Prelims
a) R34C7 = {12}
b) R45C1 = {69/78}
c) R4C89 = {15/24}
d) R56C9 = {49/58/67}, no 1,2,3
e) R6C12 = {19/28/37/46}, no 5
f) R67C3 = {14/23}
g) R7C67 = {19/28/37/46}, no 5
h) R9C45 = {18/27/36/45}, no 9
i) 20(3) cage at R5C7 = {389/479/569/578}, no 1,2
j) 9(3) cage at R7C1 = {126/135/234}, no 7,8,9
k) 30(4) cage at R8C1 = {6789}
1a. Naked pair {12} in R34C7, locked for C7, clean-up: no 8,9 in R7C6
1b. Killer pair 1,2 in R4C7 and R4C89, locked for R4
1c. Naked quad {6789} in 30(4) cage at R8C1, locked for N7
1d. Naked quad {6789} in R4589C1, locked for C1, clean-up: no 1,2,3,4 in R6C2
1e. 9(3) cage at R7C1 = {135/234}, 3 locked for N7, clean-up: no 2 in R6C3
1f. 45 rule on N7 2 innies R78C3 = 6 = [15]/{24}, no 1 in R8C3
1g. 45 rule on N1 2 innies R3C23 = 15 = {69/78}
1h. R3C23 = 15 -> R3C4 + R45C2 = 11 = {128/137/146/236/245}, no 9
1i. 8 on {128} must be in R4C2 -> no 8 in R3C4 + R5C2
1j. 45 rule on N3 2 innies R23C7 = 10 = [82/91]
1k. 16(3) cage at R2C5 = {169/178/259/268/349/358} (cannot be {367/457} because R3C7 only contains 8,9)
1l. R2C7 = {89} -> no 8,9 in R2C56
1m. 45 rule on N9 2 innies R7C89 = 11 = {38/47}/[65], no 9, no 6 in R7C8, clean-up: no 1 in R7C6
1n. 45 rule on N6 2(1+1) outies R6C6 + R7C8 = 1 innie R4C7 + 12
1o. Min R6C6 + R7C8 = 13, no 1,2,3,4 in R6C6, no 3 in R7C8, clean-up: no 8 in R7C7, no 2 in R7C6
[If I’d spotted that importance of Ed’s step 4, my step 5a, at this stage, my solving path would have been somewhat simpler.]
2. R67C3 = {14}/[32], R78C3 (step 1f) = [15]/{24} -> combined cage R678C3 = [142/324/415], 4 locked for C3
3a. 45 rule on N78 1 innie R7C6 = 2 outies R6C34 + 2
3b. Min R6C34 = 3 -> no 3,4 in R7C6, clean-up: no 6,7 in R7C7, no 4,5 in R7C8 (step 1m)
3c. R7C6 = {67} -> R6C34 = 4,5 = {13/14}/[32]
3d. Max R6C4 = 4 -> min R7C45 = 11, no 1 in R7C45
3e. R7C67 = [64/73], R7C78 (step 1m) = [38/47] -> combined cage R7C678 = [647/738], 7 locked for R7
3f. 20(3) cage at R5C8 = {389/479/578} (cannot be {569} because only 7,8 in R7C8), no 6
3g. R6C6 + R7C8 = R4C7 + 12 (step 1n)
3h. R4C7 = {12} -> R6C6 + R7C8 = 13,14 = [58/68/77] (cannot be [67] which clashes with R7C6)
3i. Combining R6C6 + R7C8 with R7C678, R6C6 + R7C678 = [5738/6738/7647], 7 in R67C6, locked for C6
4a. Consider permutations for R6C6 + R7C678 (step 3i) = [5738/6738/7647]
R6C6 + R7C678 = [5738/6738] => R7C8 = 8, 3 in N6 only in R56C8 = 12 = {39}
or R6C6 + R7C678 = [7647], R6C6 = 7 => R56C7 = 11 = {38/56}, R7C8 = 7 => R56C8 = 13 = {49} (cannot be {58} which clashes with R56C7)
-> R56C8 = {39/49}, no 5,7,8, 9 locked for C8 and N6, clean-up: no 4 in R56C9
4b. 18(3) cage at R5C7 = {378/468/567}
4c. 5 of {567} must be in R6C6 (R56C7 cannot be {56/57} which clash with R56C9), no 5 in R56C7
4d. Consider combinations for 18(3) cage
18(3) cage = {378/468}, 8 locked for N6 => R56C9 = {67}
or 18(3) cage = {567} => R6C6 = 5
-> no 5 in R6C9, clean-up: no 8 in R5C9
4e. 5 in R6 only in R6C56, locked for N5
[Looking at the 18(3) cage a different way.]
4f. Consider combinations for 18(3) cage
18(3) cage = {378/468} => R56C7 = {38/48}, killer pair 3,4 in R56C7 + R7C7, locked for C7
or 18(3) cage = {567} => R6C6 = 5, R7C6 = 7 (hidden single in C6) => R7C7 = 3
-> 3 in R567C7, locked for C7
Also R56C7 = {38/48}, locked for C7 => R2C7 = 9 or R56C7 = {67}, locked for C7 -> R89C7 cannot contain both of 6,9
4g. 16(3) cage at R8C8 = {259/358/457} (cannot be {169} because R89C7 cannot contain both of 6,9, cannot be {178} which clashes with R7C8, cannot be {349} which clashes with R7C7, cannot be {268} which clashes with R56C7, cannot be {367} which clashes with R7C78), no 1,6, 5 locked for N9
4h. Min R89C7 = 9 -> max R8C8 = 7
5a. 45 rule on R89 1 outie R7C9 = 1 innie R8C2 + 3 -> min R7C9 = 4
5b. 1 in R7 only in R7C123, locked for N7
5c. R7C9 + R8C2 = [63/85]
5d. Naked triple {678} in R7C689, 6,8 locked for R7
5e. 9 in R7 only in R7C45, locked for N8
5f. 15(3) cage at R6C4 contains 9 = {159/249}, no 3
5g. 8 in R7 only in R7C89, locked for N9, clean-up: no 3 in 16(3) cage at R8C8 (step 4g)
5h. Killer pair 6,8 in R56C9 and R7C9, locked for C9
6a. R6C34 (step 3c) = {14}/[31/32], R78C3 (step 1f) = [15]/{24}, 15(3) cage at R6C4 (step 5f) = 1{59}/{249}
6b. Consider placements for 4 in N8
4 in R7C45 => 15(3) cage = {249} = 2{49}
or 4 in R8C456 + R9C6, locked for 20(5) cage at R8C3, no 4 in R8C3 => no 2 in R7C3 => no 3 in R6C3
or 4 in R9C45 = {45}, 5 locked for N8 => 15(3) cage = {249} = 4{29}
-> no 3 in R6C3 or no 1 in R6C4
6c. R6C34 = {14}/[32] (cannot be [31]) = 5 -> R7C6 = 7 (step 3a), R7C7 = 3, R7C89 = [86], clean-up: no 7 in R56C9, no 2 in R9C45
6d. R56C9 = [58], clean-up: no 1 in R4C89, no 2 in R6C1
6e. Naked pair {24} in R4C89, locked for N6, 4 locked for R4 -> R34C7 = [21], R2C7 = 8 (step 1j)
6f. Naked pair {67} in R56C7, locked for C7, R6C6 = 5 (cage sum)
6g. 2 in R6 only in R6C45, locked for N5
6h. Naked pair {39} in R56C8, 3 locked for C8
6i. 16(3) cage at R8C8 (step 4g) = {259/457}
6j. 2,7 only in R8C8 -> R8C8 = {27}
6k. 5 in C8 only in R123C8, locked for N3
6l. 13(3) cage at R2C8 = {157/346} (cannot be {139} because 3,9 only in R3C9), no 9
6m. 3 of {346} must be in R3C9 -> no 4 in R3C9
6n. Hidden killer pair 5,6 in R1C8 and R23C8 for C8, R23C8 contains one of 5,6 -> R1C8 = {56}
7a. 45 rule on N2 1 innie R3C4 = 1 remaining outie R4C6 + 1 -> R3C4 = {47}, R4C6 = {36}
7b. R3C4 + R45C2 (step 1h) = {137/146/245} (cannot be {128/236} because R3C4 only contains 4,7), no 8
7c. R3C4 = {47} -> R45C2 = [31/52/61]
7d. 45 rule on N5 3 remaining innies R4C46 + R6C6 = 13 = [931/832] (cannot be {36}4 which clashes with R3C4 + R45C2 = [731]) -> R4C4 = {89}, R6C4 = {12}, R4C6 = 3 -> R3C4 = 4, clean-up: no 5 in R9C5
7e. R2C7 = 8 -> R2C56 = 8 = {26}/[71]
7e. R4C6 = 3 -> R3C56 = 13 = [58] (cannot be [76] which clashes with R2C56), clean-up: no 7 in R3C23 (step 1g)
7f. Naked pair {69} in R3C23, locked for N1, 6 locked for R3 and 26(5) cage at R3C2 -> R4C2 = 5, R5C2 = 2 (cage sum)
7g. R8C2 = 3 -> R7C12 = 6 = [24/51]
[I’d forgotten that I could have got R8C2 = 3 after R7C9 = 6, step 6c, because I continued looking at other things.]
7h. 15(3) cage at R6C4 (step 5f) = [159/294]
7i. Killer pair 4,5 in R7C12 and R7C45, locked for R7, clean-up: no 1 in R6C3, no 2 in R8C3 (step 1f)
7j. R6C34 (step 6c) = [32/41]
7k. Consider placement for 1 in N4
R5C3 = 1 => R7C3 = 2, R6C3 = 3 => R6C4 = 2
or R6C1 = 1 => R6C4 = 2
-> R6C4 = 2, R7C45 = [94], R7C23 = [12], R7C1 = 5, R68C3 = [34], R56C8 = [39], clean-up: no 5 in R9C4
7l. R6C15 = [41] (hidden pair in R6) -> R6C2 = 6, R56C7 = [67], R3C23 = [96], clean-up: no 9 in R45C1, no 8 in R9C4
7m. Naked pair {78} in R45C1, locked for C1 and N4
7n. R123C1 = {123} -> R1C2 = 8 (cage sum)
7o. 13(3) cage at R2C8 (step 6l) = {157} (cannot be {346} because 4,6 only in R2C8) -> R2C8 = 5, R3C89 = {17}, locked for N3, 1 locked for R3
7p. R45C3 = [91], R9C23 = [78], clean-up: no 1 in R9C4
7q. Naked pair {36} in R9C45, 6 locked for R9 and N8
7r. Naked pair {12} in R89C6, locked for C6 and N8, R2C6 = 6 -> R2C5 = 2 (cage sum)
7s. R1C67 = [94], R89C7 = {59} -> R8C8 = 2 (cage sum)
and the rest is naked singles.
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