(Unofficial) Assassin 99

[Afmob]

This assassin should warm you up on basic killer techniques.

Unofficial Assassin 99 (UA 99)



3x3::k:38405378:5378358923112569:4618:4618:53783589:4111:41112569:4618:4884:48844887:4887:41112569:4892:4884:48844887:4887:54104892:4892:4390:4390:4390:4390:4390:5410:5410:4141:4892:5935:59355170:5170:5410:5173:41415935:59355170:5170:4925:5173:4141235918594925:4925:51733144365039081871

Solution:

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

Estimated rating: 0.75 - 1.0
SS Score: 0.93

I've built this Killer around the four square cages. Some small modifications in the cage pattern can turn it into a nightmare (SS Rating: 5.0!).
A tougher version will be posted on Sunday if not requested earlier.

[frank]

Very nice Afmob. Most enjoyable. Thanx.

[Andrew]

Thanks Afmob for a fun puzzle!

This assassin should warm you up on basic killer techniques.

In at least one place I deliberately avoided a more difficult move, which gave an immediate elimination, because this puzzle didn't require such a move.

I'll rate uA99 at 1.0; there was enough work the way I did it not to rate it lower.

Here is my walkthrough. I've corrected a few typos and made the mop-up a bit quicker.

Prelims

a) R1C12 = {69/78}
b) R12C5 = {14/23}
c) R1C89 = {18/27/36/45}, no 9
d) R34C5 = {59/68}
e) R67C5 = {78} (cannot be {69} which clashes with R34C5), locked for C5, clean-up: no 6 in R34C5
f) R89C5 = {16/25/34}, no 7,8,9
g) R9C12 = {39/48/57}, no 1,2,6
h) R9C89 = {16/25/34}, no 7,8,9
i) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
j) R234C1 = {127/136/145/235}, no 8,9
k) R234C9 = {128/137/146/236/245}, no 9
l) R678C9 = {389/479/569/578}, no 1,2
m) 9(3) cage in N7 = {126/135/234}, no 7,8,9
n) 19(3) cage in N9 = {289/379/469/478/568}, no 1
o) R5C34567 = {12347/12356}, no 8,9, 1,2,3 locked for R5

1. 45 rule on C5 1 innie R5C5 = 4, clean-up: no 1 in R12C5, no 3 in R89C5
1a. 45 rule on C1234 2 innies R5C34 = 4 = {13}, locked for R5
1b. R5C34567 = {12347} (only remaining combination) -> R5C67 = {27}, locked for R5

2. Naked pair {23} in R12C5, locked for C5 and N2, clean-up: no 5 in R89C5
2a. Naked pair {16} in R89C5, locked for N8

3. R9C89 = {25/34} (cannot be {16} which clashes with R9C5), no 1,6
3a. 1 in N9 locked in R79C7, locked for C7

4. 45 rule on C1 3 innies R159C1 = 19 = {379/469/478/568}
4a. 3,4 of {379/469/478} must be in R9C1 -> no 7,9 in R9C1, clean-up: no 3,5 in R9C2

5. 45 rule on R1 2 outies R2C46 = R1C5 + 14, R1C5 = {23} -> R2C46 = 16,17 = {79/89}, 9 locked for R2 and N2 -> R34C5 = [59]
5a. No 3 in R3C2 because R2C23 = {78} clashes with R2C46
5b. No 1 in R3C8 because R2C78 = {78} clashes with R2C46

6. 21(3) cage at R1C3 = {489/579} (cannot be {678} because {67/68} clash with R1C12), no 6
6a. 7 of {579} must be in R1C4 -> no 7 in R1C3 + R2C4

7. R1C89 = {18/27/36} (cannot be {45} which clashes with R1C34 in 21(3) cage), no 4,5

8. Min R2C6 = 7 -> max R1C67 = 7, no 7,8,9 in R1C67

9. 9 in R1 locked in R1C123, locked for N1
9a. 18(3) cage in N1 = {378} (only remaining combination, cannot be {468/567} which clash with R1C12), locked for N1
9b. 3 locked in R2C23, locked for R2 -> R2C5 = 2, R1C5 = 3, clean-up: no 6 in R1C89

10. R2C46 = R1C5 + 14 (step 5), R1C5 = 3 -> R2C46 = 17 = {89}, no 7, 8 locked for R2 and N2

11. Naked pair {37} in R2C23, locked for R2 and N1 -> R3C2 = 8, clean-up: no 4 in R9C1

12. Naked pair {69} in R1C12, locked for R1 and N1

13. 21(3) cage at R1C3 (step 6) = {489/579} -> R2C4 = 9, R2C6 = 8
13a. R1C34 = 12 = [57], clean-up: no 2 in R1C89

14. Naked pair {18} in R1C89, locked for R1 and N3 -> R1C67 = [42], R5C67 = [27]

15. 16(3) cage in N3 = {457} (only remaining combination, cannot be {349/367} because 3,7,9 only in R3C8) -> R3C8 = 7, naked pair {45} in R2C78, locked for R2 and N3 -> R2C1 = 1, R23C9 = [63], R3C7 = 9, R4C9 = 2 (cage sum), clean-up: no 4,5 in R9C8
15a. R2C1 = 1 -> R34C1 = 9 = [27/45]

16. R1C9 = 1 (hidden single in C9), R1C8 = 8

17. 7 in C9 locked in R78C9
17a. R678C9 = {479/578}
17b. Killer pair 4,5 in R678C9 and R9C9, locked for C9

18. 45 rule on R9 3 outies R8C456 = 10 = {127/145} (cannot be {136} because 1,6 only in R8C5, cannot be {235} because R8C5 only contains 1,6) -> R8C5 = 1, R8C46 = [27/45], R9C5 = 6

19. 14(3) cage at R8C4 = {239/248/257/347} (cannot be {158/167/356} because R8C4 only contains 2,4, cannot be {149} because 1,9 only in R9C3), no 1
19a. 7,9 of {239/347} must be in R9C3 -> no 3 in R9C3

20. R9C7 = 1 (hidden single in R9), R89C6 = 14 = [59], R8C4 = 4 (step 18), clean-up: no 3 in R9C1
20a. R8C4 = 4 -> R9C34 = 10 = {28}/[73]

21. 9(3) cage in N7 = {126/234} (cannot be {135} because 1,5 only in R7C2), no 5, 2 locked for N7, clean-up: no 8 in R9C4 (step 20a)
21a. 1,4 of {126/234} must be in R7C2 -> R7C2 = {14}

22. 19(3) cage in N9 = {289/469/568}, no 3
22a. 4,5 of {469/568} must be in R7C8 -> no 6 in R7C8

23. 45 rule on C12 2 outies R28C3 = 9 = [36/72]

24. 45 rule on C89 2 outies R28C7 = 13 = [58], R2C8 = 4

25. R678C9 (step 17a) = {479/578}
25a. 8 of {578} only in R6C9 -> no 5 in R6C9

26. 5 in C9 locked in R79C9, locked for N9, clean-up: no 6 in R8C8 (step 22)

27. Naked pair {29} in R78C8, locked for C8 and N9 -> R9C89 = [34], R8C9 = 7, R7C9 = 5, R6C9 = 8 (step 25), R5C9 = 9, R67C5 = [78], R79C4 = [32], R7C67 = [76], R9C23 = [78], R9C1 = 5, R2C23 = [37], R5C34 = [31], R3C46 = [61], R46C4 = [85], R46C6 = [63], R6C7 = 4, R4C7 = 3, R4C1 = 7, R3C1 = 2 (step 15a), R34C3 = [41], R7C3 = 9, R6C3 = 6 (cage sum)

and the rest is naked singles

[gary w]

Thanks for a nice puzzle,Afmob.Took me 45-50 mins and I'ld rate it about 0.75.I think it's just the sort of level of difficulty that the forum needs now and then to bring in new people who may not be quite ready for the 1.25-1.5 mind-benders.

with r5c5=4 r5c12={23} thus r2c46={79 or 89} and the innies of N1=12 it (almost) immediately follows that r1c12={69}.Innies of c1234=4 ={13}.Easy to fill in options in the rest of c5-> r3c5=5 and a little more combo work allows r1 and much of N123 to be completed,including r3c2=8. .And now with the 17(5) cage N456={13}4{27} the innies of c1=19(3) must be r159c1=685 after which it comes out quickly.

Regards

Gary

[mhparker]

This was a nice relaxing puzzle. Thanks for the warm-up, Afmob! My brain cells are now fit and ready for the next variant (unless it's one of those SScore 5.0+ ones, that is! ).

I'll go along with Andrew's rating of 1.0.

Edit: Clean-up step corrected. Thanks, Andrew!

UA99 Walkthrough

Prelims

a) 15(2) at R1C1 and R6C5 = {69/78} (no 1..5)
b) 21(3) at R1C3 = {489/579/678} (no 1..3)
c) 5(2) at R1C5 = {14/23} (no 5..9)
d) 9(2) at R1C8 = {18/27/36/45} (no 9)
e) 10(3) at R2C1 = {127/136/145/235} (no 8,9)
f) 11(3) at R2C9 = {128/137/146/236/245} (no 9)
g) 14(2) at R3C5 = {59/68} (no 1..4,7)
h) 17(5) at R5C3 = {12347/12356}; {123} locked for R5
i) 20(3) at R6C9 = {389/479/569/578} (no 1,2)
j) 9(3) at R7C2 = {126/135/234} (no 7,8,9)
k) 19(3) at R7C8 = {289/379/469/478/568} (no 1)
l) 7(2) at R8C5 and R9C8 = {16/25/34} (no 7,8,9)
m) 12(2) at R9C1 = {39/48/57} (no 1,2,6)

1. Innie C5: R5C5 = 4
1a. cleanup: no 1 in R12C5; no 3 in R89C5

2. Innies C1234: R5C34 = 4(2) = {13}, locked for R5

3. Innies C6789: R5C67 = 9(2) = {27} (last combo), locked for R5

4. Naked pair (NP) at R12C5 = {23}, locked for C5 and N2
4a. cleanup: no 5 in R89C5

5. Outies R9: R8C456 = 10(3) = {127/145} (no 3,6,8,9)
(Note: {136/235} unplaceable)
5a. -> R89C5 = [16]
5b. cleanup: no 8 in R34C5; no 9 in R67C5; no 1 in R9C89

6. Innies R6789: R6C28 = 3(2) = {12}, locked for R6

7. Innie/Outie (I/O) diff. N7: R79C3 = R6C1 + 8
7a. min. R6C1 = 3 -> min. R79C3 = 11
7b. -> no 1 in R79C3

8. Hidden single (HS) in R9 at R9C7 = 1
8a. -> split 14(2) at R89C6 = [59] (last permutation)
8b. -> R8C4 = 4 (step 5)
8c. -> split 10(2) at R9C34 = {28/37} (no 5)
8d. cleanup: no 3 in R9C12

9. Outies R1: R2C456 = 19(3) = {289/379}
9a. -> R2C4 = 9, R2C6 = {78}
9b. -> R34C5 = [59]

10. Split 12(2) at R1C34 = [48/57] (no 6; no 7,8 in R1C3)
10a. -> R1C4 blocks {78} combo for 15(2) at R1C1
10b. -> 15(2) at R1C1 = {69}, locked for R1 and N1
10c. cleanup: no 3 in R1C89

11. 18(3) at R2C2 = {378} (last combo), locked for N1

12. 9(3) at R7C2 = {126/234} (no 5)
(Note: {135} unplaceable)
12a. 1 of {126} must go in R7C2
12b. -> no 6 in R7C2
12c. 4 of {234} must go in R7C2
12d. -> no 3 in R7C2
12e. {12} in R78C2 blocked by R6C2
12f. -> no 6 in R8C3

13. Outies C12: R28C3 = 9(2) = [72] (last permutation)
13a. -> R2C56 = [28] (step 9)
13c. -> R1C5 = 3
13d. cleanup: no 3,8 in R9C4 (step 8c)

14. Split 11(2) at R23C2 = [38] (last permutation)
14a. cleanup: no 4 in R9C1

15. Split 7(2) at R78C2 = [16] (last permutation)
15a. -> R1C12 = [69]

16. 19(4) at R4C2 = [4852] (last permutation)
16a. cleanup: no 7 in R9C1

17. R1C34 (step 10) = [57]

18. Split 6(2) at R1C67 = [42] (last permutation)

19. Innies C9: R159C9 = 14(3) = [194] (last permutation)
19a. -> R9C8 = 3 (cage sum)

20. 20(3) at R6C9 (Prelim i) = {578} (last combo), locked for C9

Just naked singles left now.

[Afmob]

This variant should give you a bit more trouble since it doesn't crumble after the first placements.

Unofficial Assassin 99 V1.5 (UA 99 V1.5)



3x3::k:38405378:5378:4868358923114618:4618:5378:48684111:41112569:4618:4884:4884:4868:4887:4887:41112569:4892:4884:4884:4868:4887:4887:54104892:4892:4390:4390:4390:4390:4390:5410:5410:4141:4892:5935:5935:5681:5170:5170:5410:5173:41415935:5935:5681:5170:5170:4925:5173:4141235956814925:4925:517331443650:56813908:1871

Solution:

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

Estimated rating: 1.25
SS Score: 1.28

I don't plan to post a V2 but if you want a true nightmare then merge 15(2) with 10(3) in the upper-left corner and 7(2) with 20(3) in the lower-right corner. You get a lovely Killer which SudokuSolver rates 5.0!

[mhparker]

This variant should give you a bit more trouble since it doesn't crumble after the first placements.

Indeed! Took me over twice as long to get to all singles compared to the V1! A monster of a "1.25" rating if ever there was one! Felt like at least a 1.5 to me!

Here's the (as yet unchecked) WT. Seems huge, even in TT! There must be a quicker way...


uA99 V1.5 Walkthrough

Prelims

a) 15(2) at R1C1 = {69/78} (no 1..5)
b) 21(3) at R1C3 = {489/579/678} (no 1..3)
c) 9(2) at R1C8 = {18/27/36/45} (no 9)
d) 10(3) at R2C1 = {127/136/145/235} (no 8,9)
e) 11(3) at R2C9 = {128/137/146/236/245} (no 9)
f) 17(5) at R5C3 = {12347/12356}; {123} locked for R5
g) 20(3) at R6C9 = {389/479/569/578} (no 1,2)
h) 9(3) at R7C2 = {126/135/234} (no 7,8,9)
i) 19(3) at R7C8 = {289/379/469/478/568} (no 1)
j) 7(2) at R9C8 = {16/25/34} (no 7,8,9)
k) 12(2) at R9C1 = {39/48/57} (no 1,2,6)

1. Innie C5: R5C5 = 4

2. Innies C1234: R5C34 = 4(2) = {13}, locked for R5

3. Innies C6789: R5C67 = 9(2) = {27} (last combo), locked for R5

4. Innies R6789: R6C28 = 3(2) = {12}, locked for R6

5. Innies R1234: R4C28 = 9(2) = {18/27/36/45} (no 9)

6. 19(4) at R4C2 cannot contain more than 2 of {5689}, otherwise cage sum exceeded
6a. -> no 5,6,8 in R4C2
6b. 19(4) at R4C2 cannot contain both of {12}, because {1279} unplaceable
6c. -> no 1,2 in R4C2
6b. cleanup: no 1,3,4,7,8 in R4C8 (step 5)

7. 21(4) at R4C8 cannot contain both of {12}, otherwise cage sum unreachable
7a. -> no 2 in R4C8
7b. cleanup: no 7 in R4C2

8. 21(3) at R1C3 = {489/579/678}
8a. R1C34 cannot contain either of {67/68} due to 15(2) at R1C1 (Prelim a)
8b. -> no 6 in R1C34

9. Innies N1: R1C3+R23C1+R3C3 = 12(4) = {1245} (no 3,6..9) (last combo), locked for N1
(Note: {1236} unplaceable, because none of these digits in R1C3)

10. 21(3) at R1C3 (step 8) = {489/579} ({678} now unplaceable)
10a. -> R12C4 = {789} (no 4..6)
10b. 9 locked in R12C4 for C4 and N2

11. 9(2) at R1C8 = {18/27/36} (no 4,5) ({45} combo blocked by R1C3)

12. 15(2) at R1C1, R1C4 and 9(2) at R1C8 form killer quad on {6789} within R1
12a. -> no 6..9 elsewhere in R1

13. 9(3) at R7C2 = {126/135/234} (Prelim h)
13a. {12} in R78C2 blocked by R6C2
13b. -> no 6 in R8C3

14. Outies C12: R28C3 = 9(2) = [63/72/81]
14a. -> no 3,9 in R2C3; no 4,5 in R8C3

15. 3 in N1 locked in R23C2 for C2
15a. cleanup: no 9 in R9C1

16. R4C28 = [45] (step 5)
16a. cleanup: no 8 in R9C1, no 2 in R9C9

17. 9(3) at R7C2 = {126/135} (step 13) ({234} unplaceable)
17a. 1 locked for N7

18. 1 in C1 locked in 10(3) = {127} (last combo)
(Note: {136} unplaceable, {145} blocked by R1C3)
18a. -> R4C1 = 7; R23C1 = {12}, locked for C1 and N1
18b. cleanup: no 5 in R9C2

19. Naked pair (NP) at R13C3 = {45}, locked for C3

20. Innies C1: R159C1 = 19(3) = {469/568} (no 3)
20a. can only contain 1 of {45}
20b. -> no 5 in R5C1
20c. 6 locked in R15C1 for C1

21. Hidden single (HS) in R5 at R5C2 = 5
21a. -> split 10(2) at R5C1+R6C2 = [82/91]
21b. -> no 6 in R5C1

22. HS in C1 at R1C1 = 6
22a. -> R1C2 = 9
22b. cleanup: no 3 in R1C89, no 3 in R8C3 (step 14)

23. R2C4 = 9 (HS @ 21(3))

24. 6 of R5 locked in split 16(3) at R5C89+R6C8 = {169/268}
24a. 6 locked in R5C89 for N6

25. 9(3) at R7C2 (step 17) = {126}, locked for N7

26. Innies C9: R159C9 = 14(3) = {149/158/167/239/248} ({257/347/356} unplaceable)
26a. can only contain 1 of {689}, otherwise cage sum exceeded
26b. -> no 6,8 in R19C9
26c. cleanup: no 1 in R19C8

27. 11(3) at R2C9 = {137/146/236/245} (no 8)
(Note: {128} blocked by h14(3) (step 26))

28. (Almost) locked cages: 2 in C9 locked in R1C9 or 11(3)
28a. if 2 in R1C9, then R1C8 = 7 -> no 7 in 11(3)
28b. if 2 in 11(3), then 11(3) cannot also contain a 7 (combinations 11(3))
28c. either way, 11(3) cannot contain a 7
28d. -> 11(3) at R2C8 = {146/236/245} (no 7)

29. h14(3) at R159C9 (step 26) = {149/167/239} (no 5,8)
(Note: {248} blocked by 11(3) (step 28d),
{158} blocked because it would force both of R69C8 to 2 (step 24))
29a. cleanup: no 2 in R9C8

30. Innie/Outie (I/O) diff. C9: R19C8 = R5C9 + 2
30a. -> if R5C9 = 6, then R19C8 = [26]
30b. -> (from step 24a) 6 locked in R59C8 for C8

31. I/O diff. cage at R19C8+R5C9 (step 30a) blocks {268} combo for split 16(3) at R5C89+R6C8 (step 24)
31a. -> R5C89+R6C8 = {169} (no 2,8)
31b. -> R6C8 = 1; R5C89 = {69}, 9 locked for R5 and N6

32. R5C1 = 8, R6C2 = 2 (both naked singles)

33. NP at R78C2 = {16}
33a. -> R8C3 = 2
33b. -> R2C3 = 7 (step 14)

34. NP at R23C2 = {38}, locked for C2
34a. -> 12(2) at R9C12 = [57]

35. 11(3) at R2C9 (step 28d) = {236/245} (no 1) ({146} unplaceable)
35a. 2 locked for C9
35b. cleanup: no 7 in R1C8

36. 8 in C9 locked in 20(3) at R6C9 = {389/578} (no 4,6)

37. HS in R6 at R6C7 = 4

38. 19(3) at R7C8 = {289/379/478} (no 5,6) = {(8/9)..}
(Note: {568} unplaceable, {469} blocked by 7(2) at R9C8)
38a. -> R78C9 cannot contain both of {89}
38b. -> no 3 in R6C9 (step 36)

39. 3 in N6 locked in R4C79 for R4

40. I/O diff. N3: R13C7 = R4C9 + 9
40a. R4C9 = {23} -> R13C7 = 11 or 12
40b. -> no 1 in R13C7; no 2,3,5 in R3C7

41. Outies C89: R28C7 = 13(2) = [58/67] (only possible permutations)

42. HS in N3 at R1C9 = 1
42a. -> R1C8 = 8; R59C9 = [94] (step 29, only possible permutation)
42b. -> R59C8 = [63]

43. Split 12(2) at R1C34 = [57]

44. 11(3) at R2C9 (step 35) = {236} (no 5) (only possible permutation)
44a. 6 locked in R23C9 for N3
44b. cleanup: no 7 in R8C7 (step 41)

45. R28C7 = [58]
45a. -> split 11(2) at R78C8 = [29] (only possible permutation)
45b. -> R23C8 = [47]

46. NP at R14C7 = {23}, locked for C7
46a. -> R35C7 = [97]
46b. -> R5C6 = 2, R6C9 = 8

47. NS at R3C3 = 4 (hmm, been there for some time)

48. HS in R1 at R1C6 = 4
48a. -> split 10(2) at R1C7+R2C6 = [28] (only possible permutation)
48b. -> R1C5 = 3, R4C7 = 3

49. Split 7(2) at R34C6 = {16} (no 5), locked for C6
49a. -> R9C6 = 9
49b. -> split 6(2) at R8C6+R9C7 = [51] (only possible permutation)

Only naked singles and simple cage sums left now (at last!)


Appendix

There must be a quicker way...

Afmob pointed out an alternative (technically easier) route following on from my step 25, as follows:

26. Outies C89 = 13(2) <> 1,2,3; R8C7 <> 4

27. 16(3) @ N3 <> 8{17/26} because of Killer pairs (17,28) of 9(2)
27a. 16(3) @ N3: R2C7 <> 8 because 5 only possible there
27b. R8C7 <> 5 (step 26)
27c. 19(3) @ N9 = {289/279/469/478} -> must have 7 or 9

28. 20(3) @ C9: R6C9 <> 4 because 4{79} blocked by Killer pair (79) of 19(3) @ N9

29. Hidden Single: R6C7 = 4 @ N6

Puzzle is cracked (though some non-trivial moves are still to do).

Many thanks to Afmob for pointing that out!

[Andrew]

My original walkthrough for uA99 V1.5, which I deleted, was incorrect because I thought I saw IOUs which didn't exist and then that I'd missed a possible combination in step 12b; don't know how I missed that one . Thanks Afmob and Mike for pointing out those errors. I've now reworked it, using the same steps as much as possible.

I'll rate uA99 V1.5, as reworked, as a Hard 1.25.

Here is my walkthrough. I hope it's now correct.

Prelims

a) R1C12 = {69/78}
b) R1C89 = {18/27/36/45}, no 9
c) R9C12 = {39/48/57}, no 1,2,6
d) R9C89 = {16/25/34}, no 7,8,9
e) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
f) R234C1 = {127/136/145/235}, no 8,9
g) R234C9 = {128/137/146/236/245}, no 9
h) R678C9 = {389/479/569/578}, no 1,2
i) 9(3) cage in N7 = {126/135/234}, no 7,8,9
j) 19(3) cage in N9 = {289/379/469/478/568}, no 1
k) R5C34567 = {12347/12356}, no 8,9, 1,2,3 locked for R5

1. 45 rule on C5 1 innie R5C5 = 4
1a. 45 rule on C1234 2 innies R5C34 = 4 = {13}, locked for R5
1b. R5C34567 = {12347} (only remaining combination) -> R5C67 = {27}, locked for R5

2. 45 rule on R1234 2 innies R4C28 = 9 = {18/27/36/45}, no 9

3. 45 rule on R6789 2 innies R6C28 = 3 = {12}, locked for R6

4. 45 rule on C1 3 innies R159C1 = 19 = {379/469/478/568}
4a. 3,4 of {379/469/478} must be in R9C1 -> no 7,9 in R9C1, clean-up: no 3,5 in R9C2

5. 45 rule on N1 2 innies R13C3 = 1 outie R4C1 + 2
5a. Max R4C1 = 7 -> max R13C3 = 9, no 9 in R13C3, no 6,7,8 in R3C3
5b. Min R13C3 = 5 -> min R4C1 = 3

6. R234C1 = {127/136/145/235}
6a. 7 of {127} must be in R4C1 -> no 7 in R23C1

7. 18(3) cage in N1 = {369/378/459} (cannot be {189/279/468/567} which clash with R1C12), no 1,2
7a. Killer pair 8,9 in R1C12 and 18(3) cage, locked for N1

8. 45 rule on N7 2 innies R79C3 = 1 outie R6C1 + 8
8a. Min R6C1 = 3 -> min R79C3 = 11, no 1 in R79C3

9. 45 rule on N9 1 outie R6C9 = 2 innies R79C7 + 1
9a. Min R79C7 = 3 -> min R6C9 = 4
9b. Max R6C9 = 9 -> max R79C7 = 8, no 8,9 in R79C7

10. 19(4) cage at R4C2 cannot be {1279} because no 7 in R5C12 -> no 1,2 in R4C2
10a. 21(4) cage at R4C8 cannot contain both 1 and 2 -> no 1,2 in R4C8
10b. R4C28 (step 2) = {36/45}, no 7,8

11. 19(4) cage at R4C2 = {1459/1468/2359/2368/2458} (cannot be {1369}which clashes with R5C3)
11a. 3,4 must be in R4C2 -> no 5,6 in R4C2, clean-up: no 3,4 in R4C8 (step 2)
11b. 21(4) cage at R4C8 = {1569/2568}, 5,6 locked for N6

12. 45 rule on R1 2 outies R2C46 = 1 innie R1C5 + 14
12a. Max R2C46 = 17 -> max R1C5 = 3
12b. R2C46 = 15,16,17 = {69/78/79/89}

13. 21(3) cage at R1C3 = {489/579/678}
13a. 4 of {489} must be in R1C3 -> no 4 in R1C4
13b. 6 of {678} must be in R2C4 (R1C34 cannot be {67/68} which clash with R1C12), no 6 in R1C34

14. 45 rule on N1 4 innies R1C3 + R23C1 + R3C3 = 12 = {1245} (cannot be {1236} because R1C3 only contains 4,5,7), locked for N1
[This is my original step 20 which I’ve moved here to use some of my existing steps.]
14a. 21(3) cage at R1C3 = {489/579} (cannot be {678} because R1C3 only contains 4,5), no 6, 9 locked in R12C4, locked for C4 and N2
14b. R1C3 = {45} -> no 5 in R1C4
14c. R1C89 = {18/27/36} (cannot be {45} which clashes with R1C34 in 21(3) cage), no 4,5

15. Hidden killer pair 4,5 in R1C34 and R1C67 for R1 -> R1C67 must contain one of 4,5
15a. 14(3) cage at R1C6 = {158/248/257/347/356} (cannot be {149} because R2C6 only contains 6,7,8, cannot be {167/239} which don’t contain 4 or 5), no 9
15b. R2C6 = {678} -> no 6,7,8 in R1C67

16. 45 rule on N3 2 innies R13C7 = 1 outie R4C9 + 9
16a. Max R13C7 = 14 -> no 7,8 in R4C9
16b. Min R4C9 = 1 -> min R13C7 = 10, no 1,2,3,4,5 in R3C7 (R13C7 cannot be [55])

17. 45 rule on C12 2 outies R28C3 = 9 = {36}/[72/81], no 9, no 4,5 in R8C3

18. 9(3) cage in N7 = {126/135/234}
18a. 6 of {126} must be in R78C2 (R78C2 cannot be {12} which clashes with R6C2), no 6 in R8C3, clean-up: no 3 in R2C3 (step 17)

19. 45 rule on C89 2 outies R28C7 = 13 = {49/58/67}, no 1,2,3

20. 18(3) cage in N1 (step 7) = {369/378}, 3 locked in R23C2, locked for C2 -> R4C2 = 4, R4C8 = 5 (step 2), clean-up: no 8 in R9C1, no 2 in R9C9
20a. 6 of {369} must be in R2C3 -> no 6 in R23C2
20b. 6 in N6 locked in R5C89, locked for R5
20c. 19(4) cage at R4C2 (step 11) = {1459/2458}, 5 locked for N4

21. 9(3) cage in N7 = {126/135}, 1 locked for N7

22. 1 in C1 locked in R23C1, locked for N1
22a. R234C1 = {127} (only remaining combination, cannot be {136} because 3,6 only in R4C1, cannot be {145} because R4C1 only contains 3,6,7) -> R4C1 = 7, R23C1 = {12}, locked for C1 and N1, clean-up: no 8 in R1C2
22b. Naked pair {45} in R13C3, locked for C3
22c. 4 in R2 locked in R2C789, locked for N3

23. R678C1 = {349/358}, no 6, 3 locked for C1, clean-up: no 9 in R9C2
23a. Killer pair 4,5 in R678C1 and R9C1, locked for C1
23b. R5C2 = 5 (hidden single in R5)

24. R1C1 = 6 (hidden single in C1), R1C2 = 9, clean-up: no 3 in R1C89, no 3 in R8C3 (step 17)
24a. R2C4 = 9 (hidden single in C4), clean-up: no 4 in R8C7 (step 19)

25. Naked triple {126} in 9(3) cage, locked for N7

26. R79C3 = R6C1 + 8 (step 8)
26a. R6C1 = {389} -> R79C3 = 11,16,17 = {38/79/89}
26b. Killer pair 7,8 in R2C3 and R79C3, locked for C3
26c. Killer pair 7,8 in R79C3 and R9C2, locked for N7

27. 45 rule on C9 3 innies R159C9 = 14 = {149/158/167/239/248} (cannot be {257/347} because R5C9 only contains 6,8,9, cannot be {356} because R1C9 only contains 1,2,7,8)
27a. R5C9 = {689} -> no 8 in R1C9, no 6 in R9C9, clean-up: no 1 in R19C8

28. R234C9 = {137/146/236/245} (cannot be {128} which clashes with R159C9), no 8
28a. 4 of {245} must be in R2C9 -> no 5 in R2C9

29. 16(3) cage in N3 = {169/259/349/358/367/457} (cannot be {178/268} which clash with R1C89)
29a. 9 of {169/259} must be in R3C8 -> no 1,2 in R3C8
29b. 5 of {358} must be in R2C7 -> no 8 in R2C7, clean-up: no 5 in R8C7 (step 19)

[Re-worked from here.]

30. 19(3) cage in N9 = {289/379/469/478}
30a. R678C9 = {389/479/569/578}
30b. 4 of {479} must be in R78C9 (R78C9 cannot be {79} which clashes with 19(3) cage) -> no 4 in R6C9

31. R6C7 = 4 (hidden single in R6), clean-up: no 9 in R8C7 (step 19)
31a. 19(3) cage in N9 = {289/379/469/478}
31b. 6 of {469} must be in R8C7 -> no 6 in R78C8

32. 9 in C7 locked in R34C7, locked for 19(4) cage -> no 9 in R4C6
32a. 19(4) cage at R3C6 = {1279/1369/2359} (cannot be {1459} because 4,5 only in R3C6), no 4,8
32b. 8 in R4 locked in R4C45, locked for N5
32c. 8 in N3 locked in R123C8, locked for C8

33. R8C7 = 8 (hidden single in C7), R2C7 = 5 (step 19)
33a. R78C8 (step 31a) = {29/47}, no 3
33b. R23C8 (step 29) = [29/38/47/83], no 6, no 1,7 in R2C8

34. R678C9 = {389/479/569/578}
34a. 7 of {479} must be in R78C9 (R78C9 cannot be {49} which clashes with R78C8)
34b. 8 of {578} must be in R6C9
34c. -> no 7 in R6C9

35. R5C7 = 7 (hidden single in N6), R5C6 = 2

36. R6C9 = R79C7 + 1 (step 9)
36a. R6C9 = {89} -> R79C7 = 7,8 = {16/26}, no 3, 6 locked for C7 and N9 -> R3C7 = 9, clean-up: no 2 in R2C8 (step 33b), no 1 in R9C9

37. 1 in N9 locked in R79C7, locked for C7
37a. R79C7 = {16} (hidden pair in N9) -> R6C9 = 8 (step 9), clean-up: no 5 in R78C1 (step 23)
37b. R78C9 (step 34) = {39/57}, no 4

38. R5C1 = 8 (hidden single in C1), R6C2 = 2 (step 20c), R6C8 = 1

39. R9C1 = 5 (hidden single in C1), R9C2 = 7, clean-up: no 2 in R9C8
39a. R8C3 = 2 (hidden single in C3), R2C3 = 7 (step 17), clean-up: no 9 in R7C8 (step 33a)

40. Naked pair {34} in R9C89, locked for R9 and N9, clean-up: no 7 in R78C8 (step 33a), no 9 in R78C9 (step 37b)
40a. R78C8 = [29], R5C89 = [69], clean-up: no 7 in R1C9
40b. Naked pair {57} in R78C9, locked for C9

41. Naked pair {23} in R4C79, locked for R4
41a. 19(4) cage at R3C6 (step 32a) = {1279/1369} (cannot be {2359} because R4C6 only contains 1,6), no 5
41b. 1 locked in R34C6, locked for C6
41c. R4C7 = {23} -> no 3 in R3C6

42. 15(3) cage at R8C6 = {159/168} (cannot be {348/357} because R9C7 only contains 1,6, cannot be {456} because 4,5 only in R8C6), no 3,4,7
42a. R8C6 = {56} -> no 6 in R9C67 -> R9C7 = 1, R7C7 = 6, R78C2 = [16], R8C6 = 5, R9C6 = 9 (step 42), R78C9 = [57], R9C3 = 8

43. R9C3 = 8 -> R89C4 = 6 = [42], R9C5 = 6

44. R1C3 = 5 (hidden single in R1), R1C4 = 7 (step 13), R3C3 = 4, R1C8 = 8, R1C9 = 1, clean-up: no 3 in R23C8 (step 33b)
44a. R23C8 = [47], R9C89 = [34]

45. Naked pair {16} in R34C6, locked for C6 -> R2C6 = 8, R23C2 = [38]
45a. R2C6 = 8 -> R1C67 = 6 = [42], R1C5 = 3, R8C5 = 1, R2C5 = 2, R3C5 = 5, R4C5 = 9 (cage sum)

and the rest is naked singles