Prelims
a) R1C34 = {39/48/57}, 1,2,6
b) R1C67 = {29/38/47/56}, no 1
c) R5C12 = {29/38/47/56}, no 1
d) R5C89 = {49/58/67}, no 1,2,3
e) R78C4 = {14/23}
f) R78C6 = {29/38/47/56}, no 1
g) R8C12 = {19/28/37/46}, no 5
h) R8C89 = {19/28/37/46}, no 5
i) R9C12 = {18/27/36/45}, no 9
j) R9C89 = {14/23}
k) 11(3) cage at R6C1 = {128/137/146/236/245}, no 9
l) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3
1a. 45 rule on R89 3 outies R7C456 = 7 = {124}, locked for R7 and N8
1b. R8C4 = 3 -> R7C456 = [214], R8C6 = 7, clean-up: no 9 in R1C3, no 4,7 in R1C7
1c. 14(3) cage at R4C4 = {149/158/167}, 1 locked for R4 and N5
1d. 11(3) cage at R6C1 = {137/236} (cannot be {128/146/245} because 1,2,4 only in R6C1), no 5,8, 3 locked for R7 and N7, clean-up: no 6 in R9C12
1e. 1,2 only in R6C1 -> R6C1 = {12}
1f. 15(3) cage at R7C5 = 1{59/68}
1g. 13(3) cage at R1C5 = {238/247/346} (cannot be {256} which clashes with 15(3) cage), no 5,9
1h. 17(3) cage at R4C5 = {278/359/368/467} (cannot be {269/458} which clash with 15(3) cage)
1i. 45 rule on N8 2 innies R9C46 = 14 = {59/68}
2a. Combined cage R89C89 = {19}{23}/{28}{14}/{46}{23}, 2 locked for N9
2b. 16(4) cage at R8C7 = {169/178/358/457} (cannot be {349} which clashes with R9C89, cannot be {367} because 3,7 only in R9C7)
2c. {169} must have 6 in R9C6 (R89C7 cannot be {16} which clashes with combined cage) -> no 6 in R89C7, no 9 in R9C6, clean-up: no 5 in R9C4 (step 1i)
[Ed pointed out that I can’t completely eliminate {169} at this stage, I’d been looking at the text of step 2a not the diagram; so partly reworked until that can be eliminated.]
2d. Consider combinations for R7C12 = {36/37}
R7C12 = {36} => 6 in N9 only in R8C89 = {46}, 4 locked for N9 => R9C89 = {23}, 3 locked for N9
or R7C12 = {37} => R9C7 = 7 (hidden single in N9)
-> 16(3) cage at R8C7 = {169/178/457}, no 3
2e. 7 of {178/457} must be in R9C7 -> no 4,5,8 in R9C7
2f. 3 in N9 only in R9C89 = {23}, 2 locked for R7 and N9, clean-up: no 8 in R8C89, no 7 in R9C12
2g. 4 in N9 only in R8C789, locked for R8, clean-up: no 6 in R8C12
2h. Killer pair 5,8 in R9C12 and R9C46, locked for R9, clean-up: no 6,9 in R8C5 (step 1f)
2i. 16(3) cage at R8C3 = {169/178} (cannot be {259 because 2,5 only in R8C3, cannot be{457} because no 4,5,7 in R9C4, cannot be {268} = [268] because R89C5 = [59] clashes with R9C6 = 5 or alternatively R9C34 = [68] clashes with R9C46 = [86], CCC), no 2,4,5
2j. 2 in N7 only in R8C12 = {28}, 8 locked for R8 and N9 -> R8C5 = 5, R9C5 = 9 (cage sum), clean-up: no 1 in R9C12
2k. Naked pair {68} in R9C46, 6 locked for R9
2l. 16(3) cage at R8C7 = {169/178}, no 4, 1 locked for C7 and N9, clean-up: no 9 in R8C89
2m. Naked pair {46} in R8C89, 6 locked for R8 and N9
2n. Naked pair {45} in R9C12, 5 locked for N7
2o. 1,9 in N7 only in R789C3, locked for C3
2p. 25(4) cage at R5C3 = {2689/3589/3679/4579/4678} (cannot be R567C3 = {1789} because {789} clashes with R89C3), no 1
2q. 14(3) cage at R4C6 = {239/356}, no 8, 3 locked for C6 and N5, clean-up: no 8 in R1C7
2r. 17(3) cage at R4C5 (step 1h) = {278/467}, 7 locked for C5 and N5
3a. R5C4 = 1 (hidden single in R5)
3b. 14(3) cage at R4C4 (step 1c) = {149/158}, no 6
3c. 45 rule on R1234 3 innies R4C456 = 16 = {259/268/349/358/457} (cannot be {367} because no 3,6,7 in R4C4)
3d. R4C456 = {259/268/457} (cannot be {349/358} because 13(2) in R4C45 + R4C6 = 3 clashes with the same 1 + 13(2) in 14(3) cage at R4C4, CCC), no 3 in R4C6
3e. 8 of {268} must be in R4C4, 7 of {457} must be in R4C5 -> no 4,8 in R4C5
4c. R9C46 = {68}
4b. 45 rule on C789 4 outies R1239C6 = 20 contains 1,8 = {1289/1568}
4c. 45 rule on C123 4 outies R1239C4 = 26 contains 7 = {4679/5678}
4d. R123C9 cannot be {567}8 which clashes with R1239C6 = {158}6 -> R9C46 = [68] -> R89C3 = 10 = [91], R89C7 = [17], clean-up: no 3 in R1C7
[Back to some of my original steps, renumbered and in some cases simplified.]
5a. 21(3) cage at R6C9 = {489/579} (cannot be {678} because 6,7 only in R7C89), no 6
5b. 4,7 only in R6C9 -> R6C9 = {47}
5c. 21(3) cage = {489/679}, 9 locked for N9
5d. Combined half cage R5C89 + R6C9 = {58}4/{67}4/{49}7/{58}7
5e. 45 rule on N69 3 remaining innies R4C789 = 15 = {159/168} (cannot be {456} which clashes with R5C89, cannot be {249/258/267} which clash with R4C456, step 3d, cannot be {348/357} which clash with R5C89 + R6C9), no 2,3,4,7, 1 locked for R4 and N6
5f. R6C1 = 1 (hidden single in N4) -> R7C12 = {37}, R7C3 = 6 (hidden single in R7)
5g. 45 rule on N47 3 remaining innies R4C123 = 14 contains 3 (hidden single in R4) = {239/347} (cannot be {356} which clashes with R4C789), no 5,6,8
5h. 6 in N4 only in R5C12 = {56}, locked for R5, 5 locked for N4, clean-up: no 7,8 in R5C89
5i. Naked pair {49} in R5C89, locked for R5 and N6 -> R6C9 = 7 -> R7C89 = {59}, R7C7 = 8 (hidden single in R7), clean-up: no 5 in R4C789
5j. R4C7 = 6, clean-up: no 5 in R1C6
5k. Naked pair {23} in R5C67, locked for R5
5l. 17(3) cage at R4C5 = {278} (only remaining combination), 2,8 locked for C5 and N5
5m. R5C6 = 3 -> R46C6 = 11 = [56], clean-up: no 5 in R1C7
5n. R5C7 = 2 -> R1C67 = [29], clean-up: no 3 in R1C3
5o. 4 in C7 only in R23C7, locked for N3
5p. 5 in C3 only in R123C3, locked for N1
6a. 15(4) cage at R2C9 = {1257/1356}, no 8 -> R4C9 = 1
6b. 15(4) cage = {1257/1356}, 5 locked for N3
6c. R4C78 = [68] = 14 -> R3C67 = 5 = [14]
6d. R2C67 = [93] -> R6C78 = [53], R9C89 = [23]
6e. 15(4) cage = {1257} -> R3C8 = 7, R23C9 = {25}, 5 locked for C9
6f. Naked pair {16} in R12C8, 6 locked for C8 and N3 -> R1C9 = 8, clean-up: no 4 in R1C34
6g. Naked pair {57} in R1C34, 7 locked for R1
7a. 23(4) cage at R3C3 = {2579/3578} (cannot be {3479} because R3C4 only contains 5,8, cannot be {2489} = [2894] which clashes with R4C4), no 4
7b. R4C123 (step 5g) = {239/347}
7c. 23(4) cage = {3578} (cannot be {2579} = [2597] because R4C123 cannot contain both of 7,9), no 2,9
7d. Naked pair {37} in R4C23, locked for R4 and N4, 3 locked for 23(4) cage
7e. R57C3 = [86] = 14 -> R6C23 = 11 = [92]
7f. R3C34 = [58], R1C34 = [75]
7g. R3C1 = 9 (hidden single in R3), R4C1 = 4 -> R2C1 + R3C2 = 8 = {26}
and the rest is naked singles.
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