Prelims
a) R1C12 = {16/25/34}, no 7,8,9
b) R2C12 = {18/27/36/45}, no 9
c) R3C45 = {13}
d) 13(2) cage at R4C2 = {49/58/67}, no 1,2,3
e) R5C67 = {19/28/37/46}, no 5
f) R89C6 = {19/28/37/46}, no 5
g) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9
1a. Naked pair {13} in R3C45, locked for R3 and N2
1b. 45 rule on N569 1 innie R4C8 = 1, clean-up: no 9 in R5C6
1c. 1 in N3 only in 40(7) cage at R1C6 = {1456789}, no 2,3
1d. R2C7 = 3 (hidden single in N3), clean-up: no no 6 in R2C12, 7 in R5C6
1e. 2 in N3 only in R3C78, locked for R3 and 28(6) cage at R2C6
1f. 28(6) cage = {123589/123679}, no 4
1g. 45 rule on N3 3 remaining outies R123C6 = 22 = {589/679}, 9 locked for C6 and N2, clean-up: no 1 in R89C6
1h. 45 rule on N23 2 remaining outies R12C3 = 12 = [39]/{48/57}, no 1,2,6
1i. 31(6) cage at R1C3 = {234589/234679} (cannot be {235678} because R12C3 can only contain 3 when it contains 9) -> R12C3 = [39], clean-up: no 4 in R1C12
1j. 45 rule on N8 2 outies R89C3 = R89C3 = 13 = {58/67}
1k. 45 rule on N78 1 outie R6C1 = 8, clean-up: no 1 in R2C2, no 5 in 13(3) cage at R4C2
1l. 45 rule on C6789 2 outies R6C45 = 1 innie R7C6 + 12
1m. Max R6C45 = 16 -> max R7C6 = 4
1n. Min R6C45 = 13, no 1,2,3 in R6C45
1o. Max R7C6 = 4 -> min R78C5 = 11, no 1 in R78C5
1p. 45 rule on C6789 3(1+2/2+1) innies R6C67 + R7C6 = 10, no 9 in R6C7
1q. 45 rule on N4 1 outie R3C3 = 1 innie R4C1 + 1, no 2,9 in R4C1
1r. 45 rule on N1 3 innies R3C123 = 17 = {458/467}, 4 locked for R3 and N1, clean-up: no 5 in R2C12
1s. 45 rule on N9 1 innie R7C8 = 1 outie R6C9 + 1, no 9 in R6C9, no 2,9 in R7C8
1t. 45 rule on C6 using R123C6 = 22, 4 innies R4567C6 = 13 = {1237/1246/1345}, no 8, clean-up: no 2 in R5C7
1u. 25(6) cage at R3C3 = {123469/123568/124567} (cannot be {123478} which clashes with 13(2) cage)
1v. 8 of {123568} only in R3C3, 6 of {123469} must be in R3C3 (because then R4C1 = 5, hidden single in N4), 4 of {124567} must be in R3C3 (because then R4C1 = 3, hidden single in N4) -> R3C3 = {468}, clean-up: no 4,6 in R4C1
1w. 9 in N7 only in 29(5) cage at R6C1 = {14789/23789/24689/34589} (cannot be {15689} which clashes with R89C3)
1x. 11(3) cage at R7C3 = {128/137/146/236} (cannot be {245} which clashes with 29(5) cage), no 5
2a. 11(3) cage at R7C3 (step 1x) = {128/137/146/236}, 25(6) cage at R3C3 (step 1u) = {123469/123568/124567}
2b. Consider combinations for R89C3 = {58/67}
R89C3 = {58}, locked for C3
or R89C3 = {67} => 11(3) cage = {128} => 4 in C3 only in R3456C3
-> 25(6) cage = {123469/124567}, no 8, clean-up: no 7 in R4C1 (step 1m)
2c. Consider combinations for 29(5) cage at R6C1 (step 1w) = {14789/23789/24689/34589}
29(5) cage = {14789/23789/24689}, 6 or 7 locked for N7 => R89C3 = {58}
or 29(5) cage = {34589} must have 4,9 in C1 ({345/359} clash with R4C1), locked for C1 => 13(2) cage at R4C2 = {67} => 25(6) cage = {123469} with 6 in R3C3 (step 1v)
-> R89C3 = {58}, locked for N7 and 33(6) cage at R7C4), 5 locked for C3
2d. 5 in N8 only in 15(3) cage at R7C5 = {159/258/357/456}, 5 locked for C5
2e. R7C6 = {1234} -> no 2,3,4 in R89C5
2f. 8 in N8 only in 15(3) cage = {258} or R89C6 = {28} (locking cages)
-> 2 in R789C6, locked for C6 and N8, clean-up: no 8 in R5C7
2g. Consider combinations for 11(3) cage = {137/146/236}
11(3) cage = {137/236}, 3 locked for N7 => R4C1 = 3 (hidden single in C1)
or 11(3) cage = {146} => 7 in C3 only in R456C3 => 25(6) cage = {124567}
-> 25(6) cage = {124567}, R3C4 = 4 (step 1v), placed for D\, R4C1 = 3 (hidden single in C1)
2h. 13(2) cage at R4C2 = {49} (hidden pair in N4)
2i. 5 in C1 only in R13C1, locked for N1, clean-up: no 2 in R1C1
2j. 2,8 in N5 only in 20(4) cage at R4C4 = {1289/2378/2468}, no 5
2k. 20(4) cage = {1289/2378/2468} -> R45C6 + R6C456 must contain 5 and two of {19}, {37} and {46}
2l. R6C45 = R7C6 + 12 (step 1l)
2m. R6C45 + R7C6 = {67}1/[592]/{69}3/{79}4 (cannot be {49}1 which clashes with R45C6 + R6C456 = {14569}, no 4 in R6C45
[Cracked, fairly straightforward from here.]
2n. 4 in R6 only in R6C789, locked for N6, clean-up: no 6 in R5C6
2o. Disjoint 17(3) cage at R4C6 = {269/278/458/467}
2p. 4 of {458} must be in R4C6 -> no 5 in R4C6
2q. 5 in N5 only in R6C46, locked for R6
2r. R5C2 = 5 (hidden single in N4)
2s. 5 in N6 only in disjoint 17(3) cage at R4C6 = {458} -> R4C6 = 4, placed for D/, R4C79 = {58}, 8 locked for R4 and N6, R4C2 = 9 -> R5C1 = 4, clean-up: no 6 in R5C7, no 6 in R78C5, no 6 in R89C6
2t. R4C6 = 4 -> R45C6 + R6C456 must also contain 6 in R6C456, locked for R6 and N5
2u. Naked pair {27} in R4C45, locked for R4 and N5 -> R4C3 = 6
2v. 20(4) cage = {2378} (only remaining combination), 3 locked for R5 and N5, R5C6 = 1 -> R5C7 = 9, clean-up: no 9 in R78C5
2w. R6C456 = {569} -> R6C7 = 2 (cage sum)
2x. Naked pair {67} in R5C89, 7 locked for R5 and N6
2y. R5C89 = {67} = 13 -> R67C8 = 8 = [35], R6C9 = 4, R8C5 = 5 (hidden single in N8), R89C3 = [85], clean-up: no 2 in R9C6
3a. Naked quad {2378} in R7C56 + R89C6, 3,7 locked for N8
3b. 17(3) cage at R7C7 = {467} (only remaining combination), locked for C7, 6,7 locked for N9
3c. R1C7 = 1 (hidden single in C7), R1C8 = 4 (hidden single in N3), clean-up: no 6 in R1C12
3d. R1C12 = [52], 5 placed for D\ -> R6C6 = 6, placed for D\, R7C7 = 7, placed for D\, R6C45 = [59], 5 placed for D/, R4C4 = 2, placed for D\, R2C2 = 8, placed for D\ -> R5C5 = 3, placed for both diagonals, clean-up: no 7 in R123C6 (step 1g)
3e. Naked pair {67}, locked for R3, 7 locked for N1
3f. R5C3 = 2, R7C3 = 1, naked pair {67} in R2C8 + R8C2, 6,7 locked for D/, R3C7 = 8, R1C9 = 9, placed for D/
3g. R2C45 = [42] (hidden pair in R2)
3h. Naked triple {169} in R789C4, 1,6 locked for C4 and N8 -> R9C5 = 4
3i. 11(3) cage at R7C3 = {137} (only remaining combination), R8C2 = 7, placed for D/
3j. R78C5 = [85] -> R7C6 = 2 (cage sum)
and the rest is naked singles, without using the diagonals.
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