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Assassin 74
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rcbroughton
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PostPosted: Sun Dec 02, 2007 10:11 am    Post subject: Reply with quote

mhparker wrote:
One reason for mentioning this (apart from the fact that it's a natural follow-up to my other post mentioned above) is that, despite being generally very powerful, SudokuSolver missed the above step 46, as well as missing Para's step 44. Hope you're reading this, Richard! Wink
Reading. Taking notes. Analysing. Implementing.

I'll let you know when it's ready!

Rgds
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Afmob
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PostPosted: Thu Mar 20, 2008 11:48 am    Post subject: Reply with quote

This was a project I wished to finish for quite some time but I didn't find the time or energy to tackle this Assassin again. But after being encouraged by Andrew who will also post his wt for A74 Brick Wall, I finally decided to solve it again in an appropiate way without resorting to long contradiction chains (which more match Killers of rating 2.5+) like in my original walkthrough.

The new wt is shorter than the first walkthrough since I managed to place candidates faster and there is nearly no endgame. After the last difficult steps there are only few sub-steps and then it's over which was quite surprising.

A74 Brick Wall Walkthrough (rewritten and improved):

1. R789
a) 6(3) @ N7 = {123} locked for N7
b) Innies+Outies N7: R6C1 = R7C3 -> R6C1 <> 2,3,9
c) 11(3): R6C23 <> 7,8 because R7C3 >= 4
d) Innies N8 = 10(3) <> 8,9
e) 22(3) = 9{58/67} -> 9 locked for R6+N5
f) 6(3) @ N8 = {123} locked between C4+N8 -> R89C4 <> 1,2,3
g) Hidden pair (12) in R8C17 for R8 -> R8C17 = {12}
h) 6(3) @ N7 = {123} -> 3 locked for R9
i) 9(3) = {126} locked for N9, 6 locked for R9
j) 1,2 locked in 6(3) + 9(3) for R89

2. R789
a) 15(3) = {456} -> R8C4 = 6, {45} locked for R9+N8
b) 20(3) @ N8 = {389} locked for N8, 3 locked for R8+N8
c) 6(3) @ N8 = {123} -> R6C4 = 3
d) R7C6 = 7 -> 22(3) = {679} -> 6 locked for R6+N5
e) Innies+Outies N7: R6C1 = R7C3 = (458)
f) 11(3) = 2{18/45} -> 2 locked for R6+N4
g) Innies+Outies N9: -2 = R6C7 - R7C9 -> R6C7 = (17), R7C9 = (39)
h) 19(3): R7C12 <> 4 because R6C1 <> 6,9

3. N56
a) 7 locked in R6C789 @ R6 for N6
b) 9 locked 19(3) @ N6 = 9{28/46}
c) 3 locked in 10(3) @ N6 = 3{16/25}
d) 10(3) @ N5 = 1{27/45} -> 1 locked for N5
e) 10(3) @ N5: R4C5 <> 2 since 7 only possible there

4. R123
a) 9 locked in R123C4 @ C4 for N2
b) All 15(3) @ R1 + N2 <> 5{28/46} since they are Killer triples of each other
c) 15(3) @ R2C4: R3C5 <> 7,8 because R23C4 <> 3,6
d) 15(3) @ R2C5: R2C5 <> 2,5 because 7 only possible there
e) 15(3) @ R2C5: R2C5 <> 4 because 4{38} blocked by R89C6

5. R123
a) 15(3) @ R1C1 <> {267} since it's a Killer triple of 11(3)
b) 15(3) @ R1C1 <> {348} since it's a Killer triple of 19(3)
c) 15(3) @ R1C4, R1C7 <> {159} because it's a Killer triple of 15(3) @ R1C1
d) 13(3) <> 3{28/46} since they are Killer triples of 15(3) @ N3
e) 17(3) <> 7{28/46} because they are Killer triples of 15(3) @ N3

6. C789 !
a) ! Hidden Killer pair (35) in R123C7 for C7 since only other place possible is R7C7
b) 17(3) <> {359} since it's blocked by Killer pair (35) of R123C7
c) 15(3) <> 1 because {168} is a Killer triple of 17(3)
d) ! Hidden Killer pair (26) in R123C9 for C9 because 10(3) can't have both of them
e) 13(3) <> 6 because {256} blocked by Killer pair (26) in R123C9
f) 17(3) <> 1,7 because {179} blocked by Killer pair (17) of 13(3)
g) 1 locked in R456C9 for N6
h) R6C7 = 7 -> 14(3) = 7{34} -> {34} locked for R7+N9
i) R7C9 = 9 -> 18(3) = 9{18/45}; R6C9 <> 8
j) 5 locked in R123C7 @ C7 for N3

7. N7
a) Naked triple (568) locked in R7C123
b) 19(3) = {568}
c) 11(3) must have 5 xor 8 and R7C3 = (58) -> R6C23 <> 5

8. R123+C6 !
a) 17(3) = 6{29/38} -> 6 locked for N3
b) 13(3) <> 3,9 because {139} blocked by Killer pair (39) of 15(3) @ N3
c) 1 locked in 13(3) @ N3 = 1{48/57}
d) 15(3) @ R1C4 <> {348} because of Killer pair (34) of 15(3) @ N3
e) 15(3) @ R2C4, R2C5 <> {267} since it's a Killer triple of 15(3) @ R1C4
f) 15(3) @ R2C5 = {168/348/357} <> 2
g) ! Hidden Killer pair (24) in R123C6 for C6 since 10(3) can only have one of them
h) 15(3) @ R2C4 <> 2 because {249} blocked by Killer pair (24) of R123C6
i) 2 locked in 15(3) @ R1C4 = 2{49/67} for R1
j) 15(3) @ N3 = 3{48/57} -> 3 locked for R1+N3
k) 15(3) @ N1 = 1{59/68} -> 1 locked for N1

9. R123+C19
a) 17(3) = {269} -> R2C8 = 9, {26} locked for C9
b) 10(3) must have 2 xor 6 -> R4C8 = (26)
c) 10(3) = 3{16/25} -> 3 locked for C9
d) 11(3) = 2{36/45} -> 2 locked for N1
e) 7 locked in 19(3) @ N1 = 7{39/48}
f) 2 locked in R89C1 for N7
g) 15(3) @ R2C4 <> 4 because {348} blocked by Killer pair (38) of 15(3) @ R2C5

10. C456 !
a) ! Innies+Outies C6: 22 = R2468C5 - R1C6, R1C6 = (246) -> R4C5 <> 7 because:
- R2468C5 = 24/26/28(4), now consider all combos with 7 in R4C5 = 7{359/368/458/469/489/568}
- <> 7{359/458/469/568/489} because 4,5 only possible @ R4C5
- <> {3678} because 7 in R4C5 forces 10(3) = 7{12} -> no 2 in R1C6

b) 10(3) = {145} locked for N5
c) Hidden Single: R1C6 = 2 @ C6 -> 15(3) @ R1C4 = [76/94]2
d) Hidden Single: R9C4 = 4 @ C4, R9C5 = 5
e) 10(3) = {145} -> 5 locked for C6
f) 15(3) @ R2C5 = 8{16/34} -> 8 locked for N2
g) Hidden Single: R5C5 = 7 @ C5
h) 17(3) = {278} -> 2 locked for C4
i) R7C4 = 1, R7C5 = 2
j) 15(3) @ R2C4 = 5{19/37}
k) Outies C6 = 24(4) = 89{16/34}: Outies must have 1 xor 4 and R4C5 = (14) -> R2C5 <> 1

11. R123 !
a) 19(3): R3C12 <> 3 because 7{39} blocked by Killer pair (39) of 15(3) @ R2C4
b) ! Consider both candidates of R1C5 = (46) -> R6C5 <> 6:
- i) R1C5 = 4 -> 15(3) @ R1C7 = [537] -> R9C9 = 8 -> R9C6 = 9 -> R6C6 = 6 -> R6C5 <> 6
- ii) R1C5 = 6 -> R6C5 <> 6
c) R6C5 = 9, R6C6 = 6
d) Hidden pair (26) in R3C39 for R3 -> R3C3 = (26)
e) 3 locked in R2C123 @ N1 for R2
f) Consider combos of 11(3) -> R2C1 = (37):
- i) 11(3) = {236} -> 15(3) @ N1 = {159} -> 15(3) @ R1C4 = [762] -> 15(3) @ R2C5 = [843] -> R2C1 = 7
- ii) 11(3) = {245} -> 19(3) = {379} -> R2C1 = 3
g) ! Consider placement of R2C1 -> R9C2 = 3:
- i) R2C1 = 3 -> R9C2 = 3 (HS @ N7)
- ii) R2C1 = 7 -> R2C4 = 5 -> R3C7 = 5 (HS @ R3) -> 13(3) = [157] -> R9C8 = 1 (HS @ N9) -> R9C2 = 3

12. C123
a) 6(3) = {123} -> 1 locked for C1
b) 16(3) <> {358} since it's blocked by R6C1 = (58)
c) 16(3): R5C2 <> 8 because R45C1 <> 1
d) 16(3): R5C2 <> 6 because R45C1 <> 1 and [736] blocked by R2C1 = (37)
-> 16(3) = {169/178/349/457} (because R5C2 = (1459))
e) Killer pair (14) locked in 16(3) + 11(3) for N4
f) 16(3) <> 6 because {69}1 together with R67C1 leaves no candidate for R1C1

13. C123 !
a) 6 locked in 18(3) = 6{39/57}
b) 8 locked in R456C1 for C1
c) Killer pair (37) locked in R2C1 + 16(3) for C1
d) 19(3) @ N1 must have 4 xor 9 and R3C1 = (49) -> R3C2 <> 4,9
e) 16(3): R5C2 <> 9 because {34}9 blocked by Killer pair (34) of 19(3) @ N1
f) ! R1C1 <> 9 because either 19(3) @ N1 = [397] or 19(3) = [748] -> 16(3) = {39}4
g) 15(3) @ N1 must have 5 xor 6 and R1C1 = (56) -> R1C23 <> 5,6
h) Naked pair (56) locked in R17C1 for C1
i) R6C1 = 8

14. R456
a) 16(3) = 4{39/57} -> 4 locked for N4
b) 11(3) = {128} -> R7C3 = 8; 1 locked for R6
c) Naked pair (45) locked in R6C89 for N6
d) 10(3) @ N6 = {136} -> R4C8 = 6

15. Rest is singles.

Rating: 2.0. I used lots of forcing chains and combo analysis (but not as heavy as in A60 RP) to solve this monster.
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Andrew
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PostPosted: Sat Jan 10, 2009 5:21 am    Post subject: Reply with quote

About 9 months ago Afmob wrote:
But after being encouraged by Andrew who will also post his wt for A74 Brick Wall, I finally decided to solve it again ....

A couple of weeks ago I found time to go through Para's walkthrough and Afmob's much improved second walkthrough. Great stuff from both of you! Applause

Now I've gone through my walkthrough and tried to clarify some steps.

This is by far the hardest Assassin that I've ever done by myself. The rating for the way I solved it must be at least 2.0. It's got far more combination and permutation analysis that in the walkthroughs by Afmob and Para but less chains although I now realise that my combination analysis includes implied chains for some of the clashes.

Here is my walkthrough. I've included summaries at the end of the very heavy analysis steps. I'm not sure that I'd recommend anyone to work right through it although I think some of the hidden killers are interesting and the later ones may not be in the other posted walkthroughs.

Prelims

a) 19(3) cage in N1 = {289/379/469/478/568}, no 1
b) 11(3) cage in N1 = {128/137/146/236/245}, no 9
c) 10(3) cage in N5 = {127/136/145/235}, no 8,9
d) 19(3) cage in N6 = {289/379/469/478/568}, no 1
e) 10(3) cage in N6 = {127/136/145/235}, no 8,9
f) 19(3) cage at R6C1 = {289/379/469/478/568}, no 1
g) 11(3) cage at R6C2 = {128/137/146/236/245}, no 9
h) 6(3) cage at R6C4 = {123}, CPE no 1,2,3 in R89C4
i) 22(3) cage at R6C5 = 9{58/67}
j) 6(3) cage in N7 = {123}, locked for N7
k) 20(3) cage in N7 = {479/569/578}
l) 20(3) cage in N8 = {389/479/569/578}, no 1,2
m) 9(3) cage in N9 = {126/135/234}, no 7,8,9
n) 20(3) cage in N9 = {389/479/569/578}, no 1,2

1. Hidden pair {12} in R8 -> R8C17 = {12}
1a. 3 in N7 locked in R9C12, locked for R9
1b. 9(3) cage in N9 = {126} (only remaining combination), locked for N9
1c. 6 in N9 locked in R9C78, locked for R9
1d. Hidden pair {12} in R7C45, locked for N8 and 6(3) cage -> R6C4 = 3
1e. 3 in R7 locked in R7C789, locked for N9

2. 45 rule on N8 1 remaining innie R7C6 = 7
2a. R6C56 = {69}, locked for R6 and N5
2b. 17(3) cage in N5 = {278/458}, no 1

3. 15(3) cage at R8C4 = {456} (only remaining combination) -> R8C4 = 6, R9C45 = {45}, locked for R9 and N8

4. 9 in C4 locked in R123C4, locked for N2
4a. One of the 15(3) cages at R1C4 or R2C4 must be 9{15/24}
4b. 15(3) cage at R2C5 = {168/267/348/357} (cannot be {258/456} which clash with the 15(3) cage containing 9)
4c. 7 of {267/357} must be in R2C5 -> no 2,5 in R2C5
4d. 15(3) cage at R2C4 = {159/168/249/267/348/357} (cannot be {258/456} which clash with the 15(3) cage containing 9 and, if it contains 9 it clearly cannot contain either of those combinations)
4e. 3,6 of {168/267/348/357} must be in R3C5 -> no 7,8 in R3C5
4f. 3,6 must be in different 15(3) cages and neither in the one containing 9 -> one of the two cages in step 4b must be {168/267} and the other one must be {348/357}

5. 45 rule on N7 1 outie R6C1 = 1 innie R7C3, no 2,7 in R6C1, no 6 in R7C3

6. 45 rule on N9 1 innie R7C9 = 1 outie R6C7 + 2, no 4,5,8 in R6C7, no 5,8 in R7C9

7. 11(3) cage at R6C2 = {128/245}, no 7, 2 locked in R6C23 for R6 and N4, clean-up: no 4 in R7C9 (step 5)
7a. 8 of {128} must be in R7C3 -> no 8 in R6C23
7b. 7 in R6 locked in R6C789, locked for N6
7c. R6C1 = R7C3 (step 5) -> R6C123 = {128/245}

8. 19(3) cage at R6C1 = {469/568}
8a. 4 of {469} must be in R6C1 -> no 4 in R7C12
8b. R6C1 = R7C3 (step 5) -> R7C123 = {469/568}

9. 10(3) cage in N5 = {127/145}
9a. 7 of {127} must be in R4C5 -> no 2 in R4C5

10. 9 in N6 locked in 19(3) cage = {289/469}, no 3,5
10a. 3 in N6 locked in 10(3) cage = {136/235}, no 4

11. 9 locked in R123C4, 45 rule on C4 4 outies R3579C5 = 1 innie R1C4 + 8, min R3579C5 = 10 -> min R1C4 = 2
11a. If R1C4 = 2 => R3579C5 = 10 = {1234}, R7C4 = 1, R7C5 = 2, R9C5 = 4 -> {1234} blocked by R5C5 -> no 2 in R1C4
11b. If R1C4 = 4 => R3579C5 = 12 = {1236/1245}, R9C4 = 5, R9C5 = 4 blocks {1236} but {1245} can be [5214] (cannot be [1524/2514] because 15(3) cage at R2C4 must be {19}5 to avoid clash with R9C4)
11c. If R1C4 = 5 => R3579C5 = 13 = {1237/1246/1345}, R9C4 = 4, R9C5 = 5 blocking {1237/1246}, 3 of {1345} can only be at R3C5 and that clashes with 15(3) cage at R2C4 which can only be {29}4 -> no 5 in R1C4
[Nothing useful was obtained at this stage from analysing R1C4 = 7,8,9]

12. For the same reasons as in step 4, one of the 15(3) cages in R1 must be 9{15/24} and the other two must be {168/267} and {348/357}
12a. R1C123 = {159/168/249/357} (cannot be {267/348} which clash with the 11(3) cage, they also clash with the 19(3) cage)
12b. R1C456 and R1C789 cannot be {159} which clash with R1C123

13. 1 in C4 must be in R23C4 or in R7C4
13a. If 1 in R7C4, R7C5 = 2 -> no 2 in R3C5
13b. If 1 in R23C4, 15(3) cage at R2C4 = 1{59/68} -> no 2 in R3C5
13c. -> no 2 in R3C5

14. 45 rule on C6 4 outies R2468C5 = 1 innie R1C6 + 22
14a. R2468C5 cannot be {6789}
14aa. If R2468C5 = [6798] => R45C6 = {12}, R23C6 = {18} (step 4b) clashes with R45C6
14ab. If R2468C5 = [8769] => R45C6 = {12}, R89C6 = [38], R23C6 = {16/34} (step 4b) clashes with R45C6 + R8C6
14ac. -> no 8 in R1C6

15. Hidden killer quad 3,6,8,9 in R123C6, R6C6 and R89C6 for C6 -> R123C6 contains one of 3,6,8
15a. 15(3) cage at R2C5 (step 4b) = {168/267/348/357}
15b. R23C6 cannot be {68} (step 15) -> no 1 in R2C5
15c. R23C6 cannot be {38} (step 15) -> no 4 in R2C5
Consider now the implications of step 15 when the 15(3) cages at R1C4 and R2C5 interact
15d. 15(3) at R2C5 = {168} => no 3 in R1C6 (step 15) => R1C456 = [735]/9{24}
15e. 15(3) at R2C5 = {267} => no 3 in R1C6 (step 15) => R1C456 = [834]
15f. 15(3) at R2C5 = {348} => no 6 in R1C6 (step 15) => R1C456 = [762]
15g. 15(3) at R2C5 = {357} => no 6 in R1C6 (step 15) => R1C456 = [861]/9{24}
15h. -> no 4 in R1C4, no 1,5,7,8 in R1C5, no 3,6 in R1C6
15i. R1C456 = [735/762/834/861/924/942]

16. 15(3) cage at R2C4 (step 4d) = {159/168/249/267/348/357} contains one of 7,8,9 in R23C4
16a. Hidden killer triple 7,8,9 in R1C4, R23C4 and R45C4 for C4 -> R45C4 must contain one of 7,8
16b. 17(3) cage in N5 (step 2b) = {278/458}
16c. R45C4 cannot be {78} -> no 2 in R5C5

17. R2468C5 (step 14) = R1C6 + 22, 9 in C5 locked in R68C5
17a. If R1C6 = 1 => R1C5 = 6 (step 15i) => R2468C5 = 23 = {3479} (cannot be {1589} because 1,5 only in R4C5, cannot be {1679/3569} which clash with R1C5)
17aa. {3479} can only be [7493] => R45C6 = {15} clashes with R1C6 -> no 1 in R1C6
17b. If R1C6 = 2 => R1C5 = {46} (step 15i) => R2468C5 = 24 = {1689/3489/3579} (cannot be {4569} which clashes with R1C5)
17ba. {1689} can be [6198/8169]
17bb. {3489} can be [3498/8493]
17bc. {3579} can only be [7593]
17c. If R1C6 = 4 => R2468C5 = 26 = {4679} (cannot be {3689} because R4C5 only contains 1,4,5,7, cannot be {4589} because 4,5 only in R4C5)
17ca. {4679} can only be [7469] => R23C6 = {26} (cannot be {35} which clashes with R45C6 = {15}) -> R1C456 cannot be [924]
17d. If R1C6 = 5 => R1C456 = [735] (step 15i) => R2468C5 = 27 = {4689/5679} (cannot be {3789} which clashes with R1C5)
17da. 4 of {4689} can only be in R4C5 => R45C6 = {15} clashes with R1C6 -> cannot be {4689}
17db. 7 of {5679}can only be in R2C5 which clashes with R1C456 -> cannot be {5679}
17dc. -> no 5 in R1C6
17e. Summary no 1,5 in R1C6, R1C456 = [762/834/942], no 2 in R1C5, R2468C5 = [6198/8169/3498/8493/7593/7469], no 7 in R4C5

18. R7C5 = 2 (hidden single in C5), R7C4 = 1
18a. 10(3) cage in N5 = {145} (only remaining combination), locked for N5
18b. 2 in N5 locked in R45C4, locked for C4

19. 15(3) cage at R2C4 (step 4d) = {159/348/357} (cannot be {168} because 1,6 only in R3C5), no 6
19a. 1,3 only in R3C5 -> R3C5 = {13}

20. R1C123 (step 12a) = {159/168/357} (cannot be {249} which clashes with R1C456), no 2,4
20a. 11(3) cage in N1 = {128/146/236/245} (cannot be {137} which clashes with R1C123), no 7
20b. 19(3) cage in N1 = {289/379/469/478} (cannot be {568} which clashes with R1C123), no 5

21. R3579C5 = R1C4 + 8 (step 11), R7C5 = 2 -> R359C5 = R1C4 + 6
21a. R1C4 = {789} -> R359C5 = 13,14,15
21b. If R1C4 = 7 => R45C4 = {28} => R5C5 = 7 => R359C5 = 13 = [175]
21c. If R1C5 = 8 => R45C4 = {27} => R5C5 = 8 => R359C5 = 14 = [185]
21d. If R1C6 = 9 => R1C5 = 4 (step 17e), R359C5 = 15 = [375] (cannot be [384] which clashes with R1C5)
21e. -> R359C5 = [175/185/375] -> R9C5 = 5, R9C4 = 4

22. 5 in C4 locked in R23C4, locked for N2
22a. 15(3) cage at R2C4 (step 19) = {159/357}, no 8

23. R1C789 = {168/267/348/357} (cannot be {249} which clashes with R1C456), no 9
23a. 13(3) cage in N3 = {139/148/157/247/256} (cannot be {238/346} which clash with R1C789
23b. R3C78 cannot be {13} which clashes with R3C5 -> no 9 in R2C7
23c. 17(3) cage in N3 = {179/269/359/458} (cannot be {278/368/467} which clash with R1C789)

24. 10(3) cage in N6 (step 10a) = {136/235}
24a. R45C9 may contain one, but not both, of 2,6
24b. Hidden killer pair 2,6 in R123C9 and R45C9 for C9 -> R123C9 must contain at least one of 2,6
24c. 13(3) cage in N3 = {139/148/157/247} (cannot be {256} which clashes with R123C9), no 6
24d. 5 of {157} must be in R23C7 because R23C7 = {17} clashes with R6C7 -> no 5 in R3C8
24e. 17(3) cage in N3 (step 23c) = {269/359/458} (cannot be {179} which clashes with 13(3) cage), no 1,7
[Alternatively can use hidden killer triple because the three cages in N3 each require at least one of 7,8,9 so can each only have one of 7,8,9. I saw that before the clash with the 13(3) cage but gave the clash as step 24e because itís simpler.]

25. R7C7 may contain one, but not both, of 3,5
25a. Hidden killer pair 3,5 in R123C7 and R67C7 for C7 -> R123C7 must contain at least one of 3,5
25b. 17(3) cage in N3 (step 24e) = {269/458} (cannot be {359} which clashes with R123C7), no 3
25c. 13(3) cage in N3 = {139/148/157} (cannot be {247} which clashes with 17(3) cage), no 2, 1 locked for N3

26. 1 in R1 locked in R1C123, locked for N1
26a. R1C123 = {159/168}, no 3,7
26b. 11(3) cage in N1 (step 20a) = {236/245}, no 8, 2 locked for N1
26c. Killer pair 5,6 in R1C123 and 11(3) cage, locked for N1
26d. 19(3) cage in N1 (step 20b) = {379/478}

27. 1 in C9 locked in R456C9, locked for N6 -> R6C7 = 7, R7C9 = 9 (step 6)
27a. R6C7 = 7 -> R7C78 = 7 = {34}, locked for R7 and N9, clean-up: no 4 in R6C1 (step 5)
27b. Naked triple {568} in R7C123, locked for N7
27c. R7C9 = 9 -> R6C89 = 9 = {45}/[81], no 8 in R6C9
27d. Killer pair 5,8 in R6C1 and R6C89, locked for R6
[Alternatively can eliminate 5 from R6C23 because 5 of {245} must be in R7C3]

28. 5 in C7 locked in R123C7, locked for N3, clean-up: no 4,8 in 17(3) cage (step 25b)
28a. 17(3) cage = {269} -> R2C8 = 9, R23C9 = {26}, locked for C9 and N3

29. R1C789 (step 23) = {348/357}, 3 locked for R1 and N3, clean-up: no 8 in R1C4, no 4 in R1C6 (both step 17e) -> R1C6 = 2

30. 8 in C4 locked in R45C5, locked for N5 -> R5C5 = 7

31. 10(3) cage in N6 (step 10a) = {136/235}
31a. 2,6 only in R6C8 -> R6C8 = {26}
31b. 3 locked in R45C9, locked for C9

32. R1C789 (step 29) = {348/357}
32a. 7 of {357} must be in R1C9 -> no 7 in R1C8

33. 16(3) cage in N4 = {169/178/349/367/457} (cannot be {358} which clashes with R6C1)
33a. 7 of {178} must be in R4C1 -> no 8 in R4C1

34. 2 in C1 locked in R89C1, locked for N7

35. 11(3) cage in N1 (step 26b) = {236/245}
35a. R2C23 cannot be {26} which clashes with R2C9 -> no 3 in R3C3

36. 19(3) cage in N6 (step 10) = {289/469}
36a. {289} = [298/892] (cannot be 9{28} which clashes with R5C4) -> no 2,8 in R5C7
36b. {469} => R4C8 = 2
36c. Killer pair 2,8 in R4C4 and R4C78, locked for R4

37. 18(3) cage in N4 = {189/369/378/459/567} (cannot be {468} which clashes with R6C123)
37a. 8 of {189} must be in R5C3 -> no 1 in R5C3

38. 1 in C8 locked in R39C8
38a. 5 in C7 locked in R123C7
38b. If 5 in R23C7 => R23C7 = {15}, R89C7 = [26], R3C8 = 7, R9C8 = 1
38c. 45 rule on C7 4 outies R3579C8 = 1 innies R1C7 + 10
38d. If R1C7 = {348} => R3579C8 = 13,14,18 = [7231/7241/7641] (R39C8 must be [71] (step 38b), cannot be {1278} because R7C8 only contains 3,4)
If R1C7 = 5 => R3579C8 = 15 = {1248} (cannot be {1347} because 13(3) cage in N3 can only be {148} when R1C7 = 5, cannot be {2346} which clashes with R4C8) = [1842/8241]
38e. -> R3579C8 = [1842/7231/7241/7641/8241], no 4 in R35C8, no 6 in R9C8

39. R9C7 = 6 (hidden single in R9)

40. R1C5 = {46}
40a. If R1C5 = 4 => R1C789 (step 29) = [537] => R9C9 = 8 => R9C6 = 9 => R6C6 = 6 => no 6 in R6C5
40b. If R1C5 = 6 => no 6 in R6C5
40c. -> R6C5 = 9, R6C6 = 6

41. Hidden pair {26} in R3C39
41a. 11(3) cage in N1 = {23}6/{36}2/{45}2

42. 1 in C5 locked in R34C5
42a. If R3C5 = 1 => R2C7 = 1 => R9C8 = 1 => R8C1 = 1 => no 1 in R4C1
42b. If R4C5 = 1 => no 1 in R4C1
42c. -> no 1 in R4C1

43. 16(3) cage in N4 (step 33) = {169/178/349/367/457}
43a. 3 of {349} must be in R5C12 (cannot be 3{49} which clashes with R5C7), 7 of {367} must be in R4C1 -> no 3 in R4C1
43b. 7 of {457} must be in R4C1 -> no 5 in R4C1

44. 4 in C5 locked in R14C5
44a. If R1C5 = 4 => 4 in N3 locked in R23C7 => no 4 in R4C7
44b. If R4C5 = 4 => no 4 in R4C7
44c. -> no 4 in R4C7

45. Hidden killer quad 1,3,4,5 in R4C123, R4C56 and R4C9 for R4 -> R4C123 must have one of 1,3,4,5
45a. 7 in R4 locked in R4C123
45b. 18(3) cage in N4 (step 37) = {189/369/378/459/567}
45c. 3 of {369} must be in R4C23 (from hidden killer quad because R4C1 cannot be 4 when 18(3) cage = {369}, steps 43 and 43b), 8 of {378} must be in R5C3 -> no 3 in R5C3

46. R2C4 = {57}
46a. If R2C4 = 5 => no 5 in R2C7
46b. If R2C4 = 7 => R1C45 = [94] => R1C789 (step 29) = [537] => no 5 in R2C7
46c. -> no 5 in R2C7

47. Hidden killer triple 2,6,8 in R5C123, R5C4 and R5C8 for R5 -> R5C123 must contain one of 6,8
47a. 16(3) cage in N4 (step 33) = {169/178/349/367/457}
47b. 18(3) cage in N4 (step 37) = {189/369/378/459/567}
47c. 7 in R4 locked in R4C123
47d. If 18(3) = {189} => R6C123 = {245} (step 7c) => R4C1 = 7 => R5C12 = {36} would place both of 6,8 in R5C123
47e. 18(3) cage in N4 = {369/378/459/567}, no 1
47f. If {369/459}, 7 must be in R4C1
47g. If {378} => R5C3 = 8 => R6C123 = 5{24} => 16(3) = {169} => no 6 in R5C12 (step 47) => R4C1 = 6
47h. If {567} => R6C123 = 8{12} => 16(3) = {349}, 6 must be in R5C3 (step 47), R4C23 = {57} => R4C1 = 9 (cannot be 4 because of step 45)
47i. -> R4C1 = {679}, no 4

48. 2 in C2 locked in R26C2
48a. 45 rule on C3 4 outies R2468C2 = 1 innie R1C3 + 15
48b. If R1C3 = 1 => R2468C2 = 16 = {1249/1267/2347} (cannot be {2356} because R8C2 only contains 4,7,9)
48c. If R1C3 = 5 => R2468C2 = 20 = {2369/2459/2567}
48ca. Cannot be {2459} because R26C2 = {24} => R8C2 = 9, R89C3 = [47], R4C2 = 5, R45C3 = {49/67} clash with R89C3
48cb. 5 of {2567} must be in R4C2
48d. If R1C3 = 6 => R3C3 = 2, R2C23 = {45}, R2468C2 = 21 = {2469} (cannot be {2379} because R2C2 only contains 4,5)
48e. If R1C3 = 8 => R1C12 = {16}, R3C3 = 2, R2C23 = {45}, R6C2 = 2, R2468C2 = 23 = {2579}
48ea. Cannot be {2579} because R2C2 = 5, R23C3 = [42], R6C2 = 2 (step 48), R67C3 = [18/45] clashes with R12C3
48f. If R1C3 = 9 => R2468C2 = 24 = {2679}

Summary of step 48.
From steps 48e and 48ea, no valid combinations with R1C3 = 8
From steps 48b, 48cb, 48d and 48f, no valid combinations with R2C2 = 5
-> no 8 in R1C3, no 5 in R2C2, clean-up: no 4 in R2C3 (step 41a)
R2468C2 = {1249/1267/2347/2369/2469/2567/2679}

49. 8 in C3 locked in R57C3
49a. R6C1 = R7C3 (step 5) -> 8 in N4 locked in R5C3 + R6C1 -> no 8 in R5C12

50. 45 rule on C1 4 outies R3579C2 = 1 innie R1C1 + 15
50a. If R1C1 = 1 => R9C2 = 1, R3579C2 = 16 = {1348/1357/1456}
50aa. {1348} must be [4381] (cannot be [3481] because R23C1 = [79], R45C1 = [75/93] clashes with R23C1)
50ab. Cannot be {1357} because R9C2 = 1, R89C1 = [23], R7C2 = 5, R67C1 = [86], R3C2 = 7, R23C1 = [39]/{48} clashes with R67C1 or R89C1
50ac. Cannot be {1456} because R3C2 = 4, R23C1 = {78}, R7C2 = {56}, 8 in R67C1 clashes with R23C1
50b. If R1C1 = 5 => R7C2 = 5, R3579C2 = 20 = {1568/3458} = [8453/8651]
50c. If R1C1 = 6 => R7C2 = 6, R3579C2 = 21 = {1569/3468/3567}
50ca. Cannot be {1569} because R9C2 = 1, R89C1 = [23], R3C2 = 9, R23C1 = {37} clashes with R9C1
50cb. {3468/3567} = [7563/8463]
50d. If R1C1 = 8 => R7C2 = 8, R3579C2 = 23 = {1589/3578} (cannot be {4568} because R9C2 only contains 1,3)
50da. Cannot be {1589} because R9C2 = 1, R89C1 = [23], R5C2 = 5, R3C2 = 9, R23C1 = {37} clashes with R9C1
50db. {3578} = [7583]
50e. If R1C1 = 9 => R3579C2 = 24 = {1689/3489/3579/3678} (cannot be {4569/4578} because R9C2 only contains 1,3)
50ea. Cannot be {3489/3579} because R9C2 = 3, R89C1 = {12}, R5C2 = 9, R45C1 = [61] clashes with R89C1
50eb. {1689/3678} = [7683/8961]

Summary of step 50
The only combination for R3579C2 with both of 1,3 is {1348} in step 50a when 1 must be in R9C2 -> no 1 in R5C2
3 of {1348} must be in R5C2 (step 50aa) -> no 3 in R3C2
Cannot be {1569/1589} (steps 50ca and 50da), all other combinations with 9 are for R1C1 = 9 -> no 9 in R3C2
R3579C2 = [4381/7563/7583/7683/8453/8463/8651/8961]

51. 16(3) cage in N4 (step 33) = {169/349/367/457}
51a. 1 of {169} must be in R5C1, 9 of {349} must be in R4C1 -> no 9 in R5C1

52. 19(3) cage in N1 (step 26d) = {379/478}
52a. 9 of {379} must be in R3C1 -> no 3 in R3C1
52b. 3 in R3 locked in R3C56, locked for N2

53. 15(3) cage at R2C5 (step 4b) = {168/348}
53a. 3 of {348} must be in R3C6 -> no 4 in R3C6

54. 11(3) cage in N1 (step 26b) = {236/245}
54a. If {236} => naked triple {236} in R2C239, locked for R2 => R2C5 = 8, R2C6 = 4 (step 53) => R2C1 = 7
54b. If {245} => 19(3) cage = {389} => R2C1 = 3
54c. -> R2C1 = {37}

55. 19(3) cage in N1 (step 26d) = {379/478}
55a. 7 of {379} must be in R3C2, 7 of {478} must be in R2C1 -> no 7 in R3C1

56. 16(3) cage in N4 (step 51) = {169/349/367/457}
56a. 6 of {169} must be in R4C1 (step 47g), 6 of {367} must be in R5C1 (cannot be [736] which clashes with R2C1) -> no 6 in R5C2
56b. 18(3) cage in N4 (step 47e) = {369/378/459/567}
56c. If {369} => 16(3) = {457} => 6 must be in R5C3 (step 47)
56d. If {567} => 5 must be in R4C23 (step 45) => 6 must be in R5C3
56e. Combining steps 56c and 56d -> no 6 in R4C23
56f. If {459} => 9 must be in R4C23 (cannot be {45}9 which clashes with R4C56)
56g. Combining steps 56c and 56f -> no 9 in R5C3

57. R2468C2 (step 48 summary) = {1249/2347/2369/2469/2567/2679} (cannot be {1267} because 1,2,6 only in R26C2)
57a. For {2369/2469/2567/2679} 6 must be in R2C2
57b. {1249} must have 1 in R6C2 and 2 in R2C2
57c. If {2347} => R1C3 = 1 (step 48b), {2347} = [2347/3724] (cannot be [3427/4327] because R89C3 = [49], R67C3 = [18/45] clashes with R1C3 or R89C3)
57d. Combining steps 57a, 57b and 57c -> no 4 in R2C2, clean-up: no 5 in R2C3 (step 41a)

58. Naked triple {236} in 11(3) cage in N1, locked for N1 -> R2C1 = 7, R2C4 = 5, clean-up: no 8 in R1C12 (step 26a)
58a. Naked triple {159} in R1C123, locked for R1 and N1 -> R1C4 = 7, R1C5 = 6 (step 17e), R3C4 = 9, R2C5 = 8, R8C5 = 3, R3C5 = 1, R23C6 = [43], R4C5 = 4, R2C7 = 1, R8C7 = 2, R9C8 = 1, R8C1 = 1, R9C12 = [23]
58b. Naked pair {48} in R3C12, locked for R3 -> R3C78 = [57]

59. 7 in R4 locked in R4C23, 18(4) cage (step 47e) = {378/567}, no 4,9
59a. 6,8 only in R5C3 -> R5C3 = {68}

60. 9 in N4 locked in 16(3) cage (step 51) = {349} (only remaining combination), locked for N4 -> R4C1 = 9

and the rest is naked singles

Here is the solution; I think it's only been given earlier in this thread in TT.

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