Prelims
a) R2C67 = {16/25/34}, no 7,8,9
b) R2C89 = {18/27/36/45}, no 9
c) R3C89 = {14/23}
d) R7C12 = {69/78}
e) R7C34 = {18/27/36/45}, no 9
f) R8C12 = {14/23}
g) R8C34 = {18/27/36/45}, no 9
h) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2
i) 22(3) cage at R3C5 = {589/679}
1a. 22(3) cage at R3C5 = {589/679}, 9 locked for R3
1b. 45 rule on R12 1 innie R2C1 = 7 -> R34C1 = 13 = [49/58/85], clean-up: no 2 in R2C89, no 8 in R7C2
1c. 45 rule on R9 2 outies R8C58 = 17 = {89}, locked for R8, clean-up: no 1 in R8C34
1d. Naked pair {89} in R8C58, CPE no 8,9 in R5C5 using D\
[See step 4a for the other CPE from this naked pair.]
1e. 17(3) cage at R8C8 = {179/269/278/359/368/458} (cannot be {467} because R8C8 only contains 8,9)
1f. R8C8 = {89} -> no 8,9 in R9C89
1g. 45 rule on R123 2 outies R4C12 = 12 = [57/84/93]
1h. 45 rule on N78 2 outies R89C7 = 10 = [19/28]/{37/46}, no 5, no 1,2 in R9C7
1i. 45 rule on R789 3 outies R5C8 + R6C89 = 20 = {389/479/569/578}, no 1,2
1j. 3 of {389} must be in R56C8 (R56C9 cannot be {89} which clashes with R8C8) -> no 3 in R6C9
1k. 45 rule on N6789 2 outies R46C6 = 13 = {49/58/67}, no 1,2,3
1l. 45 rule on N1234 2 outies R46C4 = 7 = {16/25/34}, no 7,8,9
1m. 45 rule on N3 1 innie R3C7 = 2 outies R12C6 + 4
1n. Min R12C6 = 3 -> min R3C7 = 7
1o. Max R3C9 = 9 -> max R12C6 = 5 = {12/13/14/23}, clean-up: no 1,2 in R2C7
1p. 45 rule on R89 3 innies R8C679 = 14 = {167/257/356} (cannot be {347} which clashes with R8C12), no 4, clean-up: no 6 in R9C7
2. R2C89 = {18/36/45}, R3C89 = {14/23} -> combined cage R23C89 = {18}{23}/{36}{14}/{45}/{23}, 3 locked for N3, clean-up: no 4 in R2C6
3a. 45 rule on R123 4 remaining innies R3C1234 = 18 = {1458/1467/2358} (cannot be {1278/1368/2457/3456} which clash with R3C89, cannot be {2367} because R3C1 only contains 4,5,8)
3b. 7 of {1467} must be in R3C4 -> no 6 in R3C4
[I ought to have seen this sooner.]
4a. Naked pair {89} in R8C58, CPE no 8,9 in R9C7, clean-up: no 1,2 in R8C7 (step 1h)
4b. Hidden killer pair 8,9 in R7C789 and R8C8 for N9, R8C8 = {89} -> R7C789 must contain one of 8,9
4c. Killer pair 8,9 in R7C12 and R7C789, locked for R7, clean-up: no 1 in R7C34
4d. 45 rule on N7 3 outies R789C4 = 15 = {249/258/267/348/357} (cannot be {159/168} because 1,8,9 only in R9C4, cannot be {456} which clashes with R46C4), no 1
[Now it got harder, but also the first key step.]
5a. The only combination for 23(4) cage at R8C5 containing both of 8,9 is {289}4
5b. R789C4 (step 4d) = {249/258/267/348/357} must contain 7 or R9C4 = {89}
5c. 17(4) cage at R7C5 = {1367/1457/2357/2456}
5d. Consider combinations for R89C7 (step 1h) = {37}/[64]
R89C7 = {37}, no 4 in R9C7 => 23(4) cage not {289}4
or R89C7 = [64] and 17(4) cage = {137}6, 7 locked for N8 => R9C4 = {89}
or R89C7 = [64] and 17(4) cage = {245}6, 2 locked for N8 => 23(4) cage not {289}4
-> no 8,9 in R9C56
[As Ed pointed out to me, these steps would have been a bit simpler as
5c. 17(4) cage at R7C5 = {1367/1457/2357} (cannot be {2456} = {245}6 which clashes with R789C4)
5d. Consider combinations for R89C7 (step 1h) = {37}/[64]
R89C7 = {37}, no 4 in R9C7 => 23(4) cage not {289}4
or R89C7 = [64] => 17(4) cage = {137}6, 7 locked for N8 => R9C4 = {89}
-> no 8,9 in R9C56]
5e. R8C5 + R9C4 = {89} (hidden pair in N8)
5f. R789C4 = {249/258/348}, no 6,7, clean-up: no 2,3 in R78C3
5g. 8,9 in R9 only in 22(4) cage at R9C1 = {1489/2389}, no 5,6,7
5h. Killer quad 1,2,3,4 in R8C12 and 22(4) cage, locked for N7, clean-up: no 5 in R78C4
5i. 5 in N7 only in R78C3, locked for C3
5j. 5 in one of R7C34 = [54] or R8C34 = [54] (locking cages) -> 4 in R78C4, locked for C4 and N8, clean-up: no 3 in R46C4 (step 1l)
5k. 17(4) cage = {1367/2357}, CPE no 3 in R8C4, clean-up: no 6 in R8C3
5l. 6 in N7 only in R7C123, locked for R7
5m. 17(4) cage = {1367} (cannot be {2357} which clashes with combined cage R8C1234 = {14}[72]/{23}[54]), 6 locked for R8, 1 locked for N8
5n. Killer pair 2,4 in R8C12 and R8C4, locked for R8
6a. Consider combinations for R8C679 (step 1p) = {167/356}
R8C679 = {167}, 7 locked for R8 => 7 in N7 only in R7C23, locked for R7 => R7C56 = {13}, R8C67 = {67} => R8C9 = 1
or R8C679 = {356} = {36}5 => R7C56 = {17}
-> R8C9 = {15}, R7C56 = {13/17}, 1 locked for R7 and N8
6b. 15(3) cage at R6C9 = {159/168/258/357/456} (cannot be {249/267/348) because R8C9 only contains 1,5)
6c. 3 of {357} must be in R7C9 -> no 7 in R7C9
6d. 6 of {456} must be in R6C9 -> no 4 in R6C9
7a. 5 in N8 only in R9C56, locked for R9
7b. 17(3) cage at R8C8 (step 1e) = {179/269/278} (cannot be {368} which clashes with R89C7), no 3,4
7c. Killer pair 1,2 in 22(4) cage at R9C1 and 17(3) cage, locked for R9
7d. R89C7 (step 5d) = [37/64/73], 17(4) cage = {1367}
7e. 23(4) cage at R8C5 = {3578/4568} (cannot be {3479} = 9{37}4 which clashes with 22(4) cage at R9C1, cannot be {3569} = 9{56}3 which clashes with 17(4) cage = {13}[67] -> R8C5 = 8, R9C4 = 9, R8C8 = 9, placed for D\, clean-up: no 4 in R4C6 (step 1k)
7f. 9 in N7 only in R7C12 = {69}
7g. R78C3 = {57} (hidden pair in N7), 7 locked for C3, clean-up: no 3 in R7C4
7h. Naked pair {24} in R78C4, 2 locked for C4, clean-up: no 5 in R46C4 (step 1l)
7i. Naked pair {16} in R46C4, locked for C4 and N5, clean-up: no 7 in R46C6 (step 1k)
7j. 27(5) cage at R1C1 = {13689/14589/23589/34569}, 9 locked for N1
7k. R5C8 + R6C89 (step 1i) = {389/479/569/578}
7l. 9 of {569} only in R6C9 -> no 6 in R6C9
7m. 15(3) cage at R6C9 (step 6b) = {159/258/357}, no 4, 5 locked for C9, clean-up: no 4 in R2C8
7n. 2 of {258} must be in R7C9 -> no 8 in R7C9
7o. 7,8,9 only in R6C9 -> R6C9 = {789}
7p. 8 in N9 only in R7C78, locked for 23(4) cage at R5C8
7q. 23(4) cage = {3578/4568} (cannot be {2678} = {67}{28} because R5C8 + R6C89 cannot contain both of 6,7), no 2
7r. R5C8 + R6C89 = {479/569/578} (cannot be {389} because 8,9 only in R6C9), no 3
7s. 8,9 only in R6C9 -> R6C9 = {89}
7t. 15(3) cage = {159/258}, no 3, 5 locked for N9
7u. 23(4) cage = {3578/4568}, 5 locked for C8 and N6, clean-up: no 4 in R2C9
7v. 23(4) cage = {56}{48}/{57}{38} -> R56C8 = {567}, R7C78 = {348}
7w. R2C67 = [25/34] (cannot be [16] which clashes with R2C89)
7x. R12C6 (step 1o) = {12/13/23}, no 4
7y. 4 in N2 only in R12C5, locked for C5 and 20(4) cage at R1C3
8a. 23(4) cage at R5C8 (step 7v) = {56}{48}/{57}{38}, R89C7 (step 5d) = {37}/[64]
8b. Combined cage 23(4) + R89C7 = {56}{48}{37}/{57}{38}[64], CPE no 6,7 in R456C7 + R9C8
8c. Killer pair 1,2 in 22(4) cage at R9C1 and R9C8, locked for R9
8d. 7 on D\ only in R5C5 + R9C9, CPE no 7 in R15C9 + R9C5 using D/
8e. 15(3) cage at R5C7 = {159/249/258/348}
8f. 5 of {258} must be in R6C6, 4 of {348} must be in R6C6 (R56C7 cannot be {34} which clashes with R89C7) -> no 4 in R56C7, no 8 in R6C6, clean-up: no 5 in R4C6 (step 1k)
8g. Killer pair 8,9 in 15(3) cage and R6C9, locked for N5
9a. 22(3) cage at R3C5 = {589/679}
9b. Consider combinations for R34C1 (step 1b) = [49]/{58}
R34C1 = [49] => R4C6 = 8, placed for D/
or R34C1 = {58} => 22(3) cage = {679} (cannot be {589} which clashes with R3C1
-> no 8 in R3C7
9c. R3C7 = R12C6 + 4 (step 1m)
9d. R3C7 = {79} -> R12C6 = 3,5 = {12/23}, 2 locked for C6 and N2
9e. Consider placements for R3C7
R3C7 = 7 => 22(3) cage at {679}
or R3C7 = 9 => R12C6 = {23}, 3 locked for N2 => R2C4 = {58} => 22(3) cage at {679} (cannot be [589] which clashes with R2C4)
-> 22(3) cage = {679}, 6,7 locked for R3, 6 locked for N2
9f. 8 in N2 only in R123C4, locked for C4
10a. 9 in R2 only in R2C35
10b. 45 rule on R1 4 outies R2C2345 = 22 = {1489/2569/3469} (cannot be {2389} which clashes with R2C6)
10c. 3 of {3469} must be in R2C4 -> no 3 in R2C245
11a. 45 rule on N23 2 innies R23C4 = 1 outie R1C3 + 7
11b. R23C4 = {35/58} (cannot be {38} = 11 because no 4 in R1C3), 5 locked for C4 and N2
11c. R23C4 = 8,13 -> R1C3 = {16}
11d. 20(4) cage at R1C3 = {1478/3467} (cannot be {1469} because R1C4 only contains {378}, cannot be {1379} = 1{379} which clashes with R1C3 + R23C4 = 1{35}) = [1874/6374/6734] -> R2C5 = 4, R1C45 = [37/73/87], 7 locked for R1 and N2
11e. R2C7 = 5 -> R2C6 = 2, R1C6 = 1 (hidden single in N2), R1C3 = 6, R1C45 = {37}, 3 locked for R1 and N2
11f. R2C234 = [198] = 18 -> R1C12 = 9 = {45}, locked for N1
11g. R3C1 = 8 -> R4C1 = 5 (cage sum), R4C2 = 7 (step 1g)
11h. R1C1 = 4, R2C2 = 1, both placed for D\, R4C4 = 6 placed for D\, R6C4 = 1 placed for D/, R9C9 = 7, placed for D\, R9C8 = 1 (cage sum)
11i. R78C9 = [25] -> R6C9 = 8 (cage sum), R7C34 = [54], R8C34 = [72] -> R8C12 = [14]
11j. R89C7 = [64] (hidden pair in N9)
11k. R6C6 = 5 -> R56C7 = 10 = [19]
11l. Naked triple {234} in R4C789, locked for R4, 3,4 locked for N6, R5C9 = 6, R56C8 = [57]
11m. Naked pair {23} in R56C5, 3 locked for C5 and N5
11n. Naked pair {23} in R3C3 + R5C5, locked for D\, CPE no 2,3 in R5C3
11o. R6C3 = 4 (hidden single in R6), R45C3 = [18], R4C4 = 6 -> R5C2 = 2 (cage total)
and the rest is naked singles, without using the diagonals.