Prelims
a) R23C9 = {29/38/47/56}, no 1
c) R3C78 = {19/28/37/46}, no 5
d) R56C1 = {39/48/57}, no 1,2,6
e) R56C5 = {15/24}
f) R78C3 = {29/38/47/56}, no 1
g) 10(3) cage at R4C2 = {127/136/145/235}, no 8,9
h) 9(3) cage at R7C1 = {126/135/234}, no 7,8,9
i) 27(4) cage at R1C1 = {3789/4689/5679}, no 1,2
j) 13(4) cage at R1C6 = {1237/1246/1345}, no 8,9
k) 33(5) cage at R5C4 = {36789/45789}, no 1,2
1a. 27(4) cage at R1C1 = {3789/4689/5679}, 9 locked for N1
1b. 45 rule on N3 2 outies R12C6 = 5 = {14/23}
1c. 45 rule on N3 2 innies R12C7 = 8 = {17/26/35}, no 4
1d. 45 rule on R12 2 innies R2C49 = 8 = [17]/{26/35}, no 4,8,9, no 7 in R2C4, clean-up: no 2,3,7 in R4C9
1e. 45 rule on R1234 2 innies R4C29 = 7 = {16/25/34}, no 7,8,9
1f. 45 rule on N7 2(1+1) outies R6C2 + R9C4 = 8 = {17/26/35}/[44], no 8,9
1g. 45 rule on N9 2 outies R6C78 = 5 = {14/23}
1h. 45 rule on N9 2 innies R7C78 = 16 = {79}, locked for R7 and N9, clean-up: no 2,4 in R8C3
1i. 33(5) cage at R5C4 = {36789/45789}, CPE no 7 in R6C6
1j. 45 rule on N69 3 innies R4C78 + R5C7 = 22 = {589/679}, 9 locked for N6
1k. 9 in C9 only in R13C9, locked for N3, clean-up: no 1 in R3C78
[I originally moved on to step 2, having spotted that important group of outies, but later came back here to work further in this area as part of step 1.]
1l. Combined cage R12C7 + R3C78 = {17}{28}/{17}{46}/{35}{28}/{35}{46} (cannot be {26}{37} which clashes with R23C9) -> R12C7 = {17/35}, no 2,6, R3C78 = {28/46}, no 3,7
1m. 16(3) cage at R1C8 = {169/178/259/349/358} (cannot be {268} which clashes with R3C78, cannot be {367/457} which clash with R12C7)
1n. Hidden killer pair 8,9 in 16(3) cage and R3C789 for N3, 16(3) cage contains one of 8,9 -> R3C789 must contain one of 8,9
1o. 18(3) cage at R2C4 = {279/369/378/459/567} (cannot be {189} = 1{89} which clashes with R3C789, cannot be {468} = 6{48} which clashes with R3C78), no 1, clean-up: no 7 in R2C9, no 4 in R3C9
1p. 2 of {279} must be in R2C4 -> no 2 in R3C45
1q. Combined cage R23C9 + R3C78 = [29]/{46}/[38]{46}/{56}{28}, 6 locked for N3
1r. 45 rule on R12 3 outies R3C459 = 21 = {489/579} (cannot be {678} which clashes with R3C78), no 3,6, 9 locked for R3, clean-up: no 5 in R2C9, no 3 in R2C4
[Simpler than my original steps for eliminating {369} and particularly {378} from 18(3) cage.]
2a. 45 rule on N78 4(1+3) outies in two different ways R5C4 + R6C234 = 29 must contain 7 in R5C4 + R6C34 for 33(5) cage = 5{789}/7{589/679}/8{579/678}/9{479/578}, no 3 -> R5C4 = {5789}, R6C2 = {57}, R9C4 = {13} (step 1c)
2b. R5C4 + R6C234 = 5{789}/7{589/679}/8{579}/9{578} (cannot be 8{678}/9{479} because R5C4 + R6C34 cannot contain two 8s or two 9s), no 4
2c. Max R9C4 = 3 -> min R9C23 = 13, no 1,2,3 in R9C23
2d. 45 rule on C12 1 innie R9C2 = 1 outie R5C3 + 3, no 7 in R5C3
2e. 45 rule on C6789 3 innies R789C6 = 21 = {489/579/678}, no 1,2,3
2f. 45 rule on C6789 3 outies R8C45 + R9C5 = 12 must contain 2 for N8 = {129/237/246}, no 5,8
2g. 9 of {129} must be in R89C5 (R89C5 cannot be {12} which clashes with R56C5), no 9 in R8C4
2h. 45 rule on N8 3 innies R7C45 + R9C4 = 12 = {138/345} (cannot be {156} because 33(5) cage at R5C4 cannot contain both of 5,6), no 6, 3 locked for N8, clean-up: no 7 in R8C45 + R9C5
2i. R789C6 = {579/678} (cannot be {489} which clashes with R7C45 + R9C4), no 4, 7 locked for C6
2j. R7C45 = {38/45} -> R5C4 + R6C34 = {679/789}, no 5
2k. R5C4 + R6C34 = {679/789}, CPE no 9 in R6C6
2l. R56C1 = {39/48} (cannot be {57} which clashes with R6C2), no 5,7
2m. 9(3) cage at R7C1 = {126/135} (cannot be {234} which clashes with R56C1), no 4, 1 locked for C1 and N7
2n. R78C3 = [29/38/47/83] (cannot be {56} which clashes with 9(3) cage), no 5,6
2o. Min R34C1 = 8 (cannot be {23/25} which clash with 9(3) cage, cannot be {24} which clashes with R56C1 + 9(3) cage, cannot be {34} which clashes with R56C1) -> max R3C2 = 4
2p. 12(3) cage at R3C1 = {129/147/237/246/345} (cannot be {138} which clashes with R56C1, cannot be {156} which clashes with 9(3) cage), no 8
2q. 6 of {246} must be in R3C12 (R3C12 cannot be {24} which clash with R3C78, or [62] but that’s not currently relevant), no 6 in R4C1
2r. 17(3) cage at R6C2 = {278/359/458/467} (cannot be {269/368} because R6C2 only contains 5,7)
2s. R6C2 = {57} -> no 5,7 in R78C2
2t. 9 of {359} must be in R8C2 -> no 3 in R8C2
2u. Consider combinations for 17(3) cage
17(3) cage = {278/359/458} => R78C3 = [29/47] (cannot be {38} which clashes with 17(3) cage
or 17(3) cage = {467}, 6 locked for N7 => 9(3) cage = {135}, 3 locked for N7
-> R78C3 = [29/47], no 3,8
2v. 16(3) cage at R9C2 = {169/178/358/367} (cannot be {349} which clashes with R78C3), no 4, clean-up: no 1 in R5C3 (step 2d)
2w. Consider combinations for 17(3) cage
17(3) cage = {278}, 8 locked for N7 => 16(3) cage = {169/367}
or 17(3) cage = {359/458}, R6C2 = 5
or 17(3) cage = {467}, 6 locked for N7 => 9(3) cage = {135}, 5 locked for N7
-> no 5 in R9C2, clean-up: no 2 in R5C3 (step 2d)
2x. Min R5C3 = 3 -> max R45C2 = 7, no 7 in R5C2
2y. 45 rule on N4 4 innies R4C13 + R6C23 = 23 must contain 7 for N4 = {1679/2579/2678} (cannot be {3479/3578} which clash with R56C1), no 3,4
[Now maybe one of Ed’s “aha” moments.]
3a. 17(3) cage at R6C2 (step 2r) = {278/359/458/467} with {359} = [539]
3b. Consider combinations for R7C45 (step 2j) = {38/45}
R7C45 = {38}, 3 locked for R7 => no 3 in R7C2
or R7C45 = {45}, 4 locked for R7 => R7C3 = 2, R8C3 = 9
-> 17(3) cage = {278/458/467}, no 3,9
3c. 3 in N7 only in 9(3) cage (step 2m) = {135}, 3,5 locked for C1, 5 locked for N7, clean-up: no 9 in R56C1
3d. Naked pair {48} in R56C1, locked for C1 and N4, clean-up: no 3 in R4C9 (step 1e), no 7 in R9C2 (step 2d)
3e. 7 in N7 only in R89C3, locked for C3
3f. 2 in C1 only in R34C1, locked for 12(3) cage at R3C1, no 2 in R3C2
3g. 27(4) cage at R1C1 = {3789/5679} (cannot be {4689} which clashes with 17(3) cage), no 4
3h. 45 rule on C1 3 outies R123C2 = 15
3i. 27(4) cage = {5679} (cannot be {3789} because R123C2 = {38}4 clashes with 17(3) cage), 5,6,7 locked for N1, 5 locked for C2, clean-up: no 2 in R4C9 (step 1e)
[Cracked. The rest is fairly straightforward.]
4a. R6C2 = 7 -> R9C4 = 1 (step 1f)
4b. R34C1 = [29] -> R3C2 = 1 (cage sum), clean-up: no 8 in R3C78, no 6 in R4C9 (step 1e)
4c. R6C3 = 6, naked pair {23} in R45C2, locked for N4, 2 locked for C2 -> R45C3 = [15], clean-up: no 1 in R6C5
4d. R6C2 = 7 -> R78C2 = 10 = {46}, locked for N7, 6 locked for C2
4e. R7C3 = 2, placed for D/, R8C3 = 9, R9C23 = [87], clean-up: no 4 in R6C5
4f. R8C45 + R9C5 (step 2f) = {246} (only remaining combination), 4,6 locked for N8 -> R789C6 (step 2e) = {579} (only remaining combination) = [579]
4g. Naked pair {38} in R7C45, locked for R7, 8 locked for 33(5) cage at R5C4 -> R56C4 = [79], 9 placed for D/
4h. Naked pair {46} in R3C78, locked for R3 and N3, clean-up: no 2 in R2C4 (step 1d), no 5 in R3C9
4i. Naked pair {38} in R3C36, locked for R3 -> R3C9 = 9, R2C9 = 2, 18(3) cage at R2C4 = [657]
[Routine clean-ups omitted from here.]
5a. R5C7 = 9 (hidden single in N6) -> R56C6 = 8 = [62], 2 placed for D\, R7C7 = 7, placed for D\, R12C1 = [67], 6 placed for D\
5b. R12C6 (step 1b) = 5 = {14}, locked for N2, 4 locked for C6, 1 locked for 13(4) cage at R1C6
5c. Naked pair {35} in R12C7, locked for C7 and N3
5d. R4C78 = [67] (hidden pair in R4)
5e. R3C7 = 4, placed for D/, R5C5 = 1, placed for both diagonals, R2C8 = 8, placed for D/, R4C6 = 3, placed for D/, R3C6 = 8, R3C3 = 3, placed for D\
5f. R9C1 =5 -> R8C8 + R9C9 = [54], both placed for D\
and the rest is naked singles, not using the diagonals.