Prelims
a) R12C1 = {89}
b) R1C23 = {18/27/36/45}, no 9
c) R2C34 = {14/23}
d) R23C9 = {79}
e) R34C1 = {29/38/47/56}, no 1
f) R3C34 = {15/24}
g) R45C4 = {14/23}
h) R5C12 = {29/38/47/56}, no 1
i) R6C12 = {18/27/36/45}, no 9
j) R89C1 = {17/26/35}, no 4,8,9
k) R89C8 = {19/28/37/46}, no 5
l) 18(5) cage at R1C6 = {12348/12357/12456}, no 9
m) 33(5) cage at R6C4 = {36789/45789}, no 1,2
Steps Resulting From Prelims
1a. Naked pair {89} in R12C1, locked for C1 and N1, clean-up: no 1 in R1C23, no 2,3 in R34C1, no 2,3 in R5C2, no 1 in R6C2
1b. 1 in N1 only in R23C23, CPE no 1 in R4C4
1c. Naked pair {79} in R23C9, locked for C9 and N3
1d. 16(3) cage at R7C9 = {268/358}, no 1,4, 8 locked for C9 and N9, clean-up: no 2 in R89C8
2. 45 rule on C1 3 innies R567C1 = 9 = {126/135/234}, no 7, clean-up: no 4 in R5C2, no 2 in R6C2
3. 45 rule on N14 1 outie R7C3 = 2 innies R23C3 + 3, min R23C3 = 3 -> min R7C3 = 6
3a. Min R7C3 = 6 -> max R56C3 = 9, no 9 in R56C3
4. 45 rule on N7 2 innies R78C3 = 1 outie R9C4 + 11
4a. Max R78C3 = 17 -> max R9C4 = 6
5. 45 rule on R9 2 outies R8C18 = 1 outie R9C9
5a. Min R8C18 = 3 -> min R9C9 = 3
5b. Max R9C9 = 8 -> max R8C18 = 8, no 9 in R8C8, clean-up: no 1 in R9C8
5c. 45 rule on R9 3 innies R9C189 = 18 = {189/369/378/468/567} (cannot be {279} because R9C9 only contains 3,5,6,8, cannot be {459} because 4,9 only in R9C8), no 2, clean-up: no 6 in R8C1
6. 45 rule on C1234 1 innie R1C4 = 1 outie R8C5 + 4 -> R1C4 = {789}, R8C5 = {345}
6a. Hidden killer triple 7,8,9 in R1C4 and R678C4 for C4, R1C4 = {789} -> R678C4 must contain two of 7,8,9
6b. 33(5) cage at R6C4 = {36789/45789}, R678C4 contains two of 7,8,9 -> R8C3 = {789}
6c. 3 of {36789} must be in R8C5 -> no 3 in R678C4
6d. Min R78C3 = 13 -> min R9C4 = 2 (step 4)
7. 45 rule on N6 2 innies R4C89 = 1 outie R7C8 + 5, IOU no 5 in R4C9
8. 18(5) cage at R1C6 = {12348/12357/12456}
8a. 8 of {12348} must be in R1C678 (R1C6789 cannot be {1234} which clashes with R1C23), no 8 in R2C7
9. 45 rule on C89 2 innies R1C89 = 1 outie R3C7 + 1, IOU no 1 in R1C89
9a. Min R1C89 = 5 -> min R3C7 = 4
9b. Max R3C7 = 8 -> max R1C89 = 9, no 8 in R1C8
9c. 1 in C9 only in R456C9, locked for N6
10. 45 rule on C12 2 outies R14C3 = 1 innie R9C2 + 8, max R14C3 = 16 -> max R9C2 = 8
11. 45 rule on C6789 2 innies R67C6 = 1 outie R9C5 + 4, IOU no 4 in R6C6
12. 45 rule on R1 3 innies R1C145 = 1 outie R2C7 + 18
12a. Min R1C145 = 19, no 1 in R1C5
12b. 1 in R1 only in R1C67, locked for 18(5) cage at R1C6
12c. Min R2C7 = 2 -> min R1C456 = 20, no 2 in R1C5
[It’s a while since I’ve done an iterative step like this, although this time only one iteration.]
13. 34(6) cage at R1C4 = {136789/145789/235789/245689} (cannot be {345679} which clashes with R8C5)
13a. 34(6) cage = {136789/235789}, 3 locked for C5
or 34(6) cage = {145789/245689}, killer triple 3,4,5 in 34(6) cage and R8C5, locked for C5
-> no 3 in R679C5
[This 45 looks important; I ought to have spotted it earlier.]
14. 45 rule on N36 2(1+1) outies R1C6 + R7C8 = 7, no 7,8 in R1C6, no 7,9 in R7C8
14a. 18(5) cage at R1C6 = {12348/12456}
14b. 1,8 of {12348} must be in R1C67 -> no 3 in R1C67, clean-up: no 4 in R7C8
14c. R1C89 = R3C7 + 1 (step 9)
14d. 3 of {12348} must be in R2C7 (18(5) cage cannot be 18{23}4 because R1C89 + R2C7 = {23}4 clashes with R1C89 + R3C7 = {23}4, 18(5) cage cannot be 18{34}2 because R1C23 = {27} (then only place for 2 in R1) clashes with R1C14 = [97]) -> no 3 in R1C89
[Thanks Ed for pointing out that I’d overlooked R1C789 = 8{34}.]
14e. Min R1C89 = {24} = 6 -> min R3C7 = 5
14f. 3 of {12348} must be in R2C7, 2,6 of {12456} must be in R2C7 (R1C6789 cannot be {1246/1256} which clash with R1C23), no 4,5 in R2C7
14g. 4 of 18(5) cage must be in R1C6789, locked for R1, clean-up: no 5 in R1C23
14h. R3C7 = {568} -> R1C89 = 6,7,9 = {24/45} (cannot be {25} because R1C89 + R2C7 of {12456} = {25}6 clashes with R1C89 + R3C7 = {25}6), no 6, 4 locked for R1 and N3, clean-up: no 3 in R7C8
14i. R1C89 = {24/45} = 6,9 -> R3C7 = {58}
15. 45 rule on N9 4 innies R7C78 + R89C7 = 19 = {1279/1459/2467/3457} (cannot be {1369/1567/2359} which clash with R789C9)
15a. Hidden killer triple 4,7,9 in 16(3) cage at R4C7 and R789C7 for C7, R789C7 must contain two of 4,7,9 (because no 4,7,9 in R7C8) -> 16(3) cage must contain one of 4,7,9
15b. 16(3) cage = {259/367} (cannot be {268/358} which don’t contain any of 4,7,9, cannot be {349/457} which contain two of 4,7,9), no 4,8
15c. 4 in C7 only in R789C7, locked for N9, clean-up: no 6 in R89C8
15d. Hidden killer pair 3,6 in R456C9 and 16(3) cage at R7C9 for C9, 16(3) cage contains one of 3,6 -> R456C9 must contain one of 3,6
15e. 16(3) cage = {259} (only remaining combination, cannot be {367} which clashes with R456C9), locked for C7 and N6
[The eliminations of 4 from R123C7 led to steps 15a-15e which crack the puzzle; the rest is fairly straightforward.]
15f. R7C78 + R89C7 = {2467/3457} no 1, 7 locked for N9, clean-up: no 6 in R1C6 (step 14), no 3 in R89C8
15g. R89C8 = [19], clean-up: no 7 in R9C1
15h. R1C7 = 1 (hidden single in C7), clean-up: no 6 in R7C8 (step 14)
15i. Naked pair {25} in R1C6 + R7C8 (step 14), CPE no 2,5 in R1C8 + R7C6 -> R1C8 = 4
15j. Naked pair {25} in R1C69, locked for R1, R1C78 = [14] -> R2C7 = 6 (cage sum), clean-up: no 7 in R1C23
15k. Naked pair {36} in R1C23, locked for R1 and N1, clean-up: no 2 in R2C4, no 5 in R4C1
15l. R789C7 = {347} -> R7C8 = 5, R1C6 = 2 (step 14), clean-up: no 4 in R3C3
15l. 16(3) cage at R7C8 = {268} (only remaining combination), 6 locked for C9
15m. R3C7 = 8, R23C8 = {23}, 3 locked for C8 and 23(5) cage at R2C8 -> R4C89 = 10 = [64], R4C1 = 7 -> R3C1 = 4, clean-up: no 1 in R2C4, no 2 in R3C3, no 1 in R5C4, no 2 in R6C1, no 5 in R6C2, no 1 in R9C1
15n. Naked pair {15} in R3C34, locked for R3
15o. 7 in N1 only in R23C2, locked for C2
15p. R9C189 = 18 (step 5c), R9C8 = 9 -> R9C19 = 9 = [36] -> R8C1 = 5, clean-up: no 9 in R1C4 (step 6), no 6,8 in R5C2, no 4,6 in R6C2
15q. Killer pair 1,3 in R6C12 and R6C9, locked for R6
15r. 3 in N9 only in R78C7, R78C7 = {34/37} = 7,10 -> R8C6 = {69}
15s. 34(6) cage at R1C4 (step 13) must contain 9, locked for C5
15t. 9 in C4 only in R678C4, locked for 33(6) cage at R6C4, no 9 in R8C3
16. 12(3) cage at R7C1 = {129/246}, no 8, 2 locked for N7
16a. R2C34 = [14] (cannot be {23} which clashes with R2C8) -> R3C34 = [51]
16b. Naked pair {23} in R45C4, locked for C4 and N5
16c. R9C4 = 5 -> R9C23 = 9 = [18]
16d. 12(3) cage at R7C1 = {246} (only remaining combination), 6 locked for N7
16e. Naked pair {27} in R23C2, 2 locked for C2 and 21(4) cage at R2C2 -> R47C3 = [39], R4C2 = 9 (cage sum)
16f. R5C2 = 5 -> R5C1 = 6
16g. R7C1 = 2, R78C9 = [82]
16h. R7C4 + R8C3 = [67], R8C6 = 9, R68C4 = [98], R8C3 = 3 (cage sum), R78C2 = [46]
16i. R6C12 = [18], R56C8 = [87]
16j. R7C56 = {17}, locked for N8 -> R6C56 = 11 = {56}, locked for R6 and N5
16k. R45C4 = [23], R89C7 = [47], R9C6 = 4
16l. Naked pair {18} in R4C56, locked for N5 -> R5C6 = 7, R7C6 = 1, R45C6 = [87] -> R23C6 = 9 = [36]
and the rest is naked singles.