Prelims
a) R1C34 = {18/27/36/45}, no 9
b) R1C89 = {69/78}
c) R34C7 = {69/78}
d) R6C67 = {18/27/36/45}, no 9
e) R9C12 = {14/23}
f) 16(5) cage at R1C1 = {12346}
g) 32(5) cage at R6C3 = {26789/35789/45689}, no 1
1. 16(5) cage at R1C1 = {12346}, locked for N1, clean-up: R1C4 = {124}
1a. 9 in N1 only in R3C123, locked for R3, clean-up: no 6 in R4C7
2. 45 rule on C12 1 innie R3C2 = 1 outie R2C3 + 7 -> R2C3 = {12}, R3C2 = {89}
3. 45 rule on C89 1 innie R7C8 = 1 outie R8C7 + 5, R7C8 = {6789}, R8C7 = {1234}
4. 45 rule on N3 3 innies R123C7 = 15 = {258/348/357/456} (cannot be {159/249} because R3C7 only contains 6,7,8, cannot be {168/267} which clash with R1C89), no 1,9
4a. R3C7 = {678} -> no 6,7,8 in R12C7
5. 45 rule on N3 3(2+1) outies R1C6 + R2C5 + R4C7 = 22
5a. Min R1C6 + R2C5 = 13, no 1,2,3 in R1C6 + R2C5
6. R123C7 (step 4) = {258/348/357/456} -> R12C7 = {25/34/35/45}
6a. 22(4) cage at R1C6 = {2569/2578/3469/3478/3568/4567} -> R1C6 + R2C5 = {67/68/69/78}, no 4,5
7. 45 rule on N7 2 outies R6C3 + R9C4 = 13 = {49/58/67}, no 1,2,3
8. 45 rule on R789 2 outies R6C35 = 1 innie R7C9
8a. Min R6C35 = 5 -> min R7C9 = 5
8b. Max R6C35 = 9, no 9 in R6C3, no 6,7,8,9 in R6C5, clean-up: no 4 in R9C4 (step 7)
9. 34(7) cage at R4C8 must contain 1, locked for N6, clean-up: no 8 in R6C6
9a. 1 in C7 only in R789C7, locked for N9
10. 45 rule on N47 2(1+1) outies R3C1 + R9C4 = 2 innies R45C3 + 9
10a. Max R3C1 + R9C4 = 18 -> max R45C3 = 9, no 9 in R45C3
[It took me a long time to spot …]
11. 32(5) cage at R6C3 = {26789/45689} (cannot be {35789} which clashes with R13C3, ALS block), no 3
11a. {26789} must be 6{289}7 (other permutations clash with R13C3), no 7 in R6789C3, clean-up: no 6 in R9C4 (step 7)
11b. 32(5) cage = {26789/45689}, 6 locked for C3
12. 3 in C3 only in R45C3, locked for N4
12a. Hidden killer triple 1,2,4 in R2C3, R45C3 and 32(5) cage at R6C3 for C3, R2C3 = {12}, 32(5) cage contains one of 2,4 -> R45C3 must contain one of 1,2,4 and also contains 3 -> R45C3 = {13/23/34}, no 5,7,8
12b. 7 in C3 only in R13C3, locked for N1
13. 16(3) cage at R3C1 = {169/178/259/268/457}
13a. 8 of {178/268} must be in R3C1 -> no 8 in R4C12
[Can also eliminate {178} combination = 8{17}, which clashes with R2C3 + R3C2 = [18] because R2C3 = 1 is hidden single in C3, but I’ll leave this for now as it doesn’t give any extra candidate eliminations.]
14. 45 rule on N7 3 innies R789C3 = 19 = {289/469/568}
14a. 32(5) cage at R6C3 = {26789/45689}
14b. 45 rule on N4 3 innies R456C3 = 1 outie R3C1 + 4
14c. Consider placements for 5 in N1
5 in R13C3 => R789C3 = {289/469}
or R3C1 = 5 => R456C3 = 9 = {135} (cannot be {234} which clashes with 16(3) cage at R3C1 = 5{29/47}) => R789C3 = {289/469}
-> R789C3 = {289/469}, no 5, 9 locked for C3 and N7
14d. R789C3 = {289/469} -> R6C1 + R9C4 = [58/67/85], no 4 in R6C1, no 9 in R9C4
14e. Killer pair 2,4 in R789C3 and R9C12, locked for N7
14f. 9 in R3 only in R3C12, CPE no 9 in R4C2
15. Consider placements for R1C3
R1C3 = 5 => naked triple {789} in R3C123, locked for R3
or R1C3 = {78} => R1C89 = {69} (cannot be {78} which clashes with R1C3), locked for N3 => R3C7 = {78} => naked quad {5789} in R3C1237, locked for R3
-> no 7,8 in R3C45689
15a. R1C3 = 5 => naked triple {789} in R3C123, locked for R3 => R3C7 = 6 => R1C89 = {78}, locked for R1
or R1C3 = {78} => naked quad {6789} in R1C3689, locked for R1
-> no 7,8 in R1C5
16. Consider combinations for R1C89 = {69/78}
R1C89 = {69}, locked for N3
or R1C89 = {78}, locked for N3 => R3C7 = 6
-> no 6 in 15(4) cage at R2C8
17. Consider combinations for R1C6 + R2C5 (step 6a) = {67/68/69/78}
R1C6 + R2C5 = {67/68/69}, 6 locked for N2
or R1C6 + R2C5 = {78} => R1C89 = {69} (cannot be {78} which clashes with R1C3), locked for R1 => 6 in N1 only in R2C12, locked for R2
-> no 6 in R1C5 + R2C46
[At last something more promising …]
18. 15(4) cage at R2C8 = {1239/1248/1257/1347}
18a. Consider placements for R1C3
R1C3 = 5 => R1C4 = 4 => R12C7 cannot contain both of 4,5 => 15(4) cage must contain at least one of 4,5 = {1248/1257/1347}
or R1C4 = {78} => R1C89 = {69} (cannot be {78} which clashes with R1C3), locked for N3
-> 15(4) cage = {1248/1257/1347}, no 9
18b. 9 in N3 only in R1C89 = {69}, locked for R1 and N3, clean-up: no 9 in R4C7
18c. 6 in N1 only in R2C12, locked for R2
18d. Naked pair {78} in R34C7, locked for C7, clean-up: no 1,2 in R6C6
19. R1C36 = {78} (hidden pair in R1), clean-up: no 4 in R1C4
19a. 5 in N1 only in R3C13, locked for R3
20. 22(4) cage at R1C6 (step 6a) = {2578/3478} -> R1C6 + R2C5 = {78}, locked for N2, R12C7 = {25/34}
21. R6C35 = R7C9 (step 8)
21a. Min R6C35 = 6 -> min R7C9 = 6
21b. Max R6C35 = 9, no 5 in R6C5
22. 45 rule on N6 3 innies R456C7 = 1 outie R7C9 + 11
22a. The value in R7C9 must be in R456C7 (because the rest of N6 is the 34(7) cage at R4C8) -> the remaining two values in R456C7 must total 11, 9 is only in R5C7 -> no 2 in R5C7
23. 39(7) cage at R4C6 = {1356789/2346789}, 7,8 locked for N5, clean-up: no 2 in R6C7
[I ought to have spotted this immediately after step 20, but then I wouldn’t have found the interesting step 22a.]
24. 15(4) cage at R2C8 (step 18a) = {1257/1347} (cannot be {1248} which clashes with R12C7), no 8, 7 locked for R2 and N3 -> R34C7 = [87], R3C2 = 9, R2C3 = 2 (step 2), R3C13 = [57], R1C3 = 8 -> R1C4 = 1, R1C6 = 7, R2C5 = 8, clean-up: no 5 in R1C7 (step 20)
24a. Naked pair {34} in R1C12, locked for R1 and N1 -> R1C7 = 2, R2C7 = 5 (cage sum), R1C5 = 5, clean-up: no 4 in R6C6
25. Naked triple {469} in R789C3, locked for C3 and N7 -> R6C3 = 5, R9C4 = 8 (cage sum), clean-up: no 4 in R6C7, no 1 in R9C12
25a. Naked pair {13} in R45C3, locked for N4
25b. Naked pair {36} in R6C67, locked for R6
26. R3C1 = 5 -> R4C12 = 11 = [92], R9C12 = [23]
27. Two values in R456C7 must total 11 (step 22a), R4C7 = 7 -> R5C7 = 4
27a. 39(7) cage at R4C6 = {2346789} (only remaining combination), no 1,5 -> R45C3 = [13]
28. R6C35 = R7C9 (step 8), R6C3 = 5, R6C5 = 1 (hidden single in N6) -> R7C9 = 6, R1C89 = [69]
28a. Naked triple {139} in R789C7, locked for C7 and N9 -> R6C67 = [36]
29. 45 rule on N9 3 remaining innies R7C78 + R9C7 = 18 = {189} (only possible combination, cannot be {378} because 7,8 only in R7C8) -> R7C8 = 8
29a. R8C7 = 3 (hidden single in C7)
30. R4C45 = [54] (hidden pair in N5)
30a. R1C5 = 5, R2C6 = 9, R4C5 = 4 -> R3C56 = 5 = [32]
31. R4C89 = [38], R45C6 = [68], R6C89 = [92], R6C4 = 7, R23C4 = [46]
32. R7C4 = 3 (hidden single in C4), R6C5 = 1 -> R7C5 = 2 (hidden single in R7), R8C4 = 9 (cage sum)
33. R89C5 = {67} = 13 -> R9C67 = 6 = [51]
and the rest is naked singles.