Prelims
a) R23C3 = {19/28/37/46}, no 5
b) Disjoint cage R5C8 + R9C7 = {18/27/36/45}, no 9
c) R67C6 = {69/78}
d) R78C9 = {89}
e) R9C45 = {19/28/37/46}, no 5
f) 20(3) cage at R4C3 = {389/479/569/578}, no 1,2
g) 6(3) cage at R6C2 = {123}
h) 21(3) cage at R8C1 = {489/579/678}, no 1,2,3
i) 39(8) cage at R5C7 = {12345789}, no 6
Steps Resulting From Prelims
1a. Naked pair {89} in R78C9, locked for C9 and N9, clean-up: no 1 in R5C8
1b. Naked triple {123} in 6(3) cage at R6C2, CPE no 3 in R45C3
[I overlooked that this CPE also eliminated 1,2,3 from R6C4 using D/. This would have simplified my solving path from step 7d onward.]
2. 45 rule on N7 3 innies R7C123 = 6 = {123}, locked for R7 and N7
2a. Max R7C12 = 5 -> min R56C1 = 13, no 1,2,3 in R56C1
3. 21(3) cage at R8C1 = {579/678} (cannot be {489} which clashes with R8C9), no 4, 7 locked for R8 and N7
3a. Killer pair 8,9 in 21(3) cage and R8C9, locked for R8
3c. 4 in N7 only in 18(3) cage at R9C1, locked for R9, clean-up: no 5 in R5C8, no 6 in R9C45
4. 45 rule on C789 2 outies R89C6 = 9 = [18/27/45] -> R8C6 = {124}, R9C6 = {578}
4a. 9 of 39(8) cage at R5C7 only in R5C7 + R6C78, locked for N6
5. 45 rule on N9 3 innies R789C7 = 13 = {157/247/256/346}
5a. 6 of {346} must be in R9C7 -> no 3 in R9C7, clean-up: no 6 in R5C8
6. 4 in N7 only in 18(3) cage at R9C1 = {459/468}, R89C6 (step 4) = [18/27/45]
6a. Consider combinations for 18(3) cage
18(3) cage = {459}, locked for R9 => R89C6 = [18/27], 2 or 8
or 18(3) cage = {468}, locked for R9
-> R9C45 = {19/37} (cannot be {28}, no 2,8
6b. 2 in R9 only in R9C789, locked for N9
6c. R789C7 (step 5) = {157/247/346} (cannot be {256} because 2,6 only in R9C7)
6d. {157} can only be [751] (cannot be [517/715] which clash with R89C6), no 5 in R7C7, no 1 in R8C7, no 5,7 in R9C7, clean-up: no 2,4 in R5C8
6e. R89C6 = [18/27] (cannot be [45] which clashes with R78C7), no 4,5
6f. Killer pair 1,7 in R89C6 and R9C45, locked for N8, clean-up: no 8 in R6C6
6g. 7 in N8 only in R9C456, locked for R9
6h. 7 in R7 only in R7C78, CPE no 7 in R6C8
6i. R67C6 = {69} (cannot be [78] which clashes with R9C6), locked for C6
6j. 16(4) cage at R2C5 = {1258/1267/1348/1357/1456/2347/2356} (cannot be {1249} which clashed with R8C6), no 9
7. Consider permutations for R789C7 (step 6c) = [436/742/751]
R789C7 = [436] => 18(3) cage at R9C1 (step 6) = {459}, locked for R9 => R9C45 = {37} => R89C6 = [18]
or R789C7 = [742/751] => R89C6 = [18] => R9C45 = {37}
-> R89C6 = [18], R9C45 = {37}, 3 locked for R9 and N8
7a. 18(3) cage at R9C1 = {459} (only remaining combination), 5,9 locked for R9 and N7
7b. Naked triple {678} in 21(3) cage at R8C1, 6,8 locked for R8 -> R78C9 = [89]
7c. 39(8) cage at R5C7 = {12345789}, CPE no 2,3,4,5,7 in R4C7
7d. 2 in N8 only in R8C45 -> 18(4) cage at R6C4 = {2349/2457} (cannot be {1269/1278/2367} because R8C45 only contain 2,4,5, cannot be {2358} because 3,8 only in R6C4), no 1,6,8
7e. 3,7 of {2349/2457} can only be in R6C4 -> R6C4 = {37}
[If I’d spotted the elimination of 1,2,3 from R6C4 in step 1b, then step 7d would have just been 2 in 18(4) cage at R6C4 = {2457} (only possible combination), and step 7e would then give R6C4 = 7, placed for D/, which would have simplified later steps.]
7f. 18(4) cage at R6C4 = {2349/2457}, 4 locked for N8
7g. 3 in R8 only in R8C78, CPE no 3 in R6C8
7h. Naked pair {37} in R69C4, locked for C4
7i. 16(4) cage at R2C5 (step 6j) = {1357/2347/2356} (cannot be {1258/1267/1348/1456} because R234C6 need to contain three of 2,3,4,5,7), no 8, CPE no 3 in R1C6
7j. 1,6 of {1357/2356} must be in R234C6 -> no 5 in R2C5
7k. Consider combinations for 16(4) cage = {1357/2347/2356}
16(4) cage = {1357/2356} => 5 in R234C6
or 16(4) cage = {2347}, CPE no 2,3,4,7 in R1C6 => R1C6 = 5
-> 5 in R1234C6, locked for C6
8. 45 rule on N58 2 innies R4C46 = 1 outie R3C5 + 3
8a. Min R4C46 = 6 -> min R3C5 = 3
8b. 35(7) cage must be missing two digits which total 10 so must contain three of the pairs 1,9, 2,8, 3,7 and 4,6
8c. Consider placements for R6C4 = {37}
R6C4 = 3 => R9C45 = [73] => no 3 in 35(7) cage
or R6C4 = 7 => R9C45 = [37] => no 7 in 35(7) cage
-> 35(7) cage = {124589} (only remaining combination, since it doesn’t contain both of 3,7), no 3,7
[Alternatively R69C4 = {37}, R9C45 = {37} -> R6C4 = R9C5
35(7) cage = {124589} (only remaining combination, other combinations clash with R6C4 + R9C5), no 3,7]
8d. R4C6 + R6C4 = {37} (hidden pair in N5), locked for D/
8e. 3 of 6(3) cage at R6C2 = {123} only in R6C12, locked for R6 and N4 -> R6C4 = 7, R4C6 = 3, R9C45 = [37]
[Now I’m back to where I should have been if I’d spotted the elimination of 1,2,3 from R6C4 in step 1b, but I wouldn’t have found some interesting steps including 8c.]
8f. 39(8) cage at R5C7 = {12345789}, 3,7 locked for C7
8g. R6C4 = 7 -> R7C4 + R8C45 = 11 = {245}, 5 locked for N8
8h. Naked pair {69} in R7C56, 6 locked for R7
8i. 5 in C6 only in R123C6, locked for N2
8j. R1C5 = 3 (hidden single in C5)
8k. 8,9 in C5 only in R34567C5, locked for 35(7) cage, no 8,9 in R5C4
9. 1,6 in N6 only in R4C789 + R5C9
9a. 45 rule on N3 4 outies R4C789 + R5C9 = 19 = {1468/1567}, no 2,3
9b. 8 of {1468} must be in R4C78, which cannot also contain 4 (because no 3 in R3C7), no 4 in R4C8
9c. R4C78 cannot total 10 -> no 5 in R3C7
10. 15(4) cage at R7C8 = {1257/1356/2346} (cannot be {1347} which clashes with R7C7)
10a. 3 of {2346} must be in R8C8 -> no 4 in R8C8
11. Hidden killer pair for 3,7 in R5C7 and R78C7 for C7, R789C7 (step 6c) must contain one of 3,7 in R78C7 -> R5C7 = {37}
11a. 39(8) cage at R5C7 = {12345789}, 2,9 locked for R6 -> R6C6 = 6, placed for D\, R7C56 = [69], clean-up: no 4 in R2C3
11b. Naked pair {13} in R6C23, 1 locked for R6, N4 and 6(3) cage at R6C2 -> R7C3 = 2, placed for D/, clean-up: no 8 in R23C3
11c. Naked pair {13} in R67C2, locked for C2
11d. R7C12 = {13} -> R56C1 = 14 = [68/95]
11e. 20(3) cage at R4C3 = {479/578} (cannot be {569} which clashes with R56C1), no 6, 7 locked for C3 and N4, clean-up: no 3 in R23C3
11f. R8C1 = 7 (hidden single in N7)
11g. 2 in N4 only in R4C12 + R5C2, locked for 23(5) cage at R2C1, no 2 in R23C1
11h. 23(5) cage = {23459/23468} (cannot be {12389} which clashes with R7C1, cannot be {12569} which clashes with R56C1), no 1, 3 locked for C1 -> R7C12 = [13], R6C23 = [13]
[Note. 23(5) cage and R56C1 form a combined 37(7) cage = {2345689}.]
11i. 16(4) cage at R2C5 (step 7h) = {1357/2347}, 7 locked for C6
12. 20(3) cage at R4C3 (step 11e) = {479/578}
12a. 9 of {479} must be in R4C4 (cannot be R45C3 + R4C4 = {79}4 which clashes with R23C3 using D\) -> no 4 in R4C4, no 9 in R45C3
12b. 45 rule on N5 1 remaining outie R3C5 = 1 innie R4C4 -> no 4 in R3C5, no 5 in R4C4
12c. R4C4 = {89} -> no 8 in R45C3
12d. 8,9 in N4 only in R45C12 + R6C1, CPE no 8,9 in R23C1
12e. Killer triple 4,5,9 in R23C3, R45C3 and R9C3, locked for C3
12f. 45 rule on N4 3(2+1) outies R23C1 + R4C4 = 16
12g. R4C4 = {89} -> R23C1 = 7,8 = {34/35}, no 6
13. R8C8 = 3 (hidden single on D\) -> R57C7 = [37] (hidden pair in C7), 7 placed for D\, clean-up: no 6 in R9C7
13a. R9C8 = 6 (hidden single in N9)
13b. 7 in N6 only in R45C89, CPE no 7 in R3C8
14. R789C7 (step 6c) = [742/751] -> R89C7 = [42/51] -> R7C8 + R89C7 = [451/542]
14a. 15(3) cage at R3C7 = {168}/[465] (cannot be {159} = [915] which clashes with R7C8 + R89C7), no 9 in R3C7, no 7 in R4C8, 6 locked for C7
14b. 45 rule on N3 3 innies R3C789 = 13 = {139/157/238/256/346} (cannot be {148} which clashes with R3C35, ALS block, cannot be {247} = [427] because no 3 in R45C9)
14c. 1 of {139/157} must be in R3C7 -> no 1 in R3C89
14d. 8 of {238} must be in R3C7 -> no 8 in R3C8
14e. R3C789 = {346} can only be [643] -> no 4 in R3C79
14f. 6 of {256/346} must be in R3C7 -> no 6 in R3C9
14e. 15(3) cage = {168} (only remaining combination), no 5 in R4C8
15. R89C7 (step 14) = [42/51]
15a. 17(3) cage at R1C7 = {179/269/458/467} (cannot be {278} because no 2,7,8 in R1C9)
15b. 17(3) cage = {179/269/458} (cannot be {467} = [476] because R1C9 + R3C7 + R8C2 = [618] and R13C7 = [41] clashes with R89C7)
15c. 1 of {179} must be in R1C9 -> no 1 in R1C78
15d. 15(3) cage at R2C7 = {249/258/348/456} (cannot be {159/168} which clash with 17(3) cage, cannot be {267/357} because 3,6,7 only in R2C9), no 1,7
15e. 17(3) cage = {179/269} (cannot be {458} which clashes with 15(3) cage), no 4,5,8
15f. 17(3) cage can only be [926/971] (cannot be [296] because then R3C7 = 1, hidden single in N3, and R13C7 = [21] clash with R9C7) -> R1C7 = 9, R6C8 = 9 (hidden single in N6)
15g. 1 in N3 only in R1C9 + R3C9, locked for D/
16. 17(4) cage at R3C8 = {1457/2357/2456} (cannot be {1367} because R3C8 only contains 2,4,5)
16a. Consider permutations for R5C8 + R9C7 = [72/81]
R5C8 + R9C7 = [72] => R9C9 = 1 => 17(4) cage = {2357/2456}
or R5C8 + R9C7 = [81] => R4C8 = 1 => 17(4) cage = {2357/2456}
-> 17(4) cage = {2357/2456}, no 1, 2 locked for R3 and N3 -> R1C8 = 7, R1C9 = 1 (cage sum)
16b. R45C8 = [18], R4C7 = 6, R3C7 = 8, placed for D/, R9C9 = 2, placed for D\, R7C8 = 4 (cage sum), R2C8 = 5, placed for D/
16c. R3C3 = 1 (hidden single on D\) -> R2C3 = 9
16d. R3C5 = 9 -> R5C5 = 4, placed for both diagonals
and the rest is naked singles, without using the diagonals.