Prelims
a) R1C45 = {18/27/36/45}, no 9
b) R12C7 = {12}
c) R1C89 = {29/38/47/56}, no 1
d) R23C9 = {49/58/67}, no 1,2,3
e) R45C9 = {17/26/35}, no 4,8,9
f) R5C12 = {39/48/57}, no 1,2,6
g) R6C12 = {29/38/47/56}, no 1
h) R6C56 = {29/38/47/56}, no 1
i) R89C6 = {39/48/57}, no 1,2,6
j) R89C7 = {29/38/47/56}, no 1
k) 27(4) cage at R4C7 = {3789/4689/5679}, no 1,2
l) 40(8) cage at R1C3 = {12346789}, no 5
Steps Resulting From Prelims
1a. Naked pair {12} in R12C7, locked for C7 and N3, clean-up: no 9 in R1C89, no 9 in R89C7
1b. 27(4) cage at R4C7 = {3789/4689/5679}, CPE no 9 in R5C8
2. 45 rule on C3456789 3 outies R789C2 = 6 = {123}, locked for C2 and N7, clean-up: no 9 in R5C1, no 8,9 in R6C1
2a. 17(3) cage at R2C2 = {458/467}, no 9, 4 locked for C2, clean-up: no 8 in R5C1, no 7 in R6C1
2b. 17(3) cage at R7C1 = {458/467}, no 9, 4 locked for C1 and N7, clean-up: no 8 in R5C2, no 7 in R6C2
2c. 9 in N7 only in R789C3, locked for C3
2d. 9 in 40(8) cage at R1C3 only in R1C6 + R2C456, locked for N2
2e. 17(4) cage at R8C2 can only contain two of 1,2,3 in R89C2 -> no 1,2,3 in R9C4
2f. R5C12 = [39]/{57}, R6C12 = [29/38]/{56} -> combined cage R56C12 = [39]{56}/[29]{57}/[38]{57}, 5 locked for N4
2g. 5 in C3 only in R789C3, locked for N7
2h. 17(3) cage at R7C1 = {467} (only remaining combination), locked for C1 and N7, clean-up: no 5 in R5C2, no 5 in R6C2
2i. 5 in N4 only in R56C1, locked for C1
2j. 8 in N7 only in R789C7, locked for C7
2k. 8 in 40(8) cage at R1C3 only in R1C6 + R2C456, locked for N2, clean-up: no 1 in R1C45
3a. 45 rule on R789 3 innies R7C23 + R8C3 = 1 outie R6C9 + 15
3b. Max R7C23 + R8C3 = 20 -> max R6C9 = 5
3c. 45 rule on N36 3 innies R45C7 + R6C9 = 19 = {289/379/469/478/568} -> R6C9 = {2345}, no 3,4,5 in R45C7
3d. 45 rule on N89 1 innie R9C4 = 1 outie R6C9 + 4, R6C9 = {2345} -> R9C4 = {6789}
3e. Min R89C2 + R9C4 = 9 -> max R9C3 = 8
3f. 9 in N7 only in R78C3, locked for 30(6) cage at R5C3
4. 45 rule on N9 2 innies R7C78 = 1 outie R6C9 + 4, max R6C9 = 5 -> max R7C78 = 9, no 9 in R7C7, no 7,8,9 in R7C8
4a. R6C9 + R7C78 cannot be [581] which clashes with R45C7 + R6C9 (step 3a) = {68}5 -> no 8 in R7C7
4b. R6C9 + R7C78 cannot be [536] because R89C7 = {38/47} and there’s no place for 5 in N9 -> no 6 in R7C8
4c. 9 in C7 only in R3456C7, CPE no 9 in R46C8
4d. 9 in N6 only in R456C7, locked for C7
4e. R45C7 + R6C9 (step 3c) = {289/379/469/478/568}
4f. Consider combinations for R89C7 = {38/47/56}
R89C7 = {38/56} => R45C7 + R6C9 cannot be {568} = {68}5
or R89C7 = {47} => R7C78 = [63] => R45C7 + R6C9 cannot be {568}
or R89C7 = {47} and R7C78 totals less than 9 => no 5 in R6C9
-> no 5 in R6C9, clean-up: no 9 in R9C4 (step 3d)
4g. Consider placements for R6C9 = {234}
R6C9 = {23} => R45C7 = {89/79}
or R6C9 = 4 => R45C7 = {69}
or R6C9 = 4 => R7C78 = 8 = [62] => R89C7 = {38/47} => R45C7 cannot be {78}
or R6C9 = 4 => R7C78 = 8 = {35} => R89C7 = {47} => R45C7 cannot be {78}
or R6C9 = 4 => R7C78 = 8 = [71] => R45C7 cannot be {78}
-> R45C7 + R6C9 = {289/379/469} (cannot be {478}, 9 locked for N6 and 27(4) cage at R4C7
[Ed pointed out that the eliminations in steps 4f and 4g could be done in the same way as step 4b.]
4h. 36(7) cage at R2C8 must contain 9 in R23C8, locked for C8 and N3, clean-up: no 4 in R23C9
4i. R1C89 = {38/47}, cannot be {56} which clashes with R23C9
4j. Killer pair 7,8 in R1C89 + R23C9, locked for N3
5. The two digits missing from 36(7) cage at R2C8 must total 9
5a. 36(7) cage must contain one or both of 1,8 and 2,7 which aren’t in R2C8 + R3C78 so must be in R456C8 + R6C7 -> R45C9 = {26/35} (cannot be {17} which clashes with 1,8 and/or 2,7), no 1,7
5b. Killer pair 5,6 in R23C9 and R45C9, locked for C9
5c. 1 in N6 only in R456C8, locked for C8 -> R456C8 + R6C7 must contain 8, locked for N6
5d. R45C7 + R6C9 (step 4g) = {379/469}, no 2, clean-up: no 6 in R9C4 (step 3d)
5e. 17(4) cage at R8C2 = {1358/2357} -> R9C3 = 5, 3 locked for N7, clean-up: no 7 in R8C6, no 6 in R8C7
5f. Naked pair {89} in R78C7, 8 locked for 30(6) cage at R5C3
5g. 45 rule on N3 3 outies R456C8 + R6C7 = 18 and contains 1,8 = {1278/1458} (cannot be {1368} which clashes with R45C9), no 3,6
5h. 3 in N6 only in R456C9, locked for C9, clean-up: no 8 in R1C8
5i. 8 in N3 only in R123C9, locked for C9
5j. Max R6C9 = 4 -> max R7C78 (step 4) = 8, no 7 in R7C7
5k. Consider placements for R6C9
R6C9 = 3 => max R7C78 = 7 => no 6 in R7C7
or R6C9 = 4 => R45C7 = {69}, locked for C7
-> no 6 in R7C7
6. 30(6) cage at R5C3 contains 8,9 in R78C3 = {123789/124689} (cannot be {134589} with R56C3 = {34}, R6C4 = 5, CPE no 3,5 in R6C1 so R6C14 = [25] + R6C39 = {34} clash with R6C56), no 5 in R6C4
[I’ve replaced my original step 6a, which was done when my mind was still focussed on getting more from the 30(6) cage, but Ed pointed out that it hadn’t eliminated R56C12 = [5738]. The new step 6a is simpler.]
6a. R5C12 = [39/57], R6C12 = [29/38/56] -> combined cage R56C12 = [3956/5729] (cannot be [5738] clashes with 17(3) cage at R2C2)
-> 9 in R56C2, locked for C2 and N4
[Ed had found 45 rule on C23456789 3 remaining innies R156C2 = 22 = {679} (cannot be [598] which would make R56C1 both 3), then cannot be [976] which would make R56C1 both 5) -> 9 locked in R56C2 for C2 and N4.]
6b. R6C12 = [29/56], no 3 in R6C1, no 8 in R6C2
6c. 8 in N4 only in R4C12, locked for R4
6d. 17(3) cage at R2C2 (step 2a) = {458} (only remaining combination, cannot be {467} which clashes with R56C2, ALS block), locked for C2
6e. 9 in R6 only in R6C12 = [29] or R6C56 = {29}, 2 locked for R6 (locking cages)
7. 45 rule on N8 3 innies R7C56 + R9C4 = 18 = {189/279/468/567} (cannot be {369} because R9C4 only contains 7,8, cannot be {378} which clashes with R89C6, cannot be {459} which doesn’t contain 7,8), no 3
7a. 7,8 of each combination must be in R9C4 -> no 7,8 in R7C56
7b. Consider combinations for R7C56 = {19/29/46/56}
R7C56 = {19/46} = 10 => R7C78 = 8 = {35}
or R7C56 = {29/56} = 11 => R7C78 = 7 = {34} (cannot be [52] which clashes with R7C56)
-> R7C78 = {34/35}, no 2, 3 locked for R7 and N9, clean-up: no 8 in R89C7
7c. Killer pair 4,5 in R7C78 and R89C7, locked for N9
7d. Killer pair 6,7 in R45C7 and R89C7, locked for C7
7e. R6C7 = 8 (hidden single in C7), clean-up: no 3 in R6C56
8. 45 rule on R6789 1 outie R5C3 = 1 remaining innie R6C8, no 2,3,6 in R5C3, no 5 in R6C8
8a. 30(6) cage at R5C3 (step 6) = {123789/124689} -> R7C2 = 2
[Cracked, the rest is straightforward.]
8b. R89C2 = {13}, R9C3 = 5 -> R9C4 = 8 (cage sum), R6C9 = 4 (step 3d), clean-up: no 7 in R1C8, no 4 in R5C3, no 7 in R6C56, no 4 in R89C6
8c. 30(6) cage = {123789} (only remaining combination), no 6
8d. R45C7 + R6C9 = 19 (step 3c), R6C9 = 4 -> R45C7 = 15 = {69}, 6 locked for C7, N6 and 27(4) cage at R5C7, clean-up: no 2 in R45C9, no 5 in R8C7
8e. Naked pair {35} in R45C9, locked for C9 and N6, clean-up: no 8 in R23C9
8f. Naked pair {67} in R23C9, locked for C9 and N3, clean-up: no 4 in R1C8
8g. Naked pair {47} in R89C7, locked for C7 and N9
8h. R1C89 = [38] -> R3C7 = 5, R7C78 = [35], clean-up: no 6 in R1C45
8i. R2C2 = 5 (hidden single in C2)
8j. R89C8 = [86] (hidden pair in C8) -> R78C3 = [89], clean-up: no 3 in R9C6
8k. Naked pair {35} in R5C19, locked for R5
8l. R45C7 = 15 -> R5C56 = 12 = {48}, 4 locked for N5
8m. R7C78 = [35] = 8 -> R7C56 = 10 = {46} (cannot be {19} which clashes with R7C9), locked for R7 and N8 -> R7C1 = 7, R89C1 = [64], R89C7 = [47], R9C6 = 9 -> R8C6 = 3, R7C49 = [19], R9C5 = 2, clean-up: no 7 in R1C4
8n. 30(6) cage = {123789}, 1 locked for C3 and N4
8o. 40(8) cage at R1C3 = {12346789}, 1 locked for N2
8p. 5 in R1 only in R1C45 = {45}, locked for R1 and N2
8q. R3C12 = [18] (hidden pair in R3), R4C2 = 4, R4C1 = 8 (hidden single in R4)
8r. R3C8 = 9 (hidden single in R3), R2C8 = 4, R3C3 = 4 (hidden single in C3)
8s. 2,6 in C3 only in R124C3, locked for 40(8) cage
8t. 3 in R3 only in R3C456, locked for N2
8u. 3 in 40(8) cage only in R24C3, locked for C3
8v. Naked pair {17} in R56C3, locked for C3, N4 and 30(6) cage at R5C3 -> R6C4 = 3, R5C2 = 9 -> R5C1 = 3, R6C2 = 6 -> R6C1 = 5
8w. 15(3) cage at R3C4 = {267} (only remaining combination), locked for C4
and the rest is naked singles.