4
20c. R12C3 = {69} (step 10) -->> locked for C3 and N1
20d. R1C2 = 8
20e. Clean up: R1C4: no 2(step 11); R2C7: no 5(step 12b); R4C5: no 5(step 9)

21. 12(3) in R6C1 = [831](last possible combination)

22. Naked Pair {78} in R79C3 -->> locked for N7

23. 11(3) in R7C8 = {45}2/[63]2 -->>R7C89 = {45}/[63]: R7C8: no 1,6
23a. 12(3) in R7C1 = [92]1: [65]1 clashes because R7C12 = [65] with R7C89 -->> R7C12 = [92]
23b. R5C2 = 9(hidden); R9C1 = 3(hidden)

Didn't see this whole cascade coming.
24. 19(3) in R6C7 = {469} (last possible combination) -->> R6C7 = 9; R6C89 = {46} -->> locked for R6 and N6
24a. R4C9 = 5; R5C9 = 8; R5C8 = 1; R4C8 = 3; R4C5 = 1; R45C3 = [25]; R45C7 = [72]
24b. R45C1 = [67]; R4C6 = 9; R3C7 = 4; R1C78 = [56]; R2C7 = 8; R12C3 = [96]
24c. R1C1 = 4; R23C4 = [45]; R23C1 = [52]; R23C8 = [29]; R2C6 = 7
24d. R3C6 = 6; R23C5 = [98]; R6C89 = [46]; R7C89 = [54]; R9C9 = 9; R8C6 = 8
24e. R89C8 = [78]; R8C4 = 9; R79C3 = [87]; R9C4 = 2; R6C4 = 7; R7C5 = 7(hidden)

And the last bit: let’s finish with a 45-test.
25. 45 on N9: 1 innie and 1 outie: R7C7 = R9C6 = 1(only common value)
25a. R89C7 = [36]; R89C2 = [65]; R89C5 = [54]; R7C46 = [63]; R1C456 = [132]
25b. R5C4 = 3; R56C5 = [62]; R56C6 = [45]


greetings

Para


Last edited by Para on Sat Jun 30, 2007 2:04 pm; edited 6 times in total
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CathyW
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PostPosted: Fri Jun 22, 2007 9:25 pm    Post subject: Reply with quote

2nd time OK Smile

In contrast to Para, I do use innies and outies as much as possible! Steps 20 and 21 should have been spotted earlier but this is in the order I did it.

1. 23(3) N1 = {689}, not elsewhere in N1

2. 6(3) r1c456 = {123}, not elsewhere in r1/N2

3. 22(3) N4 must have 9, not elsewhere in N4

4. 24(3) N9 = {789}, not elsewhere in N9

5. Innies r89: r8c19 = 3 = {12}, not elsewhere in r8

6. Outies r123: r4c258 = 8 = {125/134}, 1 not elsewhere in r4

7. Innies r6: r6c456 = 14

8. Outies r6789: r5c456 = 13 -> r4c456 = 18

9. Outies – Innies N1: r4c2 – r3c3 = 1 -> r4c2 <> 1, r3c3 = (1234)

10. O-I N3: r3c7 – r4c8 = 1 -> r3c7 = (23456)

11. O-I N4: r4c4 – r4c2 = 4 -> r4c4 = (6789)

12. O-I N6: r4c6 – r4c8 = 6 -> r4c8 = (123), r4c6 = (789) -> r3c7 = (234)

13. O-I N7: r7c3 – r9c4 = 6 -> r7c3 = (789), r9c4 = (123)

14. O-I N9: r9c6 – r7c7 = 0 -> r9c6 = r7c7 = (123456)

15. O-I c12: r1c2 – r6c3 = 7 -> r1c2 = (89), r6c3 = (12)
6 locked to r12c3, not elsewhere in c3
-> r2c7 <> 2 else conflict with r1c2

16. O-I c89: r6c7 – r1c8 = 3 -> r6c7 = (789), r1c8 = (456)

17. 19(3) in N3 can’t have 1. 14(3) r234c8 has max 3 in r4c8 -> no 1 in r23c8.
-> 11(3) r123c9 must have 1 in r23c9 (1 + {37/46}, (11(3) = {128} is blocked by r8c9) 1 not elsewhere in c9
-> r8c9 = 2, r8c1 = 1 -> r9c6 <> 2
-> 11(3) r123c1 = {245}, not elsewhere in N1/c1
-> r3c3 = (13), r23c2 = (137), 7 not elsewhere in c2 -> r4c2 = (24) -> r4c4 = (68)
-> r5c2 <> 8

18. Split 8(3) r4c258: If {125}, r4c5 <> 2; if {134}, r4c5 <> 4

19. Outies N2: r3c37 + r4c5 = 8 = [143/341/125] -> r3c7 <> 3 -> r4c8 <> 2 -> r4c6 <> 8
-> split 18(3) r4c456 = [657/639/819]

20. Innies c1234: r18c4 = 10 -> r8c4 = (789)

21. Innies c6789: r18c6 = 10 -> r8c6= {789}
-> NT {789} r8c468, not elsewhere in r8

22. 17(3) r23c6 + r3c7 = {269/278/458/467} -> r23c6 = (56789)

23. 12(3) r3c3 + r23c4 = {147/156/345} -> r23c4 = (4567)

24. 18(3) r234c5 = {189/369/378/549/567}

25. 14(3) r234c8 = {149/158/329/356} ({167/347} blocked by options for 11(3)) -> r23c8 = {29/49/56/58}, no 3 or 7
no 3 in r12c7 since max 6 in r1c8 of 19(3) (step 16)
-> 11(3) in N3 = {137}, not elsewhere in c9 -> r1c9 = 7

26. 14(3) in N6 = {149/158/248/356} -> r5c8 = (123)

27. Innies c5: r189c5 = 12 = {138/147/156/237/246/345} -> r9c5 = (45678)

28. 13(3) r89c3 + r9c4 = [391/382/481/472/571] -> r9c3 = (789)
-> NQ (6789) in r1279c3 -> other cells in c3 max 5.
-> NT (789) in r9c389 -> other cells in r9 max 6 -> 14(3) in N7 = {356}
-> 5 locked to r89c2, not elsewhere in c2 -> 22(3) in N4 = {679}
-> r8c3 = 4 -> r9c3 <> 9, r9c4 <> 3 -> r7c3 <> 9 -> NP 7/8 in r79c3 not elsewhere in c3 -> r1c2 = 8
-> r7c2 = 2, r7c1 = 9 -> r5c2 = 9
-> r9c1 = 3, r6c1 = 8, r6c2 = 3, r6c3 = 1, r4c2 = 4, r3c3 = 3 … several more singles.

Fairly straightforward from here.

Edit: Finally had time to clarify step 25 and make other minor edits. Thanks to Para and Glyn. Smile


Last edited by CathyW on Tue Jul 10, 2007 3:37 pm; edited 3 times in total
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Para
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PostPosted: Sat Jun 23, 2007 1:46 am    Post subject: Reply with quote

I didn't realise how many ways there are to use innies or outies or innie-outie differences in this cage-pattern till i tried to make a V2. Every time i try to make certain 45-tests less revealing another one pops up in the solver and the puzzle just gets less difficult.

greetings

Para
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Para
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PostPosted: Mon Jun 25, 2007 9:50 am    Post subject: Reply with quote

Hi all

Need something to do over the week? Here's a V2 i created. It's a tough one, but once you know where to look it will break slowly. When it does, you'll have a comfortable ride to the end




And a marks pic:



And here is the PS:

3x3::k:3584:3841:3841:2307:2307:2307:3590:3590:5384:3584:3850:3841:5132:3341:3598:3590:3088:5384:3584:3850:5132:5132:3341:3598:3598:3088:5384:3355:3850:4381:4381:3341:4640:4640:3088:3875:3355:3355:4381:4903:3624:5161:4640:3875:3875:3629:3629:3629:4903:3624:5161:4147:4147:4147:5942:5942:4903:4903:3624:5161:5161:2621:2621:5942:2624:3393:5698:5698:5698:4165:4678:2621:2624:2624:3393:3393:5698:4165:4165:4678:4678:

Hope you enjoy it.

greetings

Para
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sudokuEd
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PostPosted: Tue Jun 26, 2007 11:25 am    Post subject: Reply with quote

Para wrote:
It's a tough one...Hope you enjoy it.
Oh yeah. It's a ripper on both counts. You're fantastic at making these V2's Para. Thanks so much!

Cheers
Ed

[edit: Full walk-through in a following post: first 16 steps originally here have been revised for clarity]


Last edited by sudokuEd on Mon Jul 02, 2007 2:19 am; edited 1 time in total
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CathyW
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PostPosted: Thu Jun 28, 2007 12:16 pm    Post subject: Reply with quote

Quiet in here this week! I guess we've all been busy with day jobs and other commitments - I've only just got around to making amends on my walkthrough and no chance at all to look at Para's V2. Oh well - 57 to look forward to tomorrow. Very Happy
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Para
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PostPosted: Thu Jun 28, 2007 9:38 pm    Post subject: Reply with quote

Yeah well, it's my V2, can't really go and help you guys out with it. Too bad you've been too busy to do it. I hope your still gonna try it after number 57. Did some of Uwe's puzzles at the oher forum.

Para
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Andrew
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PostPosted: Fri Jun 29, 2007 2:36 am    Post subject: Reply with quote

CathyW wrote:
Bother! Gone wrong somewhere - will have to start over later. Better get on with the day job in the meantime though.

CathyW wrote:
Quiet in here this week! I guess we've all been busy with day jobs and other commitments.

I did just the same as Cathy! After unpacking and setting up my computer last night I started Assassin 56 and thought I was going to get it finished today before the guy came to install our cable TV/Internet and before Assassin 57 appeared. Then I found I'd made a mistake!

Chaos in our new house with boxes of stuff everywhere but it's good to be back on-line.

Andrew
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PostPosted: Sat Jun 30, 2007 5:57 am    Post subject: Reply with quote

Para wrote:
It would be soooooo much handier when you make a mistake the puzzle actually turns up wrong instead off ending up with the proper solution.

Guess I was luckier. My mistake did take me to an impossible position.

Second time was OK. The earlier parts of my walkthrough are more like Cathy's solution path than Para's one. There were several naked singles that I haven't included so I just added a general comment at the end of step 45.

1. R1C456 = {123}, locked for R1 and N2

2. 23(3) cage in N1 = {689}, locked for N1

3. R123C9 = {128/137/146/236/245}, no 9

4. 19(3) cage in N3 = {289/379/469/478/569}, no 1

5. 22(3) cage in N4 = 9{58/67}, 9 locked for N4

6. R6C789 = {289/379/469/478/569}, no 1

7. 10(3) cage at R8C7 = {127/136/145/235}, no 8,9

8. 24(3) cage in N9 = {789}, locked for N9
[Didn’t include the 11(3) cage as a step because the 24(3) cage does more eliminations.]

9. 13(4) cage at R5C6 = 1{237/245/345}, no 8,9

10. 26(4) cage in N8 = {2789/3689/4589/4679/5678}, no 1

11. 45 rule on N1 1 outie R4C2 – 1 = 1 innie R3C3 -> R4C2 = {234568}

12. 45 rule on N3 1 innie R3C7 – 1 = 1 outie R4C8 -> no 1 in R3C7, no 9 in R4C8

13. 45 rule on N7 1 innie R7C3 – 6 = 1 outie R9C4 -> R7C3 = {789}, R9C4 = {123}

14. 45 rule on N7 3 innies R789C3 = 19 = {289/379/469/478/569}, no 1

15. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 -> no 7 in R9C6

16. 45 rule on N4 1 outie R4C4 – 4 = 1 innie R4C2 -> R4C2 = {2345}, R4C4 = {6789}, clean-up: R3C3 = {1234} (step 11)

17. 45 rule on N6 1 outie R4C6 – 6 = 1 innie R4C8 -> R4C6 = {789}, R4C8 = {123}, clean-up: R3C7 = {234} (step 12)

18. 45 rule on R89 2 innies R8C19 = 3 = {12}, locked for R8

19. 11(3) cage in N9 = {146/236/245}, R8C9 = {12} -> no 1,2 in R7C89

20. 45 rule on C12 1 innie R1C2 – 7 = 1 outie R6C3 -> R1C2 = {89}, R6C3= {12}
20a. 6 in N1 locked in R12C3, locked for C3

21. R6C123 = {138/147/237/246} (cannot be {156} which clashes with 22(3) cage), no 5
21a. R6C3 = {12} -> no 1,2 in R6C12

22. 45 rule on C89 1 outie R6C7 – 3 = 1 innie R1C8 -> R1C8 = {456}, R6C7 = {789}

23. 19(3) cage in N3 max R1C8 = 6 -> min R12C7 = 13, no 2,3 in R2C7

24. 45 rule on N2 3 outies R3C37 + R4C5 = 8, min R3C37 = 3 -> max R4C5 = 5
24a. 45 rule on R123 3 outies R4C258 = 8 = {125/134}, 1 locked for R4

25. R234C8 = {149/158/167/239/248/257/347/356}
25a. Max R4C8 = 3 -> min R23C8 = 11, no 1

26. 1 in N3 locked in R23C9, locked for C9 -> R8C9 = 2, R8C1 = 1, clean-up: no 2 in R9C6 (step 15)
26a. R123C9 (step 3) = 1{37/46} (only remaining combinations), no 5,8
26b. 1,3 only in R23C9 -> no 7 in R23C9
26c. 1 in N9 locked in R79C7, locked for C7

27. 11(3) cage in N9 = 2{36/45} -> R789C7 = 1{36/45} -> 10(3) cage at R8C7 = 1{36/45} from R7C7 = R9C6 (step 15)
27a. 1 in 10(3) cage at R8C7 locked in R9C67, locked for R9, clean-up: no 7 in R7C3 (step 13)

28. Killer pair 8/9 in R12C3 and R7C3, locked for C3

29. 13(3) cage at R8C3 = {247} (only remaining combination) -> R9C4 = 2, R89C3 = {47}, locked for C3 and N7, clean-up: R7C3 = 8 (step 13), no 5 in R4C2 (step 11), no 9 in R4C4 (step 16)

30. R12C3 = {69} -> R1C2 = 8, clean-up: R6C3 = 1 (step 20), no 2 in R4C2 (step 11), no 6 in R4C4 (step 16)

31. 2 in N7 locked in R7C12 -> R7C12 = {29} (only remaining combination for 12(3) cage in N7), locked for R7 and N7

32. 5 in C3 locked in R45C3, locked for N4, clean-up: no 8 in 22(3) cage in N4 (step 5)
32a. 22(3) cage in N4 = {679}, locked for N4

33. R6C123 = [831] (only remaining permutation) -> R4C2 = 4, R3C3 = 3 (step 11), R4C4 = 8 (step 16), clean-up: no 5 in R1C8 (step 21), no 2 in R4C8 (step 17)

34. R23C2 = {17} (only remaining combination for R234C2), locked for C2 and N1

35. R123C1 = {245}, locked for C1 -> R7C12 = [92]

36. R45C1 = {67}, locked for C1 -> R5C2 = 9, R9C1 = 3, clean-up: no 3 in R7C7 (step 15)

37. 10(3) cage at R8C7 (step 27) = 1{36/45}
37a. 3 only in R8C7 -> no 6 in R8C7

38. R4C2 = 4 -> R4C58 = {13} (step 23a), locked for R4

39. R123C9 = [731] (only remaining permutation, {146} clashes with R1C8) -> R23C2 = [17]

40. R3C3 = 3 -> R23C4 = 9 = {45} (only remaining combination), locked for C4 and N2

41. Naked triple {245} in R3C147, locked for R3

42. 24(4) cage at R5C4 = {1689/3678} = 68{19/37}
42a. 6 locked in R567C4, locked for C4
42b. Killer pair 1/3 in R1C4 and R57C4, locked for C4

43. 7 in R7 locked in R7C456, locked for N8 -> R8C4 = 9

44. R567C4 (step 42) = {367}, locked for C4 -> R1C4 = 1

45. R6C789 = {469} (only remaining combination) -> R6C7 = 9, R6C89 = {46}, locked for R6 and N6 -> R6C4 = 7, R4C6 = 9, R6C56 = {25}, locked for N5, R45C9 = [58], R45C3 = [25], R4C7 = 7, R45C1 = [67], clean-up: R4C8 = 3 (step 17), R4C5 = 1, R3C7 = 4 (step 12), R23C4 = [45], R123C1 = [452], R1C8 = 6, R12C3 = [96], R6C89 = [46], R7C8 = 5, R7C9 = 4, R8C7 = 3, clean-up: no 4,5 in R9C6 (step 15) and a few more naked singles

46. R2C8 = 2, R5C8 = 1 (hidden singles in C8) -> R3C8 = 9 (cage sum), R9C9 = 9

47. R4C5 = 1 -> R23C5 = 17 = [98]

and the rest is naked singles
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Andrew
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PostPosted: Sat Jun 30, 2007 8:23 pm    Post subject: Reply with quote

Assassin 56 was only the 3rd V1 that didn't have any 2-cell cages.

The two earlier ones were Assassin 2, which was also notable because all its cages had 3 cells, and Assassin 29.

In contrast our next puzzle Assassin 57 has 20 2-cell cages plus a "present" but I've no doubt that it will still be an Assassin.

I've read on the forum on another website that the way to make killers harder is not to have 2-cell cages. That seems to be a very simplistic view not supported by the statistics in this message.

We know from variants on this forum that a puzzle can be changed from V1 standard to V2 or even harder by changing the cage totals while keeping the same cage pattern. Ruud has posted some excellent examples of this.
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Joined: 19 Jun 2006
Posts: 257
Location: Sydney Australia

PostPosted: Mon Jul 02, 2007 2:16 am    Post subject: Reply with quote

Here is the full Walk-through for Assassin 56V2. A very, very difficult puzzle with lots of contradiction moves and extensive combination conflicts. Had to restart many times to get a valid solution. Must have spent 15 hours on this one Very Happy. Perfect for a V2. Thanks again Para.

Please let me know of any corrections or clarifications needed. The original steps 1-16 have been revised for clarity.

Cheers
Ed

Assassin 56 V2
1. 23(3)n7 = {689}: all locked for n7

2. 10(3)n7 = {127/145/235}(no 8,9)

3. "45" n7: r789c3 = 12 = h12(3)n7
3a. = {147/237/345}

4. "45" n36: r4c6 - 6 = r3c7
4a. r4c6 = {789}, r3c7 = {123}

5. "45" n3: r4c8 - 2 = r3c7
5a. r4c8 = {345}

6. 20(3)n2: no 1,2

7. "45" n1: r4c2 + 1 = r3c3
7a. r4c2 = 2..8

8. "45" r123: r4c258 = 12 = h12(3)r4
8a. min. r4c8 = 3 -> max. r4c25 = 9
8b. min. r4c2 = 2 -> max. r4c5 = 7

9. "45" n4: r4c4 + 1 = r4c2
9a. r4c4 = 1..7

10. "45" n6: r4c6 - 4 = r4c8
10a. r4c68 = [73/84/95]

11. h12(3)r4 = {147/246/345}. Others blocked. Here's how.
i. r4c258 = {138}: blocked since can only be = [813], but including "45" moves for r4c4 (1 less than r4c2 step 9) & r4c6 (4 more than r4c8 step 10) = [87173]: but this means 2 7's r4
ii.{147} = [714] only
iii. {156}: blocked since can only be [615] but including "45" moves for r4c4 & r4c6 = [65195]: but this means 2 5's r4
iv. {237}: blocked since both [273/723] require 7 in r4c6 but this means 2 7's r4
v. {246} = [264/624]
vi. {345} = [345/354]. Others are blocked. [435/453] by 3 required in r4c4; [534/543] by 4 required in r4c4

12. In summary: h12(3) r4 = {147/246/345} = 4{..}
12a. 4 locked for r4
12b. = [714/264/624/345/354]
12c. r4c2 = {2367} -> r3c3 = {3478} (step 7) and r4c4 = {1256}(step 9)
12d. r4c5 = {12456}(no 3,7)
12e. r4c8 = {45} -> r4c6 = {89}(step 10) & r3c7 = {23}(step 5)

13. Hidden-cage in r345c3 & remembering that r789c3 = h12(3)n7 = {147/237/345}
13a. r45c3 "sees" r3c3, r4c2 and r4c4: these 3 are all linked through "45" moves and = [321/432/765/876]
13b. r3c3 + r4c2 + r4c4 = [321]
i. -> r45c3 = 16 = {79} blocked: r345c3 = [3]{79}: clashes with h12(3)n7
13c. r3c3 + r4c2 + r4c4 = [432]
i. -> r45c3 = 15 = {69} -> r345c3 = [4]{69}
ii............... = 15 = {78} blocked: r345c3 = [4]{78} but clashes with h12(3)n7
13d. r3c3 + r4c2 + r4c4 = [765]
i.-> r45c3 = 12 = {39} blocked: r345c3 = [7]{39}: clashes with h12(3)n7
ii............. = 12 = {48} blocked: r345c3 = [7]{48}: clashes with h12(3)n7
iii............ = 12 = {57} blocked by 5 in r4c4
13e. r3c3 + r4c2 + r4c4 = [876]
i. -> r45c3 = 11 = {29} -> r345c3 = [8]{29}
ii................= 11 = {38}: blocked by 8 in r3c3
iii...............= 11 = {47}: blocked by 7 in r4c2
iv................= 11 = {65}: blocked by 6 in r4c4

14. In summary: r345c3 = [4]{69}/[8]{29} = 9{..}
14a. 9 must be in r45c3: 9 locked for c3, n4 and 17(3)n4
14b. r45c3 = 9{26}
14c. r4c4 = {26}
14d. r4c2 = {37}
14e. r3c3 = {48}

15. 20(3)n2 = r3c3 + r23c4 = [4]{79}/[8]{39/57} (no 4,6,8 r23c4)
15. = {389/479/578}

16. from step 12b. h12(3)r4 = [714/345/354] = {147/345}
16a. r4c5 = {145}

Continuing on.
17. from step 16: remembering r4c4 is 1 less than r4c2 & r4c6 is 4 more than r4c8
17a. -> r4c456 = [618/249/258]

18. "45" n4: 3 innies = 18 = h18(3)n4 = 9{36/27}
18a. 14(3) n4 = {158/248/257/356} ({167/347} blocked by h18(3)n4)

19. 17(3)n4 = {269}
19a. -> no 2,6,9 r4c1

20. "45" r6: 3 innies r6c456 = 15 = h15(3)r6
20a. = {168/249/357} ({159/258/267/348/456} all blocked by r4c456! step 17a)

Now: moving across to n36 for the same tricks.
21. r3c7 + 6 = r4c6: r4c6 - 4 = r4c8 -> {r45c7} = 10/9
21a. [r3c7 = 2][r4c6 = 8][r4c8 = 4] = [284]
i. -> r45c7 = 10 = {19/37}
21b. [r3c7 = 3][r4c6 = 9][r4c8 = 5] = [395]
i. -> r45c7 = 9 = {18/27}
21c. In summary: r45c7 = {18/19/27/37}(no 456)
21d. 18(3)n5 = {189/279/378}

22. "45" n6: 3 innies = 14 = h14(3)n6 = {149/347/158/257}

Now for lots of combo. crunching.
23. 16(3)n6 = {169/349/367}(no 2,5,8). Others blocked. Here's how.
23a. {178/457} blocked by h14(3)n6 step 22
23b. {268/259/358} blocked by 14(3)n4 (step 18a.)

24. 14(3)n4 = {158/248/257}(no 3,6) ({356} blocked by 16(3)n6 step 23.)

25. deleted

26. Generalized X-wing on 2 required in 17(3)n4 (only in r45) & 2's in n6 (only in r45)
26a. -> no 2 elsewhere in r45

27. 13(3)n4 = {148/157/346}

28. "45" r12345: r5c456 = 15 = h15(3)r5 (note: h15(3)r6 shows all the blocked combinations for 15(3) by r4c456. See step 20a)
28a. r5c456 = {168/357}(no 4,9)
28b. = [1/5,1/7..]
28c. -> no 5 & 7 in r4c1 as 13(3) must be {157}
28d. but r5c12 = {17/15} will clash with h15(3)r5 (step 28b)
28e. no 5,7 r4c1

29. h15(3)r5 = [5/8..](step 28a)
29a. -> from step 17 r4c456 = [618/249](no 5) ([258] blocked by h15(3)r5)

30. 5 in r4 only in n6
30a. 5 locked for n6

31. 13(3)n4:{346} combo is the only combo with 3
31a. -> no 3 r5c12


32. 3 in n4 only in r4
32a. -> 3 locked for r4

Now to n1.
33. 15(3)r2c2 must have 3/7 in r4c2
33a. = {267/357}(no 1,4,8,9) ({348} blocked by 4 required in r3c3 when r4c2 = 3 step 7)
33b. = 7{26/35}
33c. 7 locked for c2

34. "45" n1: 3 innies = 16 = h16(3)
34a. r3c3 + r23c2 = [4]{57}/[8]{26/35}
34b. = {268/358/457}

35. 15(3)r1c2 = {159/168/249/267/357} ({258/348/456} blocked by h16(3)n1 step 34b.)

36. "45" c12: r126c3 = 14 = h14(3)c3 & remembering h12(3)n7 = {147/237/345}(step 3a)
36a. h14(3)c3 = {158/167/356} (no 2,4) ({248} blocked by r3c3; {257/347} blocked by h12(3)n7)

37. "45" c12: r6c3 + 1 = r1c2
37a. r1c2 = {2,6,8,9}

38. 15(3)r1c2 = {159/168/267}(no 3) ({357} blocked by no candidates in r1c2)

39. 3 in c7 only in n7
39a. 3 locked for n7
39b. h12(3)n7 must have 3 = 3{27/45}(no 1)

40. from step 34b. h16(3)n1 = {268/358/457}
40a. ->14(3)n1 = {149/239/347}(no 5,6,8) ({158/248/257/356} blocked by h16(3)n1; {167} blocked by 15(3)r1c2)

41. {346} combo. blocked from 13(3)n4. Here's how.
41a. {346} combo. -> r4c1 = 3 -> 14(3)n1 = {149} -> r5c1 = 6: but {69} r1235c1 clashes with r78c1.
41b. 13(3)n4 = {148/157} (no 3,6) = 1{..}
41c. 1 locked for n4
41d. -> no 2 r1c2 ("45" c12)

42. r4c2 = 3 (hsingle n4)
42a. r4c4 = 2 ("45" n4)
42b. r3c3 = 4 ("45" n1)

43. r23c2 = {57}: both locked for n1 & c2

44. r45c3 = {69}: both locked for c3

45. r12c3 = {18}: both locked for n1 & c3
45a. r1c2 = 6
45b. r6c3 = 5 ("45" c12)

46. 14(3)n1 = {239}: all locked for c1

47. r78c1 = {68}: both locked for c1 & n7
47a. r7c2 = 9

48. r6c2 = 2, r9c1 = 5 (hsingles)

49. 13(3)n4 = [148](only valid combo)

50. r6c1 = 7, r4c58 = [45]

51. r4c6 = 9, r3c7 = 3 ("45" n3,n6)

52. r45c3 = [69]

53. r23c4 = {79}: both locked for n2, c4

54. r23c6 = {56}: both locked for n2,c6

55. 9(3)n2 = {234}: all locked for n2 & r1

56. r23c8 = {16}: both locked for c8 & n3

57. "45" c89: r6c7 - 2 = r1c8
57a. r6c7 = 9, r1c8 = 7
57b. r12c7 = [52]

The rest is naked or last valid combo.
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