Prelims
a) R12C1 = {16/25/34}, no 7,8,9
b) R45C3 = {16/25/34}, no 7,8,9
c) Disjoint 15(2) cage at R6C3 = {69/78}
d) R6C78 = {17/26/35}, no 4,8,9
e) R78C6 = {16/25/34}, no 7,8,9
f) R9C56 = {16/25/34}, no 7,8,9
g) R9C89 = {19/28/37/46}, no 5
h) 24(3) cage at R1C4 = {789}
i) 8(3) cage at R2C3 = {125/134}
j) 22(3) cage at R4C1 = {589/679}
k) 20(3) cage at R5C4 = {389/479/569/578}, no 1,2
1a. Naked triple {789} in 24(3) cage at R1C4, locked for R1 and N2
1b. 22(3) cage at R4C1 = {589/679}, 9 locked for N4
1c. 45 rule on N4 3 outies R7C123 = 24 = {789}, locked for R7 and N7
1d. 25(5) cage at R5C1 cannot contain more than two of 7,8,9 -> no 7,8 in R5C1 + R6C12
1e. Hidden killer pair 7,8 in 22(3) cage and R6C3 for N4, 22(3) cage contains one of 7,8 -> R6C3 = {78}, clean-up: no 9 in R7C1
1f. R347C1 = {789} (hidden triple in C1)
1g. Naked pair {78} in disjoint 15(2) cage at R6C3, CPE no 7,8 in R4C1 + R7C3 -> R4C1 = 9, R7C3 = 9
1h. R36C3 = {78} (hidden pair in C3)
1i. Naked pair {78} in R3C13, locked for C3 and N1
1j. 45 rule on N47 1 innie R8C3 = 1, clean-up: no 6 in R45C3, no 6 in R7C6
1k. 8(3) cage at R2C3 = {125/134}, 1 locked for R2 and N2, clean-up: no 6 in R1C1
1l. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 14 = {59/68} (cannot be [77] because R6C9 + R9C6 ‘see’ all the cells in N9)
1m. 7 in R9 only in R9C789, locked for N9
1n. 45 rule on N8 1 innie R9C6 = 1 remaining outie R6C4 + 5 -> R6C4 = {34}, R9C6 = {89}, clean-up: no 8,9 in R6C9
1o. 9 in R6 only in R6C56, locked for N5
1p. 1,2 in N5 only in R4C456 + R5C56, CPE no 1,2 in R4C8
1q. 20(3) cage at R5C4 = {389/479/569/578}
1r. 9 of {389/479/569} only in R6C5 -> no 3,4,6 in R6C5
1s. 45 rule on N78 2 outies R6C49 = 9 = [36/45]
1t. R6C78 = {17/26} (cannot be {35{ which clashes with R6C49), no 3,5
1u. Hidden killer triple 7,8,9 in R9C6, R9C7 and R9C89 for R9, R9C6 contains one of 8,9, R9C7 and R9C89 can each only contain one of 7,8,9 -> R9C7 = {789}, R9C89 = {19/28/37}, no 4,6
1v. R9C6, R9C7 and R9C89 each contain one of 7,8,9, CPE no ,8,9 in R8C7
1w. R8C45 + R9C6 = {789} (hidden triple in N8)
1x. 20(3) cage = {389/479/569/578} must have one of 7,8,9 in C5 (cannot have two which would clash with R18C5
1y. Killer triple 7,8,9 in R1C5, 20(3) cage and R8C5, locked for C5
2a. 1 in N8 only in R7C6 + R9C45 -> combined cage R78C6 + R9C45 = {16}{25}/{16/34}, 6 locked for N8
2b. 7 in N8 only in 27(6) cage at R6C4 = {123579/124578}
2c R6C4 = {34} -> no 3,4 in R7C45
2d. Naked pair {25} in R7C45, locked for R7 and N8
2e. 6 in R7 only in R7C789, locked for N9
2f. 30(5) cage at R7C7 = {35679/45678} (cannot be {15789/24789} because 7,8,9 only in R9C67, cannot be {25689} because 2,5 only in R8C7, cannot be {34689} which clashes with R9C89), no 1,2, 5,6 locked for N9
2g. 30(5) cage at R7C7 = {35679/45678} -> R89C7 = [57], clean-up: no 1 in R6C8, no 3 in R9C89
2h. 45 rule on C9 3 outies R189C8 = 17 = {269/359/368/458}, no 1, clean-up: no 9 in R9C9
2i. 5,6 only in R1C8 -> R1C8 = {56}
2j. 1 in N9 only in R79C9, locked for C9
2k. Hidden killer pair 5,6 in R12345C9 and R6C9 for C9, R6C9 = {56} -> R12345C9 must contain one of 5,6
2l. 33(6) cage at R1C8 contains both of 5,6 = {235689/245679/345678}
2m. 19(4) cage at R6C9 = {1369/1459/2368/2458} (cannot be {1468/2359} which clash with R9C89)
2n. Consider combinations for R9C89 = {28}[91]
R9C89 = {28}, locked for N9, 8 locked for R9 => R9C6 = 9 => 30(5) cage = {35679}, 3 locked for N9 => 19(4) cage = {1459}
or R9C89 = [91], R9C6 = 8 => 30(5) cage = {45678}, 4 locked for N9 => 19(4) cage = {2368}
-> 19(4) cage = {1459/2368}
2o. 1 of {1459} must be in R7C9 -> no 4 in R7C9
2p. 3 of {2368} must be in R7C9 -> no 3 in R8C89
2q. 3 in N9 only in R7C789, locked for R7, clean-up: no 4 in R8C6
2r. 5,6 in C9 only in R123456C9 -> R123456C9 forms hidden 33(6) cage
2s. 45 rule on C9 3 remaining innies R789C6 = 12 must contain 1 for C9 = {129/138} = [192/381], R8C9 = {89}, R9C9 = {12}, clean-up: no 2 in R9C8
2t. Naked pair {89} in R8C9 + R9C8, locked for N9
[It took a while to find the next step.]
3a. Consider placement for 6 in N1
6 in R1C23 => R1C8 = 5 => R6C9 = 5 (hidden single in C9)
or R2C1 = 6 => R1C1 = 1 => R6C2 = 1 (hidden single in N4) => R6C78 = {26}, 6 locked for R6 => R6C9 = 5
or 6 in R23C2, R45C2 = 13 = {58}, 8 locked for C2 => R7C12 = [87], R6C3 = 7 => R6C78 = {26}, 6 locked for R6 => R6C9 = 5
-> R6C9 = 5, R6C4 = 4 (step 1s), R7C9 = 1, R8C89 = [49] (all step 2n), R9C89 = [82], R9C6 = 9, 6 in C9 only in R12345C9, locked for 33(6) cage -> R1C8 = 5, R7C6 = 4 (hidden single in R7) -> R8C6 = 3, naked pair {16} in R9C45, 6 locked for R9, clean-up: no 2 in R2C1
3b. R1C3 = 6 (hidden single in C3), clean-up: no 1 in R1C1
3c. 1 in N4 only in 25(5) cage at R5C1 = {12679/13489} (cannot be {13579} which clashes with R45C3, cannot be {12589} = 5{12}[89] which clashes with R12C1 + R9C1, killer ALS block), no 5
3d. Whichever value is in R7C2 must also be in R6C3 because of 15(2) disjoint cage at R6C3 -> R5C1 + R6C123 + R7C3 form hidden 25(5) cage with the same combination as 25(5) cage at R5C1
3e. Hidden 25(5) cage = {13489} (cannot be {12679} = {126}[79] which clashes with R6C78) -> R5C1 = 4, R6C12 = {13}, locked for R6, 3 locked for N4, R6C3 = 8 -> R7C12 = [78], clean-up: no 3 in R12C1, no 7 in R6C8
3f. Naked pair {25} in R45C3, locked for C3, 5 locked for N4
3g. R12C1 = [25] -> R9C1 = 3, R9C3 = 4, R2C3 = 3 -> R2C45 = 5 = [14], R2C2 = 9, R9C45 = [61]
3h. Naked pair {26} in R6C78, locked for R6 and N6
3i. R4C8 ‘sees’ all remaining cells in N5 except for 20(3) cage at R5C4, R4C8 = {37} -> 20(3) cage must contain one of 3,7 = {389/578}, no 6
3j. Naked triple {789} in R168C5, locked for C5 -> R5C4 = 8, R8C45 = [78], R1C456 = [978], R56C5 = [39]
4a. R6C6 = 7 -> R4C8 = 3, R7C78 = [36], R6C78 = [62]
4b. R4C8 = 3, R6C4 = 7 -> R5C678 = 12 = 2{19}, R4C456 = [561]
4c. R23C6 = [65], R2C8 = 7 -> R12C7 + R3C8 = 7 = [421]
and the rest is naked singles.
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