## Assassin 55

Our weekly <a href="http://www.sudocue.net/weeklykiller.php">Killer Sudokus</a> should not be taken too lightly. Don't turn your back on them.
CathyW
Master
Posts: 161
Joined: Wed Jan 31, 2007 5:39 pm
Location: Hertfordshire, UK
Assassin 55 V2 continuation

40. Ruling out 11(2) in N8/c5 = {38}

a) If 11(2) in N8/c5 = {38}, 6(2) = {24} -> 3,8 not elsewhere in N8, 2,4 not elsewhere in N2
-> r6c5 = 1
-> r4c46 <> 5
-> 8 locked to r4c46 not elsewhere N5/r4
-> r5c7 of 13(3) <> 5.
-> r4c4 <> 5 (additional elimination)

b) 11(3) r1c34+r2c3 = {128/137/146/236/245}. If {245}, 5 must go in r1c4 -> r12c3 <> 5.

c) Hidden pair {23} in N5 -> r5c4, r6c6 <> 4,5
-> r4c6 <> 6,7

d) O-I of N9 = 0: r7c7 <> 3 (no 1 or 2 in r6c9); r9c6 <> 7 (min 3 in r6c9)

e) 27(5) in r34 can’t have both 8 and 9 as would conflict with r4c6 -> remaining options: {14679/15678/23679/24579/24678/34569/34578}

f) All 3 of r2 locked in split cage 17(4) r2c2468
Options: {1358/1367/2357}
-> r2c6 <> 9
-> 9 locked to r3c46 -> r3c3 <> 9 -> 20(3) in N1 must have 9.

g) If r5c9 = 8, r1c8 = 8 -> r1c1 <> 8
If r5c9 = 9, r5c5 = 6, r5c1 = 8 -> r1c1 <> 8
Either option r1c1 <> 8

h) 17(4) r6c456 + r7c5 = 1 + {259/349/367}
If {259} r5c5 = 6; if {349} r5c5 = 6; if {367} r5c5 = 9 -> Killer pair of 6 and 9 -> r4c5 <> 6,9

i) If r5c9 = 8, one of r46c7 = 9 -> r7c7 <> 9
If r5c9 = 9, r5c5 = 6, r7c5 = 9 -> r7c7 <> 9
Either option r7c7 <> 9
-> r9c6 <> 6

j) Options for O-I N9 (r6c9 + r9c6 = r7c7)
If r7c7 = 4, r6c9 + r9c6 = [31]
If r7c7 = 5, r6c9 + r9c6 = [41] ([32] blocked by r6c6)
If r7c7 = 6, r6c9 + r9c6 = [42]
If r7c7 = 7, r6c9 + r9c6 = [34]
If r7c7 = 8, r6c9 + r9c6 = [35/71]

Edit - got a bit further and have now reached conflict which would rule out 11(2) c5 = {38}

k) UR move - 6(2) N2 = {24} -> r12c7 can't both be 2,4 -> r1c6 <> 7
-> options for 13(3) = {157/625} must have 5, not elsewhere in N3/c7
-> rules out r6c9 = 4, r9c6 = 1 from j above
-> r12c7 <> 4 -> 4 locked to r3c79, not elsewhere in r4.

l) UR move - 6(2) N2 = {24} -> r12c3 can't both be 2,4 -> r1c4 <> 5

m) 17(3) c9 = [278/458/719] -> r4c9 <> 3,4

n) 15(3) c1 = {159/168/249/258/267/348/456}
If {348}, r4c1 = 4 -> r4c1 <> 3

o) Resorted to JSudoku which found a Grouped Turbot Fish (would never have found this on my own! ) Hopefully I have the notation correct for the loop:
[r6c6]=3=[r23c6]-3-[r4c7]=3=[r6c79]-3-[r6c6]
-> r6c123 <> 3

Thanks to a tip from Glyn I think we've now reached conflict situation to prove the 11(2) in N8/c5 <> {38}
p) If r5c1 = 6 -> r5c5 = 9, r5c9 = 8, r4c6 = 8, r6c6 = 3, r4c5 = 5, r5c4 = 2, r5c6 = 4 CONFLICT no options for 9(3) in r5.
-> r5c1 <> 6
-> r5c5 = 6 -> r7c5 = 9 - NP {89} in r4c46 -> r4c23 <> 9
-> r3c1 <> 8, r34c1 <> 7

q) 31(5) in N478 must have 9 which is locked to r6c23, not elsewhere in N4/r6 -> r5c1 = 8 leads to several singles but then CONFLICT as no place for 3 in c6.

Thus, if 11(2) in N8/c5 = {38} leads to conflict -> 11(2) = {29} ...

Phew! With just a few hours to spare before Assassin 56 is released.

Reaching solution with 11(2) in N8/c5 = {29}

41. If 11(2) = {29}, 6(2) = {15}, r345c5 = {678}, r67c5 = {34}
-> 2, 9 not elsewhere in N8, 1,5 not elsewhere in N2
-> r4c46 = {59} not elsewhere in N5/r4
-> r6c6 = 2, r6c4 = 8 -> r4c6 = 9, r4c4 = 5, r5c5 = 6, r4c5 = 7, r3c5 = 8
-> 1 locked to r5 in N5, r5c237 <> 1

42. 27(5) in r34 must have 8 -> r4c7 = 8
-> r5c9 = 9, r5c1 = 8, r1c8 = 9

43. 7 locked to r5c78 -> r6c9 <> 7

44. HS r3c4 = 9

45. 17(3) in c9 = [53/71]9
-> 2 locked to r789c9, not elsewhere in N9
-> 4 locked to r123c7, not elsewhere in c7

46. 13(3) r1c67 + r2c7 = {247/256} -> 2 locked to r12c7 -> r3c7 <> 2
-> 2 locked to r3c23, not elsewhere in N1 -> r3c2 <> 6
-> r4c2 <> 2 (cell ‘sees’ both r3c23)

47. HS r2c1 = 9 -> 9 locked to r6c23 -> r7c3 <> 9 -> cage 31(5) must have 9.

48. Split 17(4) in r2c2468 = {1367/1457/2357} must have 7 -> r2c37 <> 7

49. 4 locked to r46c8 + r6c9 in N6. r78c8 ‘sees’ all these -> r78c8 <> 4

50. Options for 12(3) r8c7 + r9c67 = {138/147/156/345} – no 9 -> r7c7 = 9
-> r6c9 = (134), r9c6 = (568)
-> 12(3) = {138/156}no 7 and must have 1 within r89c7, not elsewhere in N9/c7

51. 12(3) r8c9 + r9c89 = {237/246/345}. Combo analysis: r89c9 <> 7.

52. 16(3) r89c1 + r9c2, Max from r89c1 = 13 -> r9c2 <> 1,2

53. Naked Quad {2457} in r1235c7 -> r689c7 <> 5, r9c6 <> 6, r6c9 <> 3

54. xy wing with r6c9 as pivot -> r6c7 <> 3 -> r6c7 = 6
-> r4c9 = 3, r6c9 = 1, r3c9 = 5, r9c6 = 8 …

Fairly straightforward from here to solution.

Last edited by CathyW on Thu Jun 21, 2007 8:34 pm, edited 4 times in total.
sudokuEd
Grandmaster
Posts: 257
Joined: Mon Jun 19, 2006 11:06 am
Location: Sydney Australia
Wow Cathy - you have been busy. Haven't looked at your steps, but I think you have solved it. Looks like you found the only way to solve this one! Good old tri-furcation . I spent hours trying different innies - and even ventured into the left side .

But... just don't quite feel ready to congratulate us yet - need to be absolutely convinced there is no other way. Will try to find some "nice loops" from where we were yesterday. Hope you don't mind if I stay back-tracked for a bit longer.

BTW: I really liked your summary (step 38) of my hypothetical. It showed that 8 in r345c5 -> {15} in r12c5 -> 5 in r4c46. But couldn't find a way to use it.

Seems I have a couple more sessions with this puzzle to feel satisfied.

Cheers
Ed
CathyW
Master
Posts: 161
Joined: Wed Jan 31, 2007 5:39 pm
Location: Hertfordshire, UK
sudokuEd wrote:But... just don't quite feel ready to congratulate us yet - need to be absolutely convinced there is no other way. Will try to find some "nice loops" from where we were yesterday. Hope you don't mind if I stay back-tracked for a bit longer.
I don't mind at all I will be very interested if you (or anyone else) can find a way without trifurcation - albeit not yet complete on proving conflict with 11(2) = {38}

Edit: Conflict now reached - see above.
Last edited by CathyW on Thu Jun 21, 2007 8:36 pm, edited 1 time in total.
sudokuEd
Grandmaster
Posts: 257
Joined: Mon Jun 19, 2006 11:06 am
Location: Sydney Australia
Getting a bit excited. Found a really nice lead to follow-up but no time now. Will post the start - can someone check it? Keep going if you want .

Starts at step 55 but follows on from step 38.
55.No 5 in r6c7 because of 7's in r5. Here's how.
55a. 7 in r5c6 -> h13(3)n6 = {256} -> 5 locked for n6
55b. 7 in r5c7 -> h13(3)n6 = {256} -> 5 locked for n6
55c. 7 in r5c8 -> r46c8 in h13(3)n6 = {247}-> r5c7 = [1/5],r6c9 = [1/3] and r4c9 = [1/3/5/]: naked triple 1/3/5/ for n6
55d. -> no 5 in r6c7

56. no 4 r456c7. Here's how.
56a. 4 in r456c7 -> 4 in n3 in r3c9 -> r4c9 = 5 (only combination)
56b. 4 in r456c7 -> h13(3)n6 = {256}
56c. But this means 2 5's in n6
56d. -> no 4 r456c7

57. no 7 r7c7. Here's how.(Actually quite proud of this one!)
57a. r789c8 must have 1 of 8/9 for c7 because of r1c8 -> r789c7 must have 1 of 8/9 for c7.
57b. if 8/9 in r89c7 then 12(3) cage = {129/138} only
57c. -> r9c6 = {123} and r7c7 = 3..7
57d. if r7c7 = 7 -> 2 outies n9 = 7 = [43]
57e. -> r89c7 = {18}
57f. 7 in r7c7 -> naked quad 1/2/4/5/ in r1235c7
57g. but this forces 2 1's into c7
57h. of course, if 8/9 is in r7c7 for c7 then r7c7 !=7
57i. -> r7c7 !=7 !!

Why stop there? Have to be careful though - mistakes happen so easily.
58. When 4 in r7c7.
58a..c. same as step 57
58d. if r7c7 = 4 -> 2 outies n9 = 4 = [13/31]
58e. but [13] is blocked the same way as step 57.(r89c7 = {18} and 4 in r7c7 -> naked quad 1/2/5/7 in r123c7 :Clash with r89c7)
58f. but [31] can have r89 = {38} (though {29} is blocked as above)

59. same trick with 5 in r7c7. This time the outies n9 = 5 = [32/41]
59a. [32] is blocked by {19} in c7: clash with 1 is naked quad
59b. [41] can have r89 = {38}

Don't have time to work out what happens for 3 and 6 yet: or what the outies will be with r7c7 = 8/9.
Should be at this marks pic: copy-paste into SudoCue

Code: Select all

``````.-----------------------.-----------------------.-----------.-----------------------.-----------------------.
| 3456789     3456789   | 12345678    12345678  | 1245      | 1246        2457      | 89          689       |
|           .-----------&#58;           .-----------&#58;           &#58;-----------.           &#58;-----------.           |
| 456789    | 123567    | 1245678   | 12345678  | 1245      | 123456789 | 2457      | 13        | 689       |
&#58;-----------&#58;           &#58;-----------'           &#58;-----------&#58;           '-----------&#58;           &#58;-----------&#58;
| 12345678  | 123567    | 123456789   123456789 | 5678      | 123456789   2457      | 13        | 2457      |
|           &#58;-----------'           .-----------'           '-----------.           '-----------&#58;           |
| 12345678  | 123456789   123456789 | 56789       56789       56789     | 13689       2456      | 13457     |
|           &#58;-----------------------'-----------.           .-----------'-----------------------&#58;           |
| 689       | 12345       12345       12345     | 689       | 1457        157         257       | 89        |
&#58;-----------+-----------------------.-----------'-----------'-----------.-----------------------+-----------&#58;
| 123456789 | 123456789   123456789 | 45678       1234        23456     | 13689       2456      | 1347      |
|           '-----------.           '-----------.           .-----------'           .-----------'           |
| 123456789   123456789 | 123456789   123456789 | 234679    | 123456789   345689    | 456789      123457    |
&#58;-----------.           &#58;-----------.           &#58;-----------&#58;           .-----------&#58;           .-----------&#58;
| 123456789 | 123456789 | 123456789 | 123456789 | 234789    | 123456789 | 123456789 | 456789    | 123457    |
|           '-----------&#58;           '-----------&#58;           &#58;-----------'           &#58;-----------'           |
| 123456789   123456789 | 123456789   123456789 | 234789    | 123456789   123456789 | 45679       123457    |
'-----------------------'-----------------------'-----------'-----------------------'-----------------------'``````
Andrew
Grandmaster
Posts: 300
Joined: Fri Aug 11, 2006 4:48 am
Location: Lethbridge, Alberta
Just managed to finish V1 before we move from Calgary to Lethbridge, about 2 hours drive south of Calgary. Will be packing up my computer as soon as I've posted this message and won't be on-line again for about a week so I don't know when I'll get to try Assassin 56.

When I first saw the comments in this thread, I was expecting this puzzle to be harder than it turned out to be. It was still a very long solution so, while it was a V1, it was definitely one of the harder ones. I hope that Ruud keeps up his excellent mixture of hard puzzles like this one and slightly easier ones.

I've worked through Para's and Mike's walkthroughs this evening but haven't had time for more than a glance at Cathy's walkthrough. Sorry Cathy! I'll look through it when my computer is in the new home.

[Edit. As promised I've now worked through Cathy's walkthrough, after our move, apart from the multi-colouring and nice loops which are techniques that I haven't yet learned.

Ed commented that Cathy and Mike were the only ones that found the narrow solution path. Not sure if I did.]

Here is my walkthrough. There is one heavy combination analysis step with a summary at the end for those who don't want to work through the details. Initially this step was even heavier but then I spotted the locked 5 in C5 which simplified the step and provided further eliminations.

1. R12C5 = {49/58/67}, no 1,2,3

2. R23C2 = {16/25/34}, no 7,8,9

3. R23C8 = {49/58/67}, no 1,2,3

4. R89C5 = {19/28/37/46}, no 5

5. 10(3) cage in N1 = {127/136/235} (cannot be {145} which clashes with R23C2), no 4,8,9

6. 19(3) cage at R1C3 = {289/379/469/478/568}, no 1

7. R345C1 = {489/579/678}, no 1,2,3

8. R5C678 = {128/137/146/236/245}, no 9

9. 22(3) cage at R8C7 = 9{58/67} -> no 9 in R9C89

10. 18(3) cage in N9 = {189/279/369/378/459/468/567}
10a. 9 only in R8C9 -> no 1,2 in R8C9

11. 19(5) cage at R6C2 = 1{2349/2358/2367/2457/3456}

12. 45 rule on R5 3 innies R5C159 = 22 = 9{58/67}, 9 locked for R5

13. 45 rule on N9 2 outies R6C9 + R9C6 – 14 = 1 innie R7C7
13a. Min R6C9 + R9C6 = 15 -> no 1,2,3,4,5 in R6C9 and R9C6
13b. Max R6C9 + R9C6 = 18 -> max R7C7 = 4

14. 45 rule on R89 4 innies R8C2468 = 13 = 1{237/246/345}, no 8,9, 1 locked for R8, clean-up: no 9 in R9C5

15. 45 rule on C12 3 innies R456C2 = 19 = {289/379/469/478/568}, no 1

16. 45 rule on C89 3 innies R456C8 = 7 = {124}, locked for C8 and N6, clean-up: no 9 in R23C8

17. 45 rule on C1234 2 innies R46C4 = 17 = {89}, locked for C4 and N5
17a. 19(3) cage at R1C3 (step 6) = {289/379/469/478/568}, 8,9 only in R12C3 -> no 2 in R12C3

18. 45 rule on C6789 2 innies R46C6 = 3 = {12}, locked for C6 and N5

19. 19(4) cage at R6C4 has R6C4 = {89}, R6C6 = {12}, valid combinations {1279/1369/1378/1459/1468/2359/2368/2458} -> no 8,9 in R7C5

20. R5C159 (step 12) = 9{58/67}
20a. 8,9 only in R5C19 -> no 5 in R5C19

21. R5C678 (step 8) = {137/146/236/245} (cannot be {128} because 1,2 only in R5C8), no 8
21a. 1,2 only in R5C8 -> no 4 in R5C8
21b. 4 only in R5C6 -> no 5 in R5C6

22. R5C234 = {138/147/237/246} (cannot be {156/345} which clash with all combinations in R5C678), no 5
22a. 1 only in R5C3 -> no 8 in R5C3

23. 5 in N5 locked in R456C5, locked for C5, clean-up: no 8 in R12C5

24. 45 rule on N1 2 innies R3C13 – 9 = 1 outie R1C4
24a. Min R1C4 = 2 -> min R3C13 = 11, no 1 in R3C3

25. 45 rule on C9 2 outies R19C8 – 6 = 2 innies R67C9
25a. Min R67C9 = 7 -> min R19C8 = 13 -> no 3 in R19C8

26. 3 in C8 locked in R78C8, locked for N9

27. 19(4) cage at R6C9 = 3{169/178/259/268/457}
27a. 1,2,4 only in R7C9 and each combination requires 1/2/4 -> R7C9 = {124}

28. 18(3) cage in N9 = {189/279/459/468/567}
28a. If 18(3) cage = {468/567} there must be a 9 in R89C7 (if R9C6 = 9 R89C7 = {58/67} which would clash with 18(3) cage) -> no 9 in R7C8

29. If R8C9 = 9, R6C9 <> 9
If R8C9 <>9, 9 in R89C7 (step 27a) -> R9C6 <>9
-> R6C9 + R9C6 <> 18 -> no 4 in R7C7 (step 13)

30. 4 in N9 locked in R789C9, locked for C9

31. 23(5) cage at R3C5 has R4C4 = {89}, R4C6 = {12}, consider the options
31a. R4C46 = [81] -> R345C5 = 14 = {257/347/356} (cannot be {149/158/167/248} which clash with R4C46, cannot be {239} because 2, 9 only in R3C5), also R6C46 = [92] -> R67C5 = 8 = [53/62/71]
31aa. R345C5 = {257} clashes with R67C5
31ab. R345C5 = {347} clashes with R12C5
31ac. R345C5 = {356} -> R67C5 = [71] -> R12C5 = {49} -> R89C5 = {28}
31b. R4C46 = [82] -> R345C5 = 13 = {157/346} (cannot be {148/238/247/256} which clash with R6C46, cannot be {139} because 1,9 only in R3C5), also R6C46 = [91] -> R67C5 = 9 = {36}/[54/72]
31ba. R345C5 = {157} -> R67C5 = {36} -> R12C5 = {49} -> R89C5 = {28}
31bb. R345C5 = {346} clashes with R12C5
31c. R4C46 = [91] -> R345C5 = 13 = {247/256/346} (cannot be {139/148/157} which clash with R6C46, cannot be {238} because 2,8 only in R3C5), also R6C46 = [82] -> R67C5 = 9 = {36}/[54]
31ca. R345C5 = {247} clashes with R12C5
31cb. R345C5 = {256} clashes with R67C5
31cc. R345C5 = {346} clashes with R67C5
31d. R4C46 = [92] -> R345C5 = 12 = {147/156/345}(cannot be {129/237/246} which clash with R6C46, cannot be {138} because 1,8 only in R3C5), also R6C46 = [81] -> R67C5 = 10 = {37/46}
31da. R345C5 = {147} clashes with R67C5
31db. R345C5 = {156} -> R67C5 = {37} -> R12C5 = {49} -> R89C5 = {28}
31dc. R345C5 = {345} clashes with R67C5

Summary R4C46 = [81/82/92] (no valid combinations for R345C5 with [91]) -> R6C46 = [81/91/92]
R345C5 = {156/157/356} -> no 2,4,8,9 in R3C5, no 4 in R4C5
For {157}, 1 only in R3C5 -> no 7 in R3C5
R67C5 = {36/37}/[71], no 2,4,5
R12C5 = {49}, no 6,7, naked pair {49} locked for C5 and N2
R89C5 = {28}, no 1,3,4,6,7,9, naked pair {28} locked for N8

32. 4 in N5 locked in R5C46, locked for R5

33. R5C234 (step 22) = {138/147/237/246}
33a. 4 only in R5C4 -> no 6 in R5C4

34. 12(3) cage at R1C6, min R1C6 = 3 -> max R12C7 = 9, no 9

35. 26(5) cage at R6C7, max R6C8 + R7C7 = 6 -> min R6C7 + R78C6 = 20 = {389/…}
35a. 8 only in R6C7 and only other 9 in R7C6 -> no 3 in R6C7 and R7C6
35b. Valid combinations for 26(5) cage with R6C8 = {124} and R7C7 = {12} {12689/14579/14678/23489/24569/24578}
35c. All combinations with 4 must have 4 in R6C8 -> no 4 in R78C6

36. R5C6 = 4 (hidden single in C6) -> R5C78 = 7 = [52/61] (step 21), no 3,7 in R5C7

37. Killer pair 5/6 in R5C159 and R5C7, locked for R5

38. R5C234 (step 22) = {138/237} = 3{18/27}

39. 3 in N6 locked in R4C79, locked for R4

40. R345C5 (step 31 summary) = {156/157/356}
40a. 1,3 only in R3C5 -> no 6 in R3C5

41. 16(3) cage in N3 = {169/259/358/367} (cannot be {178/268} which clash with R23C8)

42. R345C9 = {179/269/278/359/368}
42a. 1,2 only in R3C9 -> no 7 in R3C9
42b. 1,2 only in R3C9 and {359} requires 9 in R5C9 -> no 9 in R3C9

43. 23(5) at R2C6 has {124} = R4C8, valid combinations {12389/12569/12578/13469/13478/13568/14567/23459/23468/23567} (cannot be {12479} because 1,2,4 only in R3C7 and R4C8)
43a. All combinations with 9 require {12/14/24} which must be in R3C7 and R4C8 -> no 9 in R3C7

44. 9 in N3 locked in 16(3) cage = 9{16/25}, no 3,7,8

45. Killer pair 5/6 in 16(3) cage and R23C8, locked for N3

46. 3 in C9 locked in R34C9 -> R345C9 (step 42) = 3{59/68}, no 1,2 7
46a. 5 only in R4C9 -> no 9 in R4C9

47. R1C8 = 9 (hidden single in C8, not sure how long that has been there) -> R12C5 = [49], R12C9 = {16/25}

48. Killer pair 5/6 in R12C9 and R345C9, locked for C9

49. R7C7 = {12}, R7C9 = {124} -> 1/2/4 in 18(3) cage (step 28) = {189/279/459/468} (cannot be {567})
49a. 2 only in R9C9 and 9 only in R8C9 -> no 7 in R89C9

50. R6C9 = 7 (hidden single in C9)

51. R67C5 (step 31 summary) = {36/37} (cannot now be [71]) -> no 1 in R7C5

52. R3C5 = 1 (hidden single in C5) -> R46C6 = [21], clean-up: no 6 in R2C2

53. 19(4) cage at R6C9 (step 27) = 37{18/45}, no 2,6

54. 22(3) cage at R8C7 = {679} (cannot be {589} which must have 5,8 in R78C7 and would then clash with the part of the 19(4)cage that is in N9), no 5,8 -> no 6,7 in R9C8
54a. 6,7 in N9 locked in R89C7 = {67}, locked for C7 -> R9C6 = 9 -> R8C9 = 9 (hidden single in N9), R5C7 = 5, R5C8 = 2, R6C8 = 4, R4C8 = 1

55. Naked triple {358} in R789C8, locked for C8 and N9 -> R23C8 = {67}, locked for N3, clean-up: no 1 in R12C9 (step 47) = {25}, locked for C9 and N3

56. R7C7 = 2 (hidden single in C7)

57. 1 in C7 locked in R12C7 -> 12(3) cage at R1C6 = 1{38/47} (cannot be {156} because 5,6 only in R1C6), no 5,6

58. 26(5) cage at R7C7 = {24569/24578} (cannot be {23489} because 8,9 only in R7C7) = 245{69/78}, no 3
58a. 5 locked in R78C6, locked for C6 and N8

59. 3 in C6 locked in R123C6, locked for N2

60. R5C3 = 1 (hidden single in R5, saw that a long time ago but forgot about it) -> R5C2 = 8, R5C4 = 3, R5C159 = [976], R67C5 = [63], R4C5 = 5, R46C4 = [89] (cage sums), R34C9 = [83], R4C7 = 9, R6C7 = 8, R78C6 = {57} (step 58), locked for C6 and N8

61. R8C8 = 3 (hidden single in C8)

62. 23(5) cage at R2C6 = {13469} (only remaining combination), no 8 -> R3C7 = 4, R23C6 = {36}, locked for C6 and N2 -> R1C6 = 8, clean-up: no 3 in R2C2

63. 6 in R1 locked in R1C123, locked for N1, clean-up: no 1 in R2C2
[Alternatively X-wing in 6 on R23C68]

64. 1 in N1 locked in 10(3) cage = 1{27/36}, no 5

65. R5C1 = 9 -> R34C1 = [57/84]

66. 6 in R4 locked in R4C23 in 28(5) cage

67. R3C13 – 9 = R1C4 (step 24), R1C4 = {257} -> R3C13 = 11, 14 or 16
67a. If R3C13 = 11 -> [83]
67b. If R3C13 = 14 -> [59]
67c. R3C13 cannot total 16
-> R3C3 = {39}, R1C4 = {25}

68. 7 in C4 locked in R23C4 -> no 7 in R4C23 -> R4C1 = 7 (hidden single in R4), R3C1 = 5, R3C3 = 9 (hidden single in R3), R1C4 = 5 (step 67b), R12C9 = [25], clean-up: no 2 in R23C2 = [43], R4C23 = [64]

69. 10(3) cage in N1 = {127} (only remaining combination) -> R1C2 = 7, R12C1 = [12], R12C3 = [68], R12C7 = [31], R23C4 = [72], R23C8 = [67], R23C6 = [36], R6C1 = 3

70. R7C2 = 9 (hidden single in C2)

71. R6C1 + R7C2 = [39] -> R7C1 + R8C2 = 8 = [62], R6C23 = [52], R9C2 = 1, R9C9 = 4, R9C8 = 5 (cage sum), R7C89 = [81], R7C4 = 4, R89C5 = [82], R89C1 = [48], R89C4 = [16], R89C7 = [67], R9C3 = 3, R8C3 = 5 (cage sum), R7C3 = 7, R78C6 = [57]

and the rest is naked singles, naked pairs and cage sums in N2

I haven't had enough time to check it properly. If you find any typos or other errors, please tell me by PM and I'll make the necessary corrections. I'm sure there will be a number of things that I ought to have seen earlier so no need to point those out to me.
[Edit. A few minor corrections have been made. I didn't think they were significant enough to colour code them.]
Last edited by Andrew on Wed Jul 25, 2007 4:31 am, edited 2 times in total.
sudokuEd
Grandmaster
Posts: 257
Joined: Mon Jun 19, 2006 11:06 am
Location: Sydney Australia
Making more progress on Candy A55V2. Feels more like a mine-field. Can someone please check these next batch of steps? Can't bear the thought of having made a mistake and 'wasting' any more time ..

A big thankyou to Glyn for helping do some groundwork for n9 moves. But any mistakes are not his.

Feel free to add some more too.

59c. Glyn pointed out that 5 in r7c7 -> 5 in n3 in r3c9 and 4 in r6c9 (from step 59.)-> r45c9 = [39]: but this forces 9 in c8 into both r1 & r789
59d. no 5 r7c7

Going to be a little more systematic now.
60. 3 in r7c7 -> 2 outies n9 = [12] -> r789c7 = [3]{19}
..............................................= [3]{46} blocked since have no 8/9 (step 57a)

61. 4 in r7c7 -> 2 outies n9 = [31]
i. [13] blocked: 1 in r6c9 -> 1 in n9 in r789c7 = [4]{18}: clashes with r1235c7
ii. [31]: 3 in r6c9 -> 3 in n9 in r789c7 = [4]{38}

62. 6 in r7c7 -> 2 outies n9 = [42] (remembering can't have repeats on these 2 outies & only have {1..3} available in r9c6)
62a. -> r789c7 = [6]{19}
...............= [6]{37/46} blocked since have no 8/9 (step 57a)

63. 8 in r7c7 -> 2 outies n9 = [71/35]: others blocked.
i. [17] Blocked: 1 in r6c9 -> 1 in n9 in r789c7 = {14}: but this clashes with r1235c7
ii. [71]-> r789c7 = [8]{56} ({[8]{29} leaves no 8/9 for r789c8: [8]{47} clashes with r123c7)
iii. [35]: 3 in r6c9 -> 3 for n9 in r789c7 = [8]{34}
iv. [43] Blocked: r789c7 = [8]{27/45} both clash with r1235c7

64. 9 in r7c7 -> 2 outies n9 = [18/45]. Here's how.
i. [18] -> 1 for n9 in r89c7 -> r789c7 = [9]{13}
ii. [36] Blocked: 3 in r6c9 -> 3 in n9 in r789c7: not possible with r9c6 = 6 in a 12(3) cage: forces 2 3's in cage.
iii. [45]: 4 in r6c9 -> h13(3)n6 = {256} -> 6 for n9 in r789c7 = [9]{16}
iv. [72] Blocked: 7 in r6c9 -> h13(3) = {256} -> 6 in n9 in r789c7 = [9]{46} and 4 in c8 forced into n9: but this means 2 4's n9.

65. In summary: 2 outies n9 = [12/31/42/71/35/18/45]
65a. r9c6 = {1,2,5,8}

66. In summary: r789c7 = [3]{19}/[4]{38}/[6]{19}/[8]{34/56}/[9]{13/16} (no 2,7)
66a. 12(3)n8 = {129/138/156/345}(no 7)

67. 2 in c7 only in n3: 2 locked for n3
67a. 17(3)n3 = [458/539/548/719]
67b. no 7 r4c9

68. r789c7 = [1/2/4/5/7](step 66): -> hidden quint with r1235c7
68a. no 1 r46c7

Time to move elsewhere.
69. 17(4)n5: no {2357/2456} combo's. Here's how.
69a. Combo's with-out 1 must have {23/34} in r67c5 (step 38)
69b. -> {2456} blocked
69c. {2357} combo. must have r67c5 = {23}(step 38) -> r6c6 = 5 -> r4c6 = 6: but this leaves no 6 for c5

70. no 6 in r6c6. Here's how.
70a. combo's with 6 in 17(4)n5 = {1268/1367}
i. {1268} = [8126] ([8162]: r56c5 = [12] clashes with 6(2)n2)
ii {1367}: r6c4 = {67} -> r4c4 = {67}(h13(2)n5) -> r7c5 = {67} -> r6c6 = 3.
70b. -> no 6 in r6c6
70c. -> no 5 r4c6 (h11(2)n5)

71. "45" c5: r46c46 = 24 = h24(4)n5, and taking into account h13(2) & h11(2)
71a. from step 37d. r4c46 = {89}/{78}/{68}/[57]/[59]
71a. -> h24(4) = [8952/9843/7863/8754/6873/5784/5982]
71b. 8 locked for n5 (no 8 r45c5)
71c. no 6 in r4c6, no 5 in r6c6

72. no 8 in r789c4 because of 8's in c5. Here's how.
72a. 8 in r3c5 -> 8 in n5 in r6c4 -> no 8 r789c4
72b. 8 in r89c5 -> no 8 in r789c4

73. weak links on 8 in r5 and n3 -> no 8 r1c1

74."45" n5: -> r456c5 + r5c46 = 45 - (13+11) = 21 = h21(5)n5
74a. must have 1 for n5 and no 8
74b. h21(5) n5 = {12369/12459/12567/13467}

75.But {12459} is blocked. Here's how.
75a. h21(5)n5 = {12459} must have r45c5 = [59] -> r6c5 = 1 (step 37ii,37iii) -> r5c46 = [24]: but this forces 2 4's into r5 in 9(3)n4 and r5c6.

76. h21(5)n5 = {12369/12567/13467} = 6{..}
76a. 6 locked in r45c5 for n5 and c5
76b. no 7 r46c4 (h13(2))

77. 17(4)n5 = {1259/1349/1457/2348} ({1358} blocked since 5 & 8 only in r6c4)

78. from step 71a. h24(4) = [8952/9843/8754/5784/5982]
78a. -> when r6c6 = 3, r6c4 = 4
78b. -> [42] blocked from i/o n9 since it means r6c6 = 3: but this will require 2 4's in r6

79. from step 65. 2 outies n9 = [12/31/71/35/18/45]
79a. no 6 r7c7

80. r6c46 = [52/43/54/84/82] (step 78)
80a. -> 17(4) cage = {1259/1349/1457/2348} =
i.[5129]
ii. [4139]
iii. [5147]
iv. [8243/8342]
v. [8324/8423]

81. [12] blocked from 2 outies n9 by 17(4) cage. Here's how.
81a. r6c9 + r9c6 = [12] & [3] in r7c7 i.17(4)n6 = [5129/4139/5147]: 2 1's in r4
ii. 17(4)n6 = [8243]: 2 3's r7
iii. 17(4)n6 = [8342]: 2 2's n8
iv. 17(4)n6 = [8423]:2 2's c7
81b. 2 outies n9 = [31/71/35/18/45]
81e. no 3 r7c7, no 2 r9c6

82. from step 66: r789c7 = [4]{38}/[8]{34/56}/[9]{13/16}
82a. no 9 r89c7

83. 27(5)n6 must have 9 because of 9's in c7. Here's how.
83a. 9 in r4c7 -> 9 in c6 in r78c7 in 27(5)n6.
83b. 9 elsewhere in c7 in r67 must be in 27(5)n6

84. 27(5) = {12789/13689/14589/14679/23589/23679/24579/34569}

85. 4 in r7c7 -> r89c7 = {38}, r6c9 = 3, r9c6 = 1
i. {14589/14679} blocked by 1 in r9c6
ii. {24579} -> r6c7 = 9 (only candidate)
iii. {34569} -> r6c7 = 9 (cannot be 6 as that forces h13(3)n6 = {247}, but no 2,4,7 available for r6c8

86. 8 in r7c7 -> r6c9 + r9c6 + r89c7 = [71{56}]/[35{34}]
i. {12789} -> r6c7 = 9 (only candidate)
ii. {13689} -> r6c8 = 6 (only candidate) and 1 in r78c6 -> 2 outies n9 = [35] -> 3 in r78c6 -> r6c7 = 9
iii. {14589} -> r6c7 = 9 (only candidate)
iv. {23589} -> r6c7 = {3/9}

NOw, obviously if I can just get rid of that 3 from r6c7 then 9 will be locked in the 27(5) in r67c7. Am planning to look at I/O on n69 to see what happens.

Can anyone see an easier way?

Code: Select all

``````.-----------------------.-----------------------.-----------.-----------------------.-----------------------.
| 345679      3456789   | 12345678    12345678  | 1245      | 12467       2457      | 89          689       |
|           .-----------&#58;           .-----------&#58;           &#58;-----------.           &#58;-----------.           |
| 456789    | 123567    | 1245678   | 12345678  | 1245      | 123456789 | 2457      | 13        | 689       |
&#58;-----------&#58;           &#58;-----------'           &#58;-----------&#58;           '-----------&#58;           &#58;-----------&#58;
| 12345678  | 123567    | 123456789   123456789 | 578       | 123456789   2457      | 13        | 457       |
|           &#58;-----------'           .-----------'           '-----------.           '-----------&#58;           |
| 12345678  | 123456789   123456789 | 589         5679        789       | 3689        2456      | 1345      |
|           &#58;-----------------------'-----------.           .-----------'-----------------------&#58;           |
| 689       | 12345       12345       12345     | 69        | 1457        157         257       | 89        |
&#58;-----------+-----------------------.-----------'-----------'-----------.-----------------------+-----------&#58;
| 123456789 | 123456789   123456789 | 458         1234        234       | 3689        2456      | 1347      |
|           '-----------.           '-----------.           .-----------'           .-----------'           |
| 123456789   123456789 | 123456789   12345679  | 23479     | 123456789   489       | 456789      123457    |
&#58;-----------.           &#58;-----------.           &#58;-----------&#58;           .-----------&#58;           .-----------&#58;
| 123456789 | 123456789 | 123456789 | 12345679  | 234789    | 123456789 | 134568    | 456789    | 123457    |
|           '-----------&#58;           '-----------&#58;           &#58;-----------'           &#58;-----------'           |
| 123456789   123456789 | 123456789   12345679  | 234789    | 158         134568    | 45679       123457    |
'-----------------------'-----------------------'-----------'-----------------------'-----------------------'``````
mhparker
Grandmaster
Posts: 345
Joined: Sat Jan 20, 2007 10:47 pm
Location: Germany
sudokuEd wrote:Can anyone see an easier way?
Yes, I could (although it wasn't easy). Key moves were 87, 103a and an ALS-XZ move at step 112!

Here goes:

Assassin 55V2 Walkthrough, continued...

87. Nishio: if r7c5 = 9, then...
87a. 9 in n5 locked in r4, and
87b. 9 in n9 locked in c8
87c. -> 9 in n3 locked in c9
87d. Steps 87a and 87c -> 9 in n6 in r6c7
87e. but this would leave nowhere to place the 9 in 31/5 at r6c2
87f. Conclusion: no 9 in r7c5

88. 9 no longer available in 17(4)n5
88a. -> 17(4)n5 = {1457/2348} (see step 77) = {(5/8)..}
88b. {58} in 17(4)n5 only in r6c4
88c. -> r6c4 = {58}
88d. -> r46c4 (innies c1234, step 5) = {58}, locked for c4 and n5
88e. cleanup: no 3 in r6c6 (step 6)

89. 5 in c5 locked in n2 -> not elsewhere in n2

90. Hidden pair on {58} in 35(5)n2 at r3c5+r4c4
90a. -> r3c5 = {58}

91. 7 in 35(5)n2 locked in r4c56 -> not elsewhere in r4 and n5

92. {1457} combo for 17(4)n5 blocked by {14} in r5c6
92a. -> 17(4)n5 = {2348} (no 1,5,7) (see step 88a)
92b. -> r6c4 = 8 (at last, a placement!)
92c. -> r4c4 = 5 (step 5)
92d. -> r3c5 = 8
92e. cleanup: no 3 in 11(2)n8
92f. 8 not available in r6c23+r78c4 for 31(5)n4
92g. -> max. r6c23+r78c4 = {5679} = 27
92h. -> no 1,2,3 in r7c3

93. 1,5 in c5 locked in 6(2)n2 = {15}
93a. -> no 1 elsewhere in n2 (5 already gone)

94. 7 in r5 locked in n6 -> not elsewhere in n6 (r6c9)

95. 9(3)r5 and r5c6 form killer pair on {14} in r5 -> no 1 in r5c7

96. Naked quad on {2457} in c7 at r1235c7 -> no 4,5 elsewhere in c7 (2,7 already gone)

97. 1 in c7 locked in r89c7 -> not elsewhere in n9, and no 1 in r9c6
97a. Cleanup: no 9 in r9c8 ({129} combo now unavailable)
97b. 12(3)r8c7 = {1(38|56)} = {(3/6)..}
97c. {13} in 12(3)r8c7 only in r89c7 -> no 8 in r89c7

98. 12(3)r8c9 = {237/246/345} = {(3/6)..}
98a. {23} only in r89c9 -> no 7 in r89c9
98b. 12(3)r8c7 and 12(3)r8c9 form killer pair on {36} in n9 -> no 3,6 elsewhere in n9

99. 4 in c7 locked in n3 -> not elsewhere in n3 (r3c9)

100. 1 in n5 locked in r5 -> not elsewhere in r5

101. innies r12: r2c2468 = h17(4)r2
101a. 3 locked, 8 unavailable
101b. -> h17(4)r2 = {1349/1367/2357} = {(1/5)..}
101c. -> h17(4)r2 and r2c5 form killer pair on {15} in r2 -> no 1,5 elsewhere in r2
101d. h17(4)r2: 5 only in r2c2 -> no 2 in r2c2 -> no 6 in r3c2
101e. 9 only in r2c6 -> no 4 in r2c6

102. 11(3)n1 = {128/137/146/236/245}
102a. {13} only in r1c34 -> no 7 in r1c34
102b. {15} only in r1c3 -> no 4,8 in r1c3

103. 21(4)n69 = {1479/1578/2379/2478/3459}
103a. {2379} and {3459} both blocked by 12(3)r8c9
103b. -> 21(4)n69 = {(149/158/248)7}
103c. -> no 3 in r6c9
103d. 7 in 21(4)n69 locked in n9 -> not elsewhere in n9 (r9c8)

104. 12(3)r8c9 = {(26/35)4}
104a. 4 locked for n9

105. {14} in 21(4)n69 only available in r6c9
105a. -> {1479} combo blocked
105b. -> 21(4)n69 = {1578/2478}
105c. -> no 9 in r78c8

106. Hidden Single (HS) in c8 at r1c8 = 9

107. HS in c9 at r5c9 = 9
107a. Cleanup: no 4 in r4c9

108. Naked Single (NS) at r5c5 = 6
108a. -> r5c1 = 8
108b. Cleanup: no 2,7 in r3c1

109. HS in c7 at r7c7 = 9

110. HS in r4/c7 at r4c7 = 8

111. 9 in r6 locked in n4 -> not elsewhere in r4

Here's a neat one - the last tricky move. Haven't used ALS in a Killer before:

112. ALS-XZ: r46c9 ({134}) and r6c56 ({234}) have 4 as restricted common
112a. r6c7 can see common candidate digit 3 in both ALS's
112b. -> no 3 in r6c7
112c. -> r6c7 = 6

The rest is pretty easy now.
Cheers,
Mike
mhparker
Grandmaster
Posts: 345
Joined: Sat Jan 20, 2007 10:47 pm
Location: Germany
A few more moves, just to deliver the final blow...

Assassin 55V2 (final episode)

113. r89c7 = {13}
113a. -> r9c6 = 8
113b. -> r6c9 = 1 (outies n9, r6c9+r9c6 = 9(2))
113c. -> r4c9 = 3
113d. -> r3c9 = 5 (last digit in cage)
113e. Cleanup: no 3 in r2c2

114. HS in c7 at r5c7 = 5
114a. -> r5c68 = [17] (only remaining permutation)

115. HS in c8/n9 at r9c8 = 6
115a. -> r89c9 = {24}, 2 locked for c9/n9

116. NS at r7c9 = 7

117. Naked Pair (NP) on {24} in r6 at r6c68 -> no 2,4 elsewhere in r6

118. NS at r6c5 = 3

119. 11(2)n8 and r7c5 form killer pair on {24} in n8 -> no 2,4 elsewhere in n8

120. Split 19(4) at r23c6+r3c7+r4c8 = {2467} (only possible combo, due to {158} unavailable)
120a. -> no 3,9 in r23c6
120b. 6 only available in r23c6 -> no 6 elsewhere in n2

121. HS in c6 at r4c6 = 9
121a. -> r4c5 = 7, r6c6 = 2 (step 6)
121b. -> r7c5 = 4, r5c4 = 4
121c. -> r6c8 = 4
121d. -> r4c8 = 2

122. HS in r2 at r2c1 = 9
122a. Cleanup: no 3 in r1c2

123. HS in r3/n2 at r3c4 = 9

124. 2 in c1 locked in n7 -> not elsewhere in n7

125. HS in r7 at r7c1 = 2

126. Split 8(3) at r78c6 = {35}, 3 locked for n8

127. 5 in r9 locked in n7 -> not elsewhere in n7

128. 6 in c4 locked in 31(5)n4 = {6..}
128a. -> no 6 in r7c3

129. NS at r7c3 = 8
129a. -> r78c8 = [58]
129b. -> r78c6 = [35]

130. HS in c2 at r1c2 = 8
130a. -> r1c1 = 3 (last digit in cage)
130b. Cleanup: no 5 in r2c2

131. NS at r1c4 = 2
131a. -> r12c9 = [68]

132. HS in n1 at r1c3 = 5
132a. -> r2c3 = 4 (last digit in cage)

133. r12c5 = [15]

134. HS in c4 at r2c4 = 3
134a. -> r23c8 = [13]
134b. Cleanup: no 7 in r3c2

135. HS in c7 at r2c7 = 2 (could have also derived this by cage-splitting on 13(3)n2)

136. Split 23(4) at r6c23+r78c4 = {1679} (only combo without any of {2348}, which are unavailable)
136a. -> no 5 in r6c2
136b. {79} locked in r6c23 -> r78c4 = {16}, locked for n8

137. NS at r9c4 = 7
137a. -> r89c3 = [63] (only possible permutation)

And the rest is just naked singles.
Cheers,
Mike
CathyW
Master
Posts: 161
Joined: Wed Jan 31, 2007 5:39 pm
Location: Hertfordshire, UK
Well done guys! Perhaps you can help me out with the 57 V1.5 now!
Andrew
Grandmaster
Posts: 300
Joined: Fri Aug 11, 2006 4:48 am
Location: Lethbridge, Alberta
This ran as a “tag” solution, by Ed, Cathy and Mike, on the forum.

I never tried it at the time, we were getting ready to move as I commented when I posted by walkthrough for the V1, so I’ve now decided to have a go at it years later. I’m pleased that I did; I enjoyed this puzzle, even though it was a very tough one.

Since it was originally solved as a “tag” I didn’t hold back from using forcing chains but I tried to avoid using contradiction moves for as long as possible; originally I used four contradiction moves but I've since managed to re-write three of those steps using forcing chains. Edit. I've added an alternative step 35 which replaces the final contradiction move with a longer forcing chain.

Here is my walkthrough for A55 V2.

Prelims

a) R12C5 = {15/24}
b) R23C2 = {17/26/35}, no 4,8,9
c) R23C8 = {13}
d) R89C5 = {29/38/47/56}, no 1
e) 20(3) cage in N1 = {389/479/569/578}, no 1,2
f) 11(3) cage at R1C3 = {128/137/146/236/245}, no 9
g) 23(3) cage in N3 = {689}
h) 9(3) cage at R5C2 = {126/135/234}, no 7,8,9
i) 35(5) cage at R3C5 = {56789}

Steps resulting from Prelims
1a. Naked pair {13} in R23C8, locked for C8 and N3
1b. Naked triple {689} in 23(3) cage, locked for N3
1c. Naked quint {56789} in 35(5) cage at R3C5, CPE no 5,6,7,8,9 in R6C5
[The 35(5) cage looks potentially useful for ALS blocks.]

2. 45 rule on R5 3 innies R5C159 = 23 = {689}, locked for R5
2a. Naked triple {689} in R125C9, locked for C9
2b. Min R5C1 = 6 -> max R34C1 = 9, no 9 in R34C1
2c. 23(3) cage in N3 = {689}, R125C9 = {689} -> R1C8 = R5C9
[Added because it may be useful for analysis or clean-up.]

3. 9(3) cage at R5C2 = {135/234}, 3 locked for R5
3a. 13(3) cage at R5C6 = {157/247}

4. 45 rule on C9 4 outies R1789C8 = 28 = {4789/5689}, no 2, 8,9 locked for C8

5. 2 in C8 only in R456C8, locked for N6
5a. 45 rule on C89 3 innies R456C8 = 13 = {247/256}
5b. 4 of {247} must be in R46C8 (R5C8 cannot be 4 because R456C8 = {247} clashes with 13(3) cage at R5C6 = [274], CCC) -> no 4 in R5C8
5c. 13(3) cage at R5C6 (step 3a) = {157/247}
5d. 2 of {247} must be in R5C8 (cannot be [247] which clashes with R456C8 = {247}, CCC) -> no 2 in R5C6

6. 13(3) cage at R1C6 = {157/247/256} (cannot be {139/148/238/346} because 1,3,6,8,9 only in R1C6), no 3,8,9
6a. 1,6 of {157/256} must be in R1C6 -> no 5 in R1C6

7. 45 rule on C1234 2 innies R46C4 = 13 = {58/67}/[94], no 1,2,3,9 in R6C4

8. 45 rule on C6789 2 innies R46C6 = 11 = {56}/[74/83/92], no 1,7,8,9 in R6C6
8a. 9 in N5 only in R4C456 + R5C5, locked for 35(5) cage at R3C5 -> no 9 in R3C5

9. 45 rule on N5 2 outies R37C5 = 2 innies R5C46 + 7
9a. Min R5C46 = 3 -> min R37C5 = 10, no 1 in R7C5

10. 20(3) cage in N1 = {389/479/569/578}
10a. 3 of {389} must be in R1C12 (R12 cannot be {89} which clashes with 23(3) cage in N3, ALS block) -> no 3 in R2C1

11. 20(3) cage in N1 = {389/479/569/578}
11aa. 20(3) cage = {389/479/569} => caged X-wing for 9 in 20(3) cage in N1 and 23(3) cage in N3 for R12, no other 9 in R12
11ab. 20(3) cage = {578} => R3C3 = 9 (hidden single in N1) => no 9 in R2C4
11b. -> no 9 in R2C4
[I missed the simpler 9 in R3 only in R3C346, CPE no 9 in R2C4.]

12. R46C4 (step 7) = {58/67}/[94], R46C6 (step 8) = {56}/[74/83/92]
12a. Consider placements for R5C5
12aa. R5C5 = 6 => R46C6 cannot be {56}
12ab. R5C8 = 8 => no 3 in R89C5, R46C4 = {67}/[94]
12abi. R46C4 = {67} => R46C6 cannot be {56} which clashes with R46C4
12abii. R46C4 = [94], 3 in C5 only in R67C5 => 17(4) cage at R6C4 = {1349/2348} => no 5,6 in R6C6
12ac. R5C5 = 9 => R46C4 = {58/67} => R46C6 cannot be {56} which clashes with R46C4
12b. -> no 5,6 in R6C6, clean-up: no 5,6 in R4C6

13. R89C5 = {29/38/47} (cannot be {56} which clashes with 35(5) cage at R3C5, ALS block), no 5,6

14. 12(3) cage at R8C9 = {129/138/237/246/345} (cannot be {147/156} which clash with R1789C8, CCC because no 1 in R9C8)
14a. 7 of {237} must be in R9C8 -> no 7 in R89C9
[At one stage I eliminated {345} here, mistakenly thinking that it clashed with R1789C8, so I’ve had to do a lot of re-work from here.]

15. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 1 innie R7C7
15a. Min R6C9 + R9C6 = 2 -> min R7C7 = 2
15b. Max R7C7 = 9 -> max R6C9 + R9C6 = 9, no 9 in R9C6

16. 17(3) cage at R3C9 = {179/278/359/458/467} (cannot be {269/368} because 6,8,9 only in R5C9)
[If I’d spotted step 25 here, some of the following steps might have been simplified.]

17. 21(4) cage at R6C9 = {1389/1479/1569/1578/2469/2478/2568/3468/3567} (cannot be {2379/3459} which clash with 12(3) cage at R8C9)
17a. Hidden killer pair 5,7 in 17(3) cage at R3C9, R67C9 and R89C9 for C9, 17(3) cage at R3C9 contains one of 5,7 -> R67C9 must contain one of 5,7 or R89C9 must contain 5
17b. From the previous sub-step, if R67C9 doesn’t contain one of 5,7 then 5 must be in R89C9 -> 12(3) cage at R8C9 (step 14) = [345/543] which can block combinations for 21(4) cage at R6C9 not containing 5 or 7 in R67C9
17c. 21(4) cage at R6C9 = {1479/1569/1578/2469/2478/2568/3567} (cannot be {1389/3468} which clash with 12(3) cage at R8C9 = [345/543])
17d. 3,7 of {3567} must be in R67C9 (R67C9 cannot be {35} because R78C8 = {67} clashes with combinations for R1789C8) => R78C8 = {56}, locked for C8 => R456C8 (step 5a) = {247}, locked for N6 => 3 of {3567} must be in R6C9 -> no 3 in R7C9

18. R1789C8 (step 4) = {4789/5689}
18a. Consider the combinations for R1789C8
18aa. R1789C8 = {4789} => 21(4) cage at R6C9 cannot be {2469}
18ab. R1789C8 = {5689} => 5 in R78C8 or in R9C8
18abi. 5 in R78C8 => 21(4) cage at R6C9 cannot be {2469}
18abii. 5 in R9C8 => 12(3) cage at R8C9 = [354/453], 4 locked for C9 => 21(4) cage at R6C9 cannot be {2469}
18b. -> 21(4) cage at R6C9 cannot be {2469}
-> 21(4) cage at R6C9 (step 17c) = {1479/1569/1578/2478/2568/3567}

19. R1789C8 (step 4) = {4789/5689}
19a. Consider the combinations for R1789C8
19aa. R1789C8 = {4789}, locked for N9
19ab. R1789C8 = {5689} => R78C8 must contain at least one of 5,6 (cannot be {89} because no remaining combination for 21(4) cage at R6C9 contains both of 8,9) => no 4 in R67C9 (because no remaining combination for 21(4) cage at R6C9 contains 4 and one of 5,6)
19b. -> no 4 in R7C9

20. R1789C8 (step 4) = {4789/5689}
20a. 4 or 7 of {4789} must be in R9C8 (R78C8 cannot be {47} because 21(4) cage at R6C9 doesn’t contain 8 or 9 in C9) => 12(3) cage at R8C9 = {237/345}, 3 and either 2 or 5 locked for C9
[This might be useful later.]

[I’d been concentrating so much on C89, particularly before the error and re-work, that I’ve only just thought of looking at C5 again.]

21. R12C5 = {15/24}
21a. Consider the combinations for R12C5
21aa. R12C5 = {15} => hidden killer triple 2,3,4 in R6789C5, R6C5 = {234}, R89C5 must contain one of 2,3,4 => R7C5 = {234}, R6C56 + R7C5 = {234} = 9 => R6C4 = 8
21ab. R12C4 = {24}, locked for C5 => R89C5 = {38}, locked for C5, R7C5 = 1 (hidden single in C5), 8 in 35(5) cage at R3C5 only in R4C46, locked for N5, no 8 in R6C4 => no 5 in R4C4 => 5 in 35(5) cage at R3C5 only in R34C5
21b. From steps 21aa and 21ab
21ba. 8 must be in R6C4 or R89C5, CPE no 8 in R457C5
21bb. 5 must be in R12C5 or R34C5, locked for C5
21c. 17(4) cage at R6C4 must be 8{234} or contain 1 at R7C5 -> 17(4) cage = {1259/1349/1367/1457/2348} (cannot be {1358} because 5,8 only in R6C4, cannot be {1268} because 8 must be in R4C46 and also in R89C5 when R7C5 = 1)

[I hope the next step is acceptable as a hidden killer. I first saw it as a contradiction move and then tried to find a way to write it in a more acceptable way.]
22. Hidden killer pair 2,4 in R5C46 and R6C456 for N5, R5C46 cannot contain both of 2,4 (because 2,4 in R5 must both be in 9(3) cage at R5C2 or both in 13(3) cage at R5C6) -> R6C456 must contain at least one of 2,4
22a. 17(4) cage at R6C4 (step 21c) = {1259/1349/1457/2348} (cannot be {1367} which doesn’t contain either of 2,4), no 6, clean-up: no 7 in R4C4 (step 7)
22b. 6 in C5 only in R345C5, locked for 35(5) cage at R3C5, no 6 in R4C4, clean-up: no 7 in R6C4 (step 7)
22c. 6 in N5 only in R45C5, locked for C5

23. Hidden killer pair 2,3 in R5C4 and R6C456, R5C4 cannot contain more than one of 2,3 -> R6C456 must contain at least one of 2,3
23a. 17(4) cage at R6C4 (step 22a) = {1259/1349/2348} (cannot be {1457} which doesn’t contain either of 2,3), no 7
23b. Consider the combinations for the 17(4) cage
23ba. 17(4) cage = {1259} => R6C4 = 5
23bb. 17(4) cage = {1349} => R5C4 = 2 (hidden single in N5)
23bc. 17(4) cage = {2348} = 8{234} => R4C4 = 5 (step 7)
23c. -> no 5 in R5C4

24. 13(3) cage at R5C6 (step 3a) = {157/247}, R456C8 (step 5a) = {247/256}
24a. Consider the combinations for the 13(3) cage
24aa. 13(3) cage = {157} => R5C8 = {57}
24aai. R5C8 = 5 => R456C8 = {256} => no 7 in R46C8
24aaii. R5C8 = 7 => no 7 in R46C8
24ab. 13(3) cage = {247} => R456C8 = {256} (cannot be {247} which clashes with 13(3) cage, CCC) => no 7 in R46C8
24b. -> no 7 in R46C8

25. R456C8 (step 5a) = {247/256}
25a. Consider the combinations for R456C8
25aa. R456C8 = {247}, locked for N6 => no 4,7 in R4C9
25ab. R456C8 = {256}, locked for N6 => no 6 in R5C9
25b. -> 17(3) cage at R3C9 (step 16) = {179/278/359/458} (cannot be {467}), no 6, clean-up: no 6 in R1C8 (step 2c)
[I saw this as a short forcing chain but it can probably be considered to be a combo blocker.]

26. R1789C8 (step 4) = {4789/5689}
26a. Consider the combinations for R1789C8
26aa. R1789C8 = {4789} => R9C8 = {47} (step 20a)
26ab. R1789C8 = {5689} => R9C8 = {56} or R78C8 = {56} = 11 => R67C9 = 10 = [37] => 12(3) cage at R8C9 cannot be [183/381]
26b. -> no 8 in R9C8

27. R1789C8 (step 4) = {4789/5689}
27a. Consider the combinations for R1789C8
27aa. R1789C8 = {4789} => R9C8 = {47} (step 20a)
27aai. R78C8 = {48/49} => R67C9 = {17/27} (from combinations for 21(4) cage at R6C9, step 18b), no 5 in R67C9
27aaii. R9C8 = 4 => R89C9 = {35} (step 14), 5 locked for C9
27ab. R1789C8 = {5689}, 5 locked for N9
27b. -> no 5 in R7C9

28. 13(3) cage at R5C6 (step 3a) = {157/247}
28a. Consider combinations for the 13(3) cage
28aa. 13(3) cage = {157} => R5C8 = {57}, R456C8 (step 5a) = {247/256}
28aai. R456C8 = {247} => R46C7 + R5C9 = {689} (hidden triple in N5)
28aaii. R456C8 = {256}, locked for C8 => R1789C8 = {4789} => R9C8 = {47} and 3 locked for C9 (step 20a) => R46C7 + R5C9 = {389} (hidden triple in N5)
28ab. 13(3) cage = {247} => R456C8 (step 5a) = {256} (cannot be {247} which clashes with 13(3) cage, CCC), locked for C8 => R1789C8 = {4789} => R9C8 = {47} and 3 locked for C9 (step 20a) => R46C7 + R5C9 = {389} (hidden triple in N5)
28b. -> R46C7 + R5C9 = {389/689}, no 1,4,5,7 in R46C7

[I originally did the next three steps as contradiction moves, two of which ended with no possible combination for the 17(3) cage at R3C9 and the other one with an ALS block involving the 17(4) cage at R6C4. I’ve now re-written all three steps as forcing chains based on these 17(3) and 17(4) cages.]

29. 17(3) cage at R3C9 (step 25b) = {179/278/359/458}
29a. Consider combinations for the 17(3) cage
29aa. 17(3) cage = {179} = [719], naked triple {245} in R123C7, locked for C7 => R5C7 = 7, R456C8 (step 5a) = {256} => R6C9 = 4 (hidden single in N6)
29ab. 17(3) cage = {278/359/458}, 2 or 5 locked for C9 => R67C9 cannot be [52]
29b. -> R67C9 cannot be [52] -> 21(4) cage at R6C9 (step 18b) = {1479/1569/1578/2478/3567} (cannot be {2568} = [52]{68})

30. 17(3) cage at R3C9 (step 25b) = {179/278/359/458}
30a. Consider combinations for the 17(3) cage
30aa. 17(3) cage = {179/359/458}, 1 or 5 locked for C9 => R67C9 cannot be [51]
30ab. 17(3) cage = {278} = [278] => R456C8 (step 5a) = {256}, locked for C8 => no 6 in R78C8
30b. -> R67C9 cannot be [51] or no 6 in R78C8 -> 21(4) cage at R6C9 (step 29b) = {1479/1578/2478/3567} (cannot be {1569} = [51]{69})

31. 17(4) cage at R6C4 (step 23a) = {1259/1349/2348}
31a. Consider combinations for the 17(4) cage
31aa. 17(4) cage = {1259} = [5129] => R5C4 = 3 (hidden single in N4) => R5C6 = 4 (hidden single in N4) => 13(3) cage at R5C4 (step 3a) = {247} = [472] => R456C8 (step 5a) = {256}, locked for C8 => no 6 in R78C8
31ab. 17(4) cage = {1349} => R7C5 = 9, 3 locked for R6 => no 3 in R6C9
31ac. 17(4) cage = {2348} can have 3 in R6C56 or R7C5
31aci. 3 in R6C56, locked for R6 => no 3 in R6C9
31acii. 3 in R7C5 => R6C56 = {24}, locked for R6 => R6C8 = {56} => R456C8 (step 5a) = {256}, locked for C8 => no 6 in R78C8
31b. -> no 3 in R6C9 or no 6 in R78C8 -> 21(4) cage at R6C9 (step 30b) = {1479/1578/2478} (cannot be {3567} = [37]{56}), no 3,6

32. R1789C8 (step 4) = {4789/5689}
32a. 4 or 7 of {4789} must be in R9C8 (step 20a), 6 of {5689} only in R9C8 -> R9C8 = {467}, no 5,9
32b. 12(3) cage at R8C9 (step 14) = {237/246/345}, no 1

33. R1789C8 (step 4) = {4789/5689}
33a. Consider the combinations for R1789C8
33aa. R1789C8 = {4789}, locked for C8 => R456C8 (step 5a) = {256}, locked for N6, no 5 in R6C9
33ab. R1789C8 = {5689} => 5 must be in R78C8 => 5 in 21(4) cage at R6C9 = {1578} (step 31b) must be in R78C8 => no 5 in R6C9
33b. -> no 5 in R6C9
[Alternatively there’s a short contradiction move starting with R6C9 = 5 ...]

34. 17(3) cage at R3C9 (step 25b) = {179/278/359/458}
34a. Consider combinations of 17(3) cage which contain 5
34aa. 17(3) cage = {359} = [539]
34ab. 17(3) cage = {458}, locked for C9 => R89C9 = {23}, R9C8 = 7 (step 32b) => R456C8 (step 5a) = {256}, locked for N6
34b. -> no 5 in R4C9
34c. 4 of {458} must be in R4C9 -> no 4 in R3C9
34d. 4 in N3 only in R123C3, locked for C3

[I’ve managed to replace all the contradiction moves except for this one, which can probably be considered to be “brute force” although the clash at the end of the chain is an interesting one. I tried looking at forcing chains starting from 17(3) cage at R3C9, 17(4) cage at R6C4 and from R46C7 + R5C9 = {389/689}, step 28b, but couldn’t make any of them work except possibly as contradictions. After further thought I managed to come up with alternative step 35, which is quite a lot longer but is a forcing chain rather than a contradiction move.]

35. 13(3) cage at R5C6 (step 3a) = {157/247} cannot be {247}, here’s how
35a. 13(3) cage = {247} => R5C8 = 2, R5C6 = 4, R5C7 = 7, R3C9 = 7 (hidden single in N3) => 17(3) cage at R3C9 (step 25b) = {179} => R4C9 = 1, R5C9 = 9, R46C7 = {38} (hidden pair in N6), 17(4) cage at R6C4 (step 23a) = {1259} (cannot be {2348} = 8{23}4 clashes with R6C7) => R6C456 = [512], R4C4 = 8 (step 7), R5C4 = 3 (hidden single in N5), R23C6 = {38} (hidden pair in N2) clashes with R4C7
35b. -> 13(3) cage at R5C6 = {157}, locked for R5
35c. 9(3) cage at R5C2 = {234}

[Alternative step 35.
35. 13(3) cage at R5C6 (step 3a) = {157/247}, R456C8 (step 5a) = {247/256}
35a. Consider the combinations for 17(4) cage at R6C4 (step 23a) = {1259/1349/2348}
35aa. 17(4) cage = {1259} = [5129], R4C4 = 8 (step 7), R5C4 = 3 (hidden single in N5) => R23C6 = {38} (hidden pair in N2), locked for 27(5) cage at R2C6 => R4C7 = {69}
35aai. R4C7 = 6 => R456C8 = {247} => 13(3) cage at R5C6 = {157} (cannot be {247}, CCC)
35aaii. R4C7 = 9 => R5C9 = 8 => 17(3) cage at R3C9 (step 25b) = {278/458} => R3C9 = {25} => 7 in N3 only in R123C7, locked for C7 => R5C7 = {15} => 13(3) cage at R5C6 = {157} (only remaining combination)
35ab. 17(4) cage = {1349}, 4 locked for N5 => R5C6 = {157} => 13(3) cage at R5C6 = {157}
35ac. 17(4) cage = {2348} = 8{23}4/8{24}3/8{34}2
35aca. 17(4) cage = 8{23}4, 3,8 locked for R6 => R6C7 = {69}
35acai. R6C7 = 6 => R456C8 = {247} => 13(3) cage at R5C6 = {157} (cannot be {247}, CCC)
35acaii. R6C7 = 9 => R5C9 = 8 => 17(3) cage at R3C9 (step 25b) = {278/458} => R3C9 = {25} => 7 in N3 only in R123C7, locked for C7 => R5C7 = {15} => 13(3) cage at R5C6 = {157} (only remaining combination)
35acb. 17(4) cage = 8{24}3/8{34}2, 4 locked for N5 => R5C6 = {157} => 13(3) cage at R5C6 = {157}
35b. -> 13(3) cage at R5C6 = {157}, locked for R5
35c. 9(3) cage at R5C2 = {234}]

36. R456C8 (step 5a) = {247/256}
36a. R5C8 = {57} -> no 5 in R46C8

[Now for a “two-pronged attack” which I found interesting.]
37. 31(5) cage at R6C2 must contain 9
37a. First consider if 12(3) cage at R8C7 = {129}, 9 locked for C7 and N9 => R5C9 = 9 (hidden single in N6) => 17(3) cage at R3C9 (step 25b) = {179/359} => R3C9 = {57}, 2 in N3 only in R123C7, locked for C7 => R89C7 = {19}, locked for C7 => R5C6 = 1 (hidden single in R5) => 17(4) cage at R6C4 (step 23a) = {2348} => no 9 in R7C5
37b. Now consider placements for 9 in 31(5) cage
37ba. 9 in 31(5) cage in R6C23 => 9 in R5 only in R5C59
37bai. R5C5 = 9 => no 9 in R7C5
37baii. R5C9 = 9 => 9 in C7 only in R789C7 => either R7C7 = 9 => no 9 in R7C5 or 9 in 12(3) cage at R8C7 => no 9 in R7C5 (step 37a)
37bb. 9 in 31(5) cage in R7C3 + R78C4 => no 9 in R7C5
37c. -> no 9 in R7C5
[Looks like the puzzle may now be cracked.]

38. 17(4) cage at R6C4 (step 23a) = {2348} (only remaining combination), no 1 -> R6C4 = 8, R4C4 = 5 (step 7), R3C5 = 8 (hidden single in 35(5) cage at R3C5), clean-up: no 3 in R6C6 (step 8), no 3 in R89C5

39. R12C5 = {15} (hidden pair in C5), locked for N2

40. R5C6 = 1 (hidden single in N5)
40a. Naked pair {57} in R5C78, locked for N6
40b. 1 in C7 only in R89C7, locked for N9

41. 17(3) cage at R3C9 (step 25b) = {179/359/458} (cannot be {278} because 2,7 only in R3C9), no 2
41a. 2 in C9 only in R789C9, locked for N9

42. 21(4) cage at R6C9 (step 31b) = {1479/1578/2478}, 7 locked for N9
42a. 12(3) cage at R8C9 (step 32b) = {246/345}, 4 locked for N9
42b. 17(3) cage at R3C9 (step 41) = {179/359} (cannot be {458} which clashes with 12(3) cage at R8C9, ALS block because 12(3) cage must contain 4 or 5 in R89C9) => R5C9 = 9, R5C5 = 6, R5C1 = 8, R1C8 = 9 (step 2c), R4C9 = {13}
42c. Naked pair {79} in R4C56, locked for R4
42d. R4C7 = 8 (hidden single in R4)

43. Naked quad {2346} in R6C5678, locked for R6 -> R6C9 = 1, R4C9 = 3, R3C9 = 5 (step 42b), R6C7 = 6, clean-up: no 3 in R2C2

44. Naked pair {24} in R89C9, locked for N9 => R7C9 = 7, R9C8 = 6

45. Naked pair {58} in R78C8, locked for C8 and N9 -> R5C8 = 7, R5C7 = 5

46. 1 in N9 only in 12(3) cage at R8C7 = {129/138} (cannot be {147} because 4,7 only in R9C6), no 4,5,7
46a. 2,8 only in R9C6 -> R9C6 = {28}

47. R6C5 = 3 (hidden single in R6)
47a. Killer pair 2,4 in R7C5 and R89C5, locked for N8 -> R9C6 = 8 -> 12(3) cage at R8C7 (step 46) = {138}, no 9

49. R7C7 = 9 (hidden single in C7)
49a. 5 in N8 only in R78C6 -> 27(5) cage at R6C7 = 69{345} (only remaining combination) -> R6C8 = 4, R78C6 = {35}, locked for C6 and N8, R4C8 = 2, R6C6 = 2, R7C5 = 4, clean-up: no 7 in R89C5

50. Naked pair {29} in R89C5, locked for C5 and N8 -> R4C56 = [79]
50a. Naked triple {467} in R123C6, locked for N2

51. Naked pair {23} in R12C4, locked for C4, CPE no 2,3 in R2C3
51a. R3C4 = 9, R5C4 = 4

52. R2C1 = 9 (hidden single in N1)
52a. R1C12 = 11 = [38]/{47/56}, no 3 in R1C2

53. R5C1 = 8 -> R34C1 = 7 = [16/34/61], no 2,4,7 in R3C1

54. 6 in N8 only in R78C4 -> 31(5) cage at R6C2 = {16789} (only remaining combination, cannot be {35689} because 3,8 only in R7C3) -> R7C3 = 8, R6C23 = {79}, R78C4 = {16}, R78C8 = [58], R78C6 = [35]
54a. R6C1 = 5 (hidden single in R6), clean-up: no 6 in R1C2 (step 53)

55. R9C4 = 7 (hidden single in C4) -> R89C3 = 9 = [45/63], no 1,2,9, no 3 in R8C3, no 4 in R9C3

56. R6C3 = 9 (hidden single in C3), R6C2 = 7, clean-up: no 4 in R1C1 (step 52a), no 1 in R23C2
56a. Killer pair 2,3 in R23C2 and R5C2, locked for C2

57. R8C1 = 7 (hidden single in R8) -> R9C12 = 9 = [45], clean-up: no 6 in R1C1, no 4 in R1C2 (both step 52a), no 3 in R3C1 (step 53)

58. R1C12 = [38], R1C4 = 2, R12C3 = 9 = {45}, locked for C3

and the rest is naked singles.

Here is the solution, since it hasn't been posted on this forum.

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

Rating Comment. I won't try to decide whether my solving path was harder or easier than that used in the "tag" solution (steps 1 to 38 and 55 onward); I'll leave that to anyone who may decide to work through both of them. However I will say that they should probably both be in the same rating range.