Prelims
a) R1C12 = {16/25/34}, no 7,8,9
b) R12C9 = {16/25/34}, no 7,8,9
c) R45C1 = {29/38/47/56}, no 1
d) R78C9 = {79}
e) R89C5 = {59/68}
f) R89C6 = {17/26/35}, no 4,8,9
g) 20(3) cage at R1C7 = {389/479/569/578}, no 1,2
h) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2
i) 21(3) cage at R9C2 = {489/579/678}, no 1,2,3
j) 18(5) cage at R3C5 = {12348/12357/12456}, no 9
Steps resulting from Prelims
1a. Naked pair {79} in R78C9, locked for C9 and N9
1b. 18(5) cage at R3C5 = {12348/12357/12456}, CPE no 1,2 in R6C5
2. 45 rule on R9 3 innies R9C156 = 9 = {126/135} (cannot be {234} because no 2,3,4 in R9C5), no 4,7,8,9, clean-up: no 5,6 in R8C5, no 1 in R8C6
3. 7,9 in R9 only in 21(3) cage at R9C2 = {579}, locked for R9 -> R9C5 = 6, R8C6 = 8, clean-up: no 2,3 in R8C6, no 2 in R9C6
4. 8 in R9 only in 15(3) cage at R9C7 = {348}, locked for R9 and N9 -> R9C6 = 1, R8C6 = 7, R78C9 = [79], R9C1 = 2, clean-up: no 5 in R1C2, no 9 in R45C1
5. 21(5) cage at R6C5 = {12459/23457} (cannot be {12369/12567/13467} because 1,6 only in R8C3), no 6, 2 locked for N8
5a. 7 of {23457} must be in R6C5 -> no 3 in R6C5
6. 25(6) cage at R5C2 = {123478/123568/124567} (cannot be {123469} which clashes with R45C1), no 9
[Alternatively 25(6) cage and R45C1 form a 36(8) cage -> no 9 in 25(6) cage; would also have eliminated 9 from R45C1 if this hadn’t been done in step 4.]
7. 9 in C1 only in R23C1, locked for N1
7a. 20(3) cage at R2C1 = {389/479/569}
8. 15(3) cage at R3C6 = {249/258/348/456}
8a. 20(5) cage at R6C6 = {12359/12368/23456} (cannot be {12458} which clashes with 15(3) cage), 3 locked for C6
8b. 20(5) cage = {12359/12368/23456} -> R67C5 = {39/38/34}, no 2,5,6
[I was going to use 45 on N3 next, then I spotted the much more powerful …]
9. R67C5 (step 8b) = {39/38/34}
9a. 45 rule on N69 3(2+1) outies R3C9 + R67C6 = 18 -> R3C9 = 6, R67C6 = 12 = {39}, locked for C6, clean-up: no 1 in R12C9
9b. 20(5) cage at R6C6 (step 8a) = {12359} (only remaining combination) -> R7C8 = 6 (hidden single in N9)
9c. 1 in C9 only in R456C9, locked for N6
9d. 9 in N5 only in R6C456, locked for R6
9e. 6 in R8 only in R8C12, CPE no 6 in R56C2
10. 6 in N6 only in 18(3) cage at R5C7 = {369/468/567}, no 2
11. 45 rule on R123 3 remaining innies R3C256 = 17 = {278/458}, no 1,3, 8 locked for R3
11a. 18(5) cage at R3C5 = {12348/12357/12456}, 1 locked for N5
12. 1 in N3 only in R2C7 + R3C78
12a. 45 rule on N3 2 outies R12C6 = 10 = {28/46}, no 5
12b. 45 rule on N3 3 remaining innies R2C7 + R3C78 = 12 = {129/138/147}, no 5
12c. 8 of {138} must be in R2C7 -> no 3 in R2C7
13. 4 in N8 only in R7C45 + R8C4, locked for 21(5) cage at R6C5, no 4 in R6C5 + R8C4
14. 21(5) cage at R6C5 (step 5) = {12459/23457}
14a. 3 in N8 only in 21(5) cage = {23457} or in R67C6 = [93] -> 9 of {12459} must be in R7C45 (blocking cages), no 9 in R6C5
15. 18(5) cage at R3C5 = {12348/12456} (cannot be {12357} which clashes with R6C5), no 7
15a. 7 in N5 only in R6C45, locked for R6
16. 19(4) cage at R3C9 contain 6 = {1369/1468/1567/2368/2467}
16a. 9 in N6 only in 19(4) cage = {1369} or in 18(3) cage at R5C7 (step 10) = {369} -> 19(4) cage = {1369/1468/1567/2467} (cannot be {2368}, locking-out cages)
16b. 1 of {1369/1468/1567} must be in R4C9, 2 or 4 of {2467} must be in R4C9 -> R4C9 = {124}
17. 20(4) cage at R5C9 contains 6 = {1568/2468}, no 3, 8 locked for N6
18. 18(3) cage at R5C7 (step 10) = {369/567}, no 4
18a. 45 rule on R6789 4 outies R5C2789 = 25 = {2689/3589/4678} (cannot be {1789/3679/4579} because 18(3) cage only contains one of 7,9), no 1, 8 locked for R5, clean-up: no 3 in R4C1
18b. 3,9 of {3589} must be in R5C78 (18(3) cage cannot contain both of 5,9), no 3 in R5C2, no 5 in R5C78
18c. [67] of {4678} must be in R5C78, no 7 in R5C27
19. 20(4) cage at R5C9 (step 17) = {1568/2468}
19a. 1 of {1568} must be in R6C9 -> no 5 in R6C9
20. 45 rule on C9 2 innies R49C9 = 1 remaining outie R6C8 + 2
20a. R49C9 cannot total 6 (because R49C9 = [24] clashes with R12C9) -> no 4 in R6C8
20b. R6C8 = {258} -> R49C9 = 4,7,10 = [13/43/28], no 4 in R9C9
20c. R12C9 = {25/34}, R49C9 = [13/43/28] -> combined cage R1249C9 = {25}[13]/{25}[43]/{34}[28], 2 locked for C9
21. 25(6) cage at R5C2 (step 6) = {123568} (only remaining combination), 6 locked for C1, clean-up: no 1 in R1C2
21a. 25(6) cage at R5C2 = {123568}, CPE no 3,5,8 in R45C1
21b. Naked pair {47} in R45C1, locked for C1 and N4, clean-up: no 3 in R1C2
22. 20(3) cage at R2C1 (step 7a) = {389/569} (cannot be {479} because 4,7 only in R2C2), no 4,7
22a. 6 of {569} must be in R2C2 -> no 5 in R2C2
23. R5C2789 (step 18a) = {3589} (only remaining combination, cannot be {4678} which clashes with R5C1) -> R5C29 = {58}, locked for R5, R5C78 = {39}, locked for R5 and N5, R6C7 = 6 (cage sum)
23a. R8C1 = 6 (hidden single in C1)
24. R5C1 = 7 (hidden single in R5), R4C1 = 4
25. 19(4) cage at R3C9 (step 16a) = {1567} (only remaining combination) -> R4C9 = 1, R4C78 = {57}, locked for R4 and N6 -> R5C9 = 8, R5C2 = 5, R6C89 = [24], R9C9 = 3
25a. Naked pair {25} in R12C9, locked for N3
25b. R2C7 + R3C78 (step 12b) = {138/147}, no 9
26. Naked triple {138} in R6C123, locked for R6 and N4 -> R67C6 = [93]
26a. 25(6) cage at R5C2 (step 21a) = {123568}, 3 locked for N4
27. R4C23 + R5C3 = {269} = 17 -> R3C2 = 8 (cage sum)
27a. 20(3) cage at R2C1 (step 22) = {569} (only remaining combination) -> R2C2 = 6, R23C1 = {59}, locked for N1, clean-up: no 1 in R1C1, no 2 in R1C2
27b. R1C12 = [34]
27c. Naked triple {127} in R123C3, locked for C3 -> R45C3 = [96], R4C2 = 2, R4C5 = 3, R6C3 = 8, R67C1 = [18], R6C2 = 3, R78C2 = [91], R789C3 = [435], R6C4 = 5 (cage sum), R78C4 = [24], R5C4 = 1, R7C5 = 5
28. R3C6 = 5 (hidden single in C6), R23C1 = [59], R12C9 = [52]
29. 18(5) cage at R3C5 (step 15) = {12348} -> R4C4 = 8, R4C6 = 6, R5C6 = 4 (cage sum), R12C6 = [28]
30. 8 in N3 only in 20(3) cage at R1C7 = {389} (only remaining combination) -> R2C8 = 3
and the rest is naked singles.