SudokuSolver Forum

3x3::k:4096:4096:2305:2305:2818:2818:7427:4100:3845:4096:10502:10502:2311:2311:7427:7427:4100:3845:8712:10502:7177:7177:2311:7427:5386:5386:3845:8712:10502:7177:2571:2571:7427:5386:1292:3853:8712:10502:7177:2062:3087:3087:5386:1292:3853:8712:10502:7177:2062:5136:5136:3857:3857:3857:8712:10502:10502:4882:5136:5136:3857:3603:3857:8712:8712:4882:4882:4882:5908:6677:3603:6677:2326:2326:5908:5908:5908:5908:6677:6677:6677:

Don't know whether I used any of the techniques Ed referred to, unless my interactions along R1 come into that category; my killer ALS blocks seem to be related to blocking cages. Maybe Ed will show that I missed something important. I used a fair amount of combination analysis and my solving path definitely wasn't short.

I'll rate my walkthrough for A233 at 1.75.

Prelims

a) R1C34 = {18/27/36/45}, no 9
b) R1C56 = {29/38/47/56}, no 1
c) R12C8 = {79}
d) R4C45 = {19/28/37/46}, no 5
e) R45C8 = {14/23}
f) R45C9 = {69/78}
g) R56C4 = {17/26/35}, no 4,8,9
h) R5C56 = {39/48/57}, no 1,2,6
i) R78C8 = {59/68}
j) R9C12 = {18/27/36/45}, no 1
k) 9(3) cage at R2C4 = {126/135/234}, no 7,8,9
l) 15(5) cage at R6C7 = {12345}, no 6,7,8,9
m) 41(8) cage at R2C2 = {12356789}, no 4

1. Naked pair {79} in R12C8, locked for C8 and N3, clean-up: no 5 in R78C8
1a. Naked pair {68} in R78C8, locked for C8 and N9

2. Naked quad {12345} in R45C8 + R6C789, locked for N6
2a. 5 in N6 only in R6C789, locked for R6 and 15(3) cage at R3C7, no 5 in R7C79, clean-up: no 3 in R5C4

3. 34(7) cage at R3C1 must contain 1, CPE no 1 in R9C1, clean-up: no 8 in R9C2

4. 45 rule on N9 2 innies R7C79 = 5 = {14/23}
4a. 45 rule on N9 3 outies R6C789 = 10 contains 5 = {145/235}

5. 45 rule on N69 2 outies R3C78 = 6 = {15/24}

6. 45 rule on C789 2 innies R12C7 = 8 = {26/35}, no 1,4,8
6a. Killer pair 2,5 in R12C7 and R3C78, locked for N3

7. 8 in N3 only in 15(3) cage at R1C9, locked for C9, clean-up: no 7 in R45C9

8. Naked pair {69} in R45C9, locked for C9 and N6
8a. Naked pair {78} in R45C7, locked for C7

9. 15(3) cage at R1C9 = {348} (only remaining combination), locked for C9 and N3, clean-up: no 5 in R12C7 (step 6), no 2 in R3C78 (step 5), no 1,2 in R7C7 (step 4)
9a. Naked pair {15} in R3C78, locked for R3

10. Naked pair {26} in R12C7, locked for C7 and 29(5) cage at R1C7, no 2,6 in R234C6
10a. 45 rule on C789 3 outies R234C6 = 21 = {489/579}, no 1,3, 9 locked for C6, clean-up: no 2 in R1C5, no 3 in R5C5
10b. 1 in C6 only in R6789C6, CPE no 1 in R7C5

11. 45 rule on C12 2 outies R27C3 = 10 = {19/28/37}, no 5,6
11a. 5,6 in 41(8) cage at R2C2 only in R234567C2, locked for C2, clean-up: no 3,4 in R9C1

12. 45 rule on R1 3 innies R1C789 = 1 outie R2C1 + 9
12a. Min R1C789 = 12 -> min R2C1 = 3

13. 45 rule on N5 3 innies R4C6 + R6C56 = 15 = {159/168/249/258/267/348/357/456}
13a. 5 of {159} must be in R4C6, 9 of {249} must be in R6C5 (cannot be 9{24} which clashes with R6C789), no 9 in R4C6
13b. 9 in C6 only in R23C6, locked for N2, clean-up: no 2 in R1C6

14. 45 rule on N5 2 outies R7C56 = 1 innie R4C6 + 5, IOU no 5 in R7C5

15. R4C6 + R6C56 (step 13) = {159/168/249/258/267/348/357/456}
15a. Consider combinations for R6C789 (step 4a) = {145/235}
R6C789 = {145}, 1 locked for R6
or R6C789 = {235}, 3 locked for R6 => R56C4 = {17/26} => R4C6 + R6C56 cannot be {168} which clashes with R56C4
-> no {168} in R4C6 + R6C56 = {159/249/258/267/348/357/456}
15b. 9 of {159} must be in R6C5 -> no 1 in R6C5
15c. 5 of {258} must be in R4C6, 8 of {348} must be in R6C56 (R6C56 cannot be {34} which clashes with R6C789) -> no 8 in R4C6
[Apologies for using a short forcing chain so early. I’ve been using them a lot when working on puzzles in my Unfinished folder; at least they are better than contradiction moves.]

16. R234C6 (step 10a) = {489/579}
16a. 4 of {489} must be in R4C6 -> no 4 in R23C6

17. R1C34 = {18/27/36} (cannot be {45} which clashes with R1C56 + R1C9, killer ALS block), no 4,5

18. 1 in R1 only in 16(3) cage at R1C1 or R1C34
18a. Consider combinations for 16(3) cage
16(3) cage = {169} = {16}9 => R1C7 = 2 => no {27} in R1C34
or 16(3) cage = {169} = {19}6 => R1C8 = 7 => no {27} in R1C34
or 16(3) cage = {178} => caged X-Wing for 7 in 16(3) cage and R12C8, no other 7 in R12 => no {27} in R1C34
or any other combination in 16(3) cage => 1 in R1 must be in R1C34 = {18}
-> R1C34 = {18/36}, no 2,7

19. R1C56 = {47/56} (cannot be {38} which clashes with R1C34), no 3,8

20. 8,9 in N2 only in R13C4 + R23C6
20a. 45 rule on N2 4 innies R13C4 + R23C6 = 25 = {1789/2689/3589}, no 4
20b. 2 of {2689} must be in R3C4 -> no 6 in R3C4

21. 16(3) cage at R1C1 = {169/178/259/349/358/367/457} (cannot be {268} = [628] which clashes with R1C7, cannot be {268} = {28}6 which clashes with R1C34 + R1C7, killer ALS block)
21a. 6 of {169} must be in R2C1 (R12C1 cannot be [61] which clashes with R1C34), 6 of {367} must be in R2C1 (R12C1 cannot be [63] which clashes with R1C34 + R1C56 + R1C9, killer ALS block, R12C1 cannot be [67] which clashes with R1C56) -> no 6 in R1C1
21b. 8 of {178} must be in R1C12 (R12C1 cannot be {17} which clashes with R1C34 + R1C56, killer ALS block), 8 of {358} must be in R1C12 (R12C1 cannot be [53] which clashes with R1C34 + R1C56 + R1C9, killer ALS block) -> no 8 in R2C1

22. 16(3) cage at R1C1 (step 21) = {169/178/259/349/358/367/457}
22a. Consider placements for 6 in R12C7
R1C7 = 6 => R1C34 = {18}, R1C56 = {47} => no 1,7 in R1C12
or R2C7 = 6 => no 6 in R2C1
-> no {169/367} in 16(3) cage at R1C1
-> 16(3) cage at R1C1 = {178/259/349/358/457}, no 6
22b. 16(3) cage = {178/259/349/457} (cannot be {358} which clashes with R1C34 = {18}, only other place for 1 in R1)
22c. 1 in R1 only in 16(3) cage = {18}7 or in R1C34 = {18}, locking cages, 1,8 locked for R1

23. Consider placements for R1C9
R1C9 = 3 => R1C34 = {18}
or R1C9 = 4 => R1C56 = {56} => R1C34 = {18}
-> R1C34 = {18}, locked for R1

24. R13C4 + R23C6 (step 20a) = {1789/3589}, no 2
24a. 2 on N2 only in 9(3) cage at R2C4 = {126/234}, no 5

25. 16(3) cage at R1C1 (step 22b) = {259/349/457}
25a. 5 of {457} must be in R2C1 (R1C12 cannot be {45/57} which clash with R1C56) -> no 7 in R2C1

26. Deleted. I’d originally incorrectly omitted one combination from R4C6 + R6C56; then step 26 had depended on that error.

27. Consider combinations for R13C4 + R23C6 (step 24) = {1789/3589}
R13C4 + R23C6 = {1789} => R1C4 = 1
or R13C4 + R23C6 = {3589} => R23C6 = [59] => R4C6 = 7 (step 10a)
-> R56C4 = [26/53/62] (cannot be {17} because R1C4 = 1 or R4C6 = 7), no 1,7

[Reworked from here, using some of my later steps …]

28. R4C6 + R6C56 (step 15a) = {159/249/267/348/357} (cannot be {258/456} which clash with R56C4)
28a. 1 in N5 only in R4C45 = {19} or R4C6 + R6C56 = {159}, 9 locked for N5 (locking cages), clean-up: no 3 in R5C6
28b. 1 in N5 only in R4C45 = {19} or R4C6 + R6C56 = {159} => R4C6 + R6C56 = {159/267/348/357} (cannot be {249}, locking-out cages)
28c. Killer pair 7,8 in R5C56 and R5C7, locked for R5
28d. 3 in C6 only in R6789C6, CPE no 3 in R7C5

29. 1 in N5 only in R4C45 = {19}
or R4C6 + R6C56 = {159} => R56C4 = {26} => 3 in N5 only in R4C45 = {37}
-> R4C45 = {19/37}, no 2,4,6,8

30. 1 in N1 only in R1C3 + R2C23, CPE no 1 in R7C3, clean-up: no 9 in R2C3 (step 11)
30a. 1 in C1 only in 34(7) cage at R3C1, no 1 in R8C2
30b. 41(8) cage at R2C2 = {12356789}, CPE no 9 in R8C2

31. Consider combinations for 23(5) cage at R8C6 = {12389/12479/12569/12578/13469/13478/13568/14567/23459/23468/23567}
23(5) cage = {12389/12479/12578/13478/23459} => 6 in R9 only in R9C12 = [63]
or 23(5) cage = {12569/13469/13568/14567/23468/23567} contains at least one of 4,5 => variable caged X-Wing for 4 and/or 5 in 23(5) cage at 26(5) cage at R8C7 => no 4 and/or 5 in R89 => R9C12 cannot be [54]
-> R9C12 = [27/63/72/81], no 4,5

32. 4 in C12 only in 16(3) cage at R1C1 and 34(7) cage at R3C1 => 16(3) cage and 34(7) cage must both contain 4
32a. 16(3) cage (step 25) = {349/457}, no 2, 4 locked for N1
32b. 9 of {349} must be in R1C12 (R1C12 cannot be {34} which clashes with R1C9), no 9 in R2C1
32c. 34(7) cage = {1234789/1245679/1345678}, CPE no 7 in R9C1, clean-up: no 2 in R9C2

33. R1C7 = 2 (hidden single in R1), R2C7 = 6

34. 6 in R1 only in R1C56 = {56}, locked for R1 and N2

35. 9(3) cage at R2C4 (step 24a) = {234} (only remaining combination), locked for N2

36. R1C4 = 1 (hidden single in N2), R1C3 = 8, clean-up: no 2 in R28C3 (step 11), no 9 in R4C5

37. R234C6 (step 10a) = {489/579}
37a. 4,5 only in R4C6 -> R4C6 = {45}
37b. Killer pair 4,5 in R4C6 and R5C56, locked for N5, clean-up: no 3 in R6C4

38. Naked pair {26} in R56C4, locked for C4 and N5
38a. 2 in N1 only in R23C5, locked for C5

39. 5 in C4 only in R789C4, locked for N8

40. 41(8) cage at R2C2 = {12356789}, 2,8 locked for C2
40a. 1 in N1 only in R2C23, locked for 41(8) cage, no 1 in R4567C2

41. R4C6 + R6C56 (step 28b) = {159/348/357}, R6C789 (step 4a) = {145/235} -> killer pair 1,3 locked for R6

42. R7C56 = R4C6 + 5 (step 14)
42a. Max R4C6 = 5 -> max R7C56 = 10, max R7C6 = 6

43. 45 rule on N58 2 outies R89C3 = 1 innie R4C6 + 2
43a. R4C6 = {45} -> R89C3 = 6,7 = {15/24/16/25/34}, no 7,9 in R89C3

44. 5 in R7 only in R7C124, CPE no 5 in R8C3

45. 5 in N5 only in R4C6 or R5C56 -> R234C6 = {579}, 7 locked for C6 or R5C56 = {57}, locked for N5
-> no 7 in R6C6

[I originally saw the next step as
46. R7C56 = R4C6 + 5 (step 14), R4C6 = {45} -> R7C56 = 9,10
46a. R4C6 + R6C56 (step 28b) = {159/348/357}
46b. Consider combinations for R6C789 (step 4a) = {145/235}
R6C789 = {145}, 1,4 locked for 15(3) cage at R7C3 => R7C79 = [32] => R7C56 = {46}/[81/91] => R7C56 = [81] => R6C56 = [73] => R4C6 = 5 or R7C56 = {46}/[91] = 10 => R4C6 = 5
or R6C789 = {235}, 3 locked for R6 => R6C56 = [91] => R4C6 = 5
-> R4C6 = 5, R1C56 = [56], clean-up: no 7 in R5C56
However the following may be simpler …]

46. R7C56 = R4C6 + 5 (step 14), R4C6 = {45} -> R7C56 = 9,10
46a. R4C6 + R6C56 (step 28b) = {159/348/357}
46b. R6C789 (step 4a) = {145/235} -> 2,3 in 15(5) cage at R6C3 must be in the same row
46c. 20(4) cage at R6C5 = {1289/1379/1469/1478/2369/2468/3467} (cannot be {2378} which must have 3 in R6 and 2 in R7 so clashes with 15(5) cage at R6C3) -> no remaining combinations containing both of 3,8 -> R4C6 = 5, R1C56 = [56], clean-up: no 7 in R5C56

47. R4C6 = 5 -> R234C6 (step 10a) = {579} (only remaining combination), 7 locked for C6 and N2 -> R3C4 = 8
47a. R2C9 = 8 (hidden single in C9)
47b. Naked pair {79} in R2C68, locked for R2, clean-up: no 3 in R7C3 (step 11)
47c. 41(8) cage at R2C2 = {12356789}, CPE no 7 in R89C2, clean-up: no 2 in R9C1

48. Naked pair {48} in R5C56, locked for R5 and N5 -> R45C7 = [87], clean-up: no 1 in R4C8
48a. R4C6 + R6C56 (step 28b) = {159/357}, R6C6 = {13} -> no 3 in R6C5
48b. R7C56 = R4C6 + 5 (step 14), R4C6 = 5 -> R7C56 = 10, no 4 in R7C5

49. R89C3 = R4C6 + 2 (step 43)
49a. R4C6 = 5 -> R89C3 = 7 = {25/34} (cannot be {16} which clashes with R9C12), no 1,6
49b. 2 of {25} must be in R8C3 -> no 2 in R9C3

50. 28(5) cage at R3C3 contains 8 = {14689/24589/25678/34678} (cannot be {13789} which clashes with R7C3, cannot be {23689} which clashes with R89C3)
50a. 5 of {24589/25678} must be in R5C3 -> no 2 in R5C3

51. 45 rule on R9 3 outies R8C679 = 13 = {139/148/157/238/247}
51a. 8 of {148} must be in R8C6, 4 of {247} must be in R8C7 -> no 4 in R8C6
51b. 7 of {157} must be in R8C9 -> no 5 in R8C9
51c. 4,5,9 of {139/148/157} must be in R8C7 -> no 1 in R8C7

52. 34(7) cage at R3C1 (step 32c) = {1234789/1245679} (cannot be {1345678} which clashes with R9C1), 9 locked for C1

53. 4 in C2 only in R18C2
53a. 45 rule on C1 3 outies R189C2 = 14 = {149/347}
53b. 7,9 only in R1C2 -> R1C2 = {79}
53c. 16(3) cage at R1C1 (step 32a) = {349/457}
53d. R1C2 = {79} -> no 7 in R1C1

54. R8C2 = 4 (hidden single in C2), clean-up: no 3 in R89C3 (step 49a)

55. R89C3 = [25]

56. R8C7 = 5 (hidden single in N9), R3C78 = [15]

57. Naked pair {34} in R67C7, locked for C7 and 15(5) cage at R6C3, no 3,4 in R6C8 -> R9C7 = 9
57a. Naked triple {127} in R789C9, locked for C9 and N9 -> R6C9 = 5

58. R8C7 = 7 -> R8C679 (step 51) = {157} (only remaining combination) -> R8C69 = [17]

59. R6C6 = 3 -> R6C5 = 7 (step 28b), R4C45 = [91], R67C7 = [43], R9C8 = 4, clean-up: no 1 in R5C8
59a. Naked pair {23} in R45C8, locked for N6 -> R6C8 = 1, R7C9 = 2, R9C9 = 1, R9C2 = 3, R9C1 = 6

60. R7C6 = 4, R7C5 = 6 (step 46c)

61. R2C2 = 1 (hidden single in C2), R2C3 = 3, R7C3 = 7 (step 11)

and the rest is naked singles.

1. c789

16/2@r1c8 = {79}
-> 14/2@r7c8 = {68}

Innies n9 r7c79 = +5 = {14} | {23}
-> r6c789 = +10 = {5(23|14)}
-> Innies n6 r45c7 = +15
-> r3c78 = +6 (no 6,8)
-> Innies n3 r12c7 = +8 (no 8)
-> 8 in n1 in r123c9
-> 15/2@r4c9 = {69}
-> r45c7 = {78}
Also 6 in n3 in c7 -> r12c7 = {26}
-> r3c78 = {15}
-> 15/2@r1c9 = {348}
Also 7 in n9 in r89c9 and 9 in n9 in r89c7
Also r7c79 = [32] | [41]


2. c123

41/8@r2c2 - no 4
Outies c12 = +10 -> r27c3 not 456
Whatever goes in r7c3 must go in r345c1 in n4 and r1c2 in n1. -> r1c2 not 456.
Whatever goes in r2c3 must go in r345c1 in n4 and r9c2 in n7. -> r9c2 not 456.
-> HS 4 in c2 -> r8c2 = 4.
-> 4 in n4 in r456c3 and 4 in n1 in r12c1.
-> 16/2@r1c1 = {439} | {457}. r1c2 one of (379).
-> HS 1 in r1 in r1c34 -> 9/2@r1c3 = {18}


3. n12

Either r1c2 from (79) -> r1c28 = {79} -> 11/2@r1c5 = {56}
Or r1c2 = 3 -> r2c3 = 7 -> r12c8 = [79] -> r1c1 = 9 -> 11/2@r1c5 = {56}
Either way 11/2@r1c5 = {56}

-> 9/3@r2c4 = {234}
-> HS 1 in n2 in r1c4 -> 9/2@r1c3 = [81]
-> (15) in n1 in r2c123. 5 in r2c12 and 1 in r2c23.

5 in r2c1 puts 7 in r1c2 and 3 in r2c3.
-> r2c123 = [513] or [x51]. (x from (34)).
-> r2c3 from (13)
-> r28c3 = [19] | [37]
-> r19c2 = [91] | [73]
-> 9/2@r9c1 = [81] | [63]


4. "5"

r234c6 = +21 must be one of {489} | {579} -> 9 in r23c6 and r4c6 from (45).
-> Innies n5 r6c56 = +11 or +10.
1 in n5 must be in 10/2@r4c4 or hidden +10/2@r6c56 along with a 9.
-> 9 not in 12/3@r5c5
-> (45) both in (r4c5,r5c56) in n5.
-> 5 in n8 in c4 in r789c4.
-> 8/2@r5c4 = {26}
-> r2c5 = 2

Given r3c4 from 78 and r1c4 = 1.
-> Innies - Outies c123 r89c3 = +6 or +7.
if +6 must be {15}. (4 already in n7)
if +7 must be {25}. (4 already in n7 and either 1 or 6 in 9/2@r9c1).
Either way 5 in r89c3.

-> HS 5 in r7 r7c4 = 5.
-> r9c23 = [15] or [25].
-> 5 in n9 in r8c79.


5. n58
[r3c4,r4c6,r8c2] = [741] or [852].
But the former leaves no solution for 19/4@r7c4.
(r8c45 would = +13. (45) already in r8 and (67) in c4.)

-> [r3c4,r4c6,r8c2] = [852]
-> r23c6 = {79}, 11/2@r1c5 = [56]
Also 12/2@r5c5 = {48} -> r45c7 = [87]
Also r8c45 = +12 = {39} -> r9c7 = 9
-> HS 3 in c6 r6c6 = 3 -> r6c5 = 7.
-> 10/2@r4c4 = [91]
-> 15/2@r4c9 = [69] and r8c45 = [39]
-> 9/3@r2c4 = [423]
-> 15/3@r1c9 = [384] and r1c1 = 4.



6. Finish

Also HS 3 in n6 5/2@r4c8 = {23}
-> r6c789 = {145}
-> r7c79 = [32]
-> r7c56 = [64]
-> 12/2@r5c5 = [48]
-> r89c6 = [12]
-> r9c45 = [78]
-> 9/2@r9c1 = [63]
-> r1c2 = 7, r27c3 = [37], r12c1 = [45]
etc., etc.

Andrew's end step 11; use "Paste Into" A233 in SudokuSolver

.-------------------------------.-------------------------------.-------------------------------.
| 123456789 1234789   12345678  | 12345678  3456789   2345678   | 26        79        348       |
| 123456789 12356789  123789    | 123456    123456    45789     | 26        79        348       |
| 2346789   236789    2346789   | 2346789   2346      4789      | 15        15        348       |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 12356789  123456789 | 12346789  12346789  45789     | 78        1234      69        |
| 123456789 12356789  123456789 | 12567     45789     34578     | 78        1234      69        |
| 12346789  1236789   12346789  | 12367     12346789  1234678   | 1345      12345     125       |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 12356789  123789    | 123456789 23456789  12345678  | 34        68        12        |
| 123456789 1234789   123456789 | 123456789 123456789 12345678  | 13459     68        1257      |
| 25678     12347     123456789 | 123456789 123456789 12345678  | 13459     12345     1257      |
'-------------------------------.-------------------------------.-------------------------------'

12. r7c3 sees all c2 except r1c2 -> r1c2 = r7c3: no 4
12a. 4 in c2 only in n7: locked for n7

The following steps are very new to me - my walkthrough for A230 step 24 is the first time I used this. Also see wellbeback's A232 WT, step 2, line 11, for a nice one.
13. since r1c2 = r7c3 and since r28c3 = 10 (step 11)-> h10(2) at r1c2 + r2c3
13a. from overlap of this second h10(2) and the 16(3)r1c1 -> r2c3 + 6 = r12c1
13b. -> no 6 in r12c1 since it would force r2c3 to equal the other one of r12c1 (IOU)

14. "45" on c1: 3 outies r189c2 = 14
14a. r8c2 sees all c1 except r12c1 -> r8c2 is cloned in one of r12c1: ie, 2 cells of the h14(3)r189c2 overlaps 2 cells of the 16(3)r1c1
-> r9c2+2 = one of r12c1. No 6 in r12c1 -> no 4 r9c2!
14b. r8c2 = 4 (hsingle in c2)
14c. no 5 in r9c1

15. 4 in c1 in 16(3) = {349/457}(no 1,2,8)
15a. 4 locked for n1
15b. no 2,8,9 in r2c3 (h10(2)r1c2+r2c3)
15c. no 1,2,8 in r7c3 (h10(2)r27c3)
15d. no 5 in r1c4

16. 1 in r1 only in 9(2)r1c3 = {18} only: 8 locked for r1
16a. no 3 in 11(2)n2

17. "45" on n5: 3 innies r4c6+r6c56 = 15
17a. combos with 9 are {159/249}
17b. {159} must have 5 in r4c6
17c. {249} can't be [9]{24} because r6c789 = [2/4..](step 4a)
17d. -> no 9 in r4c6
17e. -> 9 in c6 only in n2: 9 locked for n2
17f. no 2 in r1c6

18. r1c7 = 2 (hsingle r1)
18a. r2c7 = 6

18. 6 in r1 only in 11(2)n2 = {56} only: locked for r1 and n2

19. 9(3)n2 = {234} only: locked for n2
19a. r1c4 = 1 (hsingle n2)
19b. r1c3 = 8
19c. no 7 in 8(2)n5
19d. no 9 in r4c5

20. h21(3)r234c6 = {489/579}
20a. must have 4/5 -> r4c6 = (45)

21. h15(3)n5 must have 4/5 for r4c6 = {159/249/258/348/357/456} = [1->9..]
21a. 1 in n5 in h15(3) or 10(2) -> 9 locked for n5 (Locking cages)
21b. no 3 in r5c6

22. 12(2)n5 = {48/57} = [4/5..]
22a. Killer pair 4,5 with r4c6: both locked for n5
22b. no 6 in 10(2)n5
22c. & 8(2)n5 = {26} only: both locked for c4 and n5
22d. no 8 in 10(2)n5

23. 5 in c4 only in n8: locked for n8

24. 5 in r7 only in r7c1234 -> no 5 in r8c3 (CPE)

25. 9 in r1c2 -> r12c1 = 7 = {34} or h10(2)r1c2+r2c3 = {37}
25a. 3 locked in 16(3)+r2c3 for n1

26. hidden pair {34} in r3 in r3c59
26a. r2c5 = 2 (hsingle n2)
26b. r2c9 = 8 (hsingle n3)

27. naked pair {79} in r2c68: both locked for r2
27a. no 3 in r1c2 (h10(2)r1c2+r2c3)

28. "45" on c1: 2 remaining outies r19c2 = 10 = [73/91]
28a. 9(2)n7 = [63/81] = [1/6..]

29. "45" on n369+n58: 1 innie r4c6 + 2 = 2 outies r89c3
29a. -> 2 outies = 6/7
29b. but {16} blocked by 9(2)n7 (step 28a)
29c. = [15/25]
29d. r9c3 = 5
29e. r7c4 = 5 (hsingle r7)

30. "45" on r9: 3 outies r8c679 = 13 and must have 5 for r8
30a. = {157/256}(no 3,8,9) = [1/2,6/7..]

31. Killer pair 1,2 in r8 in h13(3) (step 30a) and r8c3: both locked for r8

32. Split 14(3)r8c345, {167} blocked by h13(3)r8c679 (step 30a)
32a. = {239} only
32b. r8c3 = 2
32c. 3 & 9 locked for r8 & n8

33. r89c3 = 7 -> r4c6 = 5 (step 29)
33a. r23c6 = 16 = {79} only: 7 locked for n2 & c6

Much easier now.

That was certainly true. Both wellbeback and Ed showed what I'd missed. They both spotted a "clone" which I ought to have seen. Then wellbeback neatly found that there was a second "clone" by using a "variable clone" in R456C1. Ed used a longer way to get this result but it had the benefit of also getting a useful hidden cage in N1.

I'll have to remember to look for "clones" when there are large cages as in this puzzle, particularly when they occupy a particular area of the puzzle grid.

Even though I missed these useful "clones", which gave an early placement followed by some useful simplifications, I was pleased to see that some of my later steps were similar to ones used by Ed.