Prelims
a) 9(3) cage at R1C4 = {126/135/234}, no 7,8,9
b) 21(3) cage at R1C9 = {489/579/678}, no 1,2,3
c) 20(3) cage at R2C4 = {389/479/569/578}, no 1,2
d) 23(3) cage at R7C1 = {689}
e) 10(3) cage at R7C8 = {127/136/145/235}, no 8,9
f) 10(3) cage at R8C2 = {127/136/145/235}, no 8,9
1a. Naked triple {689} in 23(3) cage at R7C1, locked for N7
1b. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 1 -> no 1,7,9 in R9C4
1c. 45 rule on R89 2 innies R8C19 = 11 = [65/83/92]
1d. 45 rule on R89 4 outies R7C1289 = 22 = {1489/1678/2389/2569/3469/3568} (cannot be {1579/2479/2578/3478/4567} because R7C12 must contain two of 6,8,9) -> R7C89 = {14/17/23/25/35} (cannot be {34} because 10(3) cage at R7C8 cannot be {34}3), no 6
2a. 45 rule on N36 1 outie R4C6 = 1 innie R3C7 + 6 -> R3C7 = {123}, R4C6 = {789}
2b. 45 rule on N36 3 outies R234C6 = 20 = {389/479/569/578}, no 1,2
2c. 45 rule on N3 3 innies R2C8 + R3C78 = 10 = {127/136/145/235}, no 8,9
2d. 45 rule on N6 1 outie R4C6 = 1 innie R4C8 + 4 -> R4C8 = {345}
3a. 45 rule on C1234 2 innies R18C4 = 7 = {16/25/34}, no 7,8,9
3b. 45 rule on C6789 2 innies R18C6 = 6 = {15/24}
3c. 45 rule on C5 3 innies R1C189C5 = 18
3d. Max R1C5 = 6 -> min R89C5 = 12, no 1,2 in R89C5
4a. 45 rule on N2 3(2+1) outies R3C37 + R4C5 = 11
4b. Min R3C37 = 4 -> max R4C5 = 7
4c. 45 rule on R123 3 outies R4C258 = 12
4d. Min R4C8 = 3 -> max R4C25 = 9, no 9 in R4C2
4e. 45 rule on N4 1 innie R4C2 = 1 outie R4C4 + 1, no 1 in R4C2, no 8,9 in R4C4
4f. R4C258 = 12 = {138/147/246/345} (cannot be {156} = [615] which clashes with R4C24 = [65], cannot be {237} = {27}3 which clashes with R4C68 = [73], step 2d)
4g. 1 of {147} must be in R4C5 -> no 7 in R4C5
4h. R4C258 = {147/246/345} (cannot be {138} = [813] because R4C24 = [87] clashes with R4C68 = [73], step 2d), no 8, 4 locked for R4, clean-up: no 5 in R4C2, no 7 in R4C4
4i. 45 rule on N1 1 innie R3C3 = 1 outie R4C2 + 1, R4C2 = {23467} -> R3C3 = {34578}
5a. R4C2 = R4C4 + 1 (step 4e), R4C6 = R4C8 + 4 (step 2d), R4C258 (step 4h) = {147/246/345}
5b. 45 rule on N5789 3 innies R4C456 = 15 = {159/168/249/258/267/348/357} (cannot be {456} because R4C6 only contains 7,8,9)
5c. R4C456 = {168/249/258/267} (cannot be {159} which clashes with R4C68 = [95], cannot be {348} = [348] which clashes with R4C68 = [84], cannot be {357} which clashes with R4C68 = [73]), no 3, clean-up: no 4 in R4C2, no 5 in R3C3 (step 4i)
5d. 4 in R4 only in R4C58 -> R4C456 must contain one of 4,8 = {168/249/258} (locking-out cages), no 7, clean-up: no 3 in R4C8, no 1 in R3C7 (step 2a)
5e. Max R4C4 = 6 -> min R45C3 = 11, no 1 in R45C3
5f. 20(3) cage at R2C4 = {389/479/578} (cannot be {569} because no 5,6,9 in R3C3), no 6
5g. R2C8 + R3C78 (step 2c) = {127/136/235} (cannot be {145} because R3C7 only contains 2,3), no 4
6a. R18C4 = 7 (step 3a), R18C6 = 6 (step 3b) -> R1C4 cannot be 1 greater than R1C6 which would make R8C4 and R8C6 equal
6b. 9(3) cage at R1C4 = {126/135/234}
6c. {126} = [162/612/621] (cannot be [261])
6d. R4C258 (step 4h) = {147/246/345}, R4C2 = R4C4 + 1 (step 4e)
6f. R4C258 = {147} can only be [714] giving R4C2458 = [7614]
6g. Consider combinations for 9(3) cage
9(3) cage = [162] => R8C4 = 6 blocking R4C2458 = [7614]
or 9(3) cage = [612/621] blocking R4C2458 = [7614]
or 9(3) cage = {135/234}, only other place for 1,2 in N2 in R23C5, 13(3) cage cannot contain both of 1,2 => no 1,2 in R4C5
-> R4C258 = {246/345}, no 1,7, clean-up: no 8 in R3C3 (step 4i), no 6 in R4C4 (step 4e)
6h. R4C456 (step 5d) = {168/249/258}
6i. R4C456 = {168} can only be [168]
6j. Consider placement for 6 in N2
6 in R1C4 => R8C4 = 1 blocking [168]
or 6 in R123C5 blocking [168]
or 6 in R23C6 => R23C6 can only be {56} => R3C7 = 3 => R4C6 = 9 (step 2a)
-> R4C456 = {249/258}, no 1,6, 2 locked for R4 and N5, clean-up: no 3 in R3C3 (step 4i)
6k. 20(3) cage at R2C4 (step 5f) = {479/578} (cannot be {389} because R3C3 only contains 4,7), no 3
6l. 45 rule on N14 3 outies R234C4 = 18 = {279/459}, no 8, 9 locked for C4 and N2
6m. R4C4 = {25} -> no 5 in R23C4
6n. 20(3) cage = {479}, CPE no 4,7 in R3C56
6o. 14(3) cage at R2C6 = {257/356} (cannot be {248} = [482] which clashes with R3C7 + R4C6 (step 2a) = [28], cannot be {347} because 4,7 only in R2C6), no 4,8, 5 locked for C6 and N2, clean-up: no 2 in R8C4 (step 3a), no 1 in R18C6 (step 3b)
6p. R3C7 = {23} -> no 3 in R23C6
6q. Naked pair {24} in R18C6, locked for C6
6r. Naked triple {245} in R4C458, 5 locked for R4
6s. R3C37 + R4C5 = 11 (step 4a)
6t. Consider placements for R3C3 = {47}
R3C3 = 4 => R23C4 = {79}, 7 locked for N2 => R23C6 = {56} = 11 => R3C7 = 3 (cage total) => R4C5 = 4
or R3C4 = 7 => R3C7 + R4C5 = 4 = [22]
-> R4C5 = {24}
6u. 8 in N2 only in 13(3) cage at R2C5 = {148/238} -> R23C5 = {18/38}, 8 locked for C5
6v. Max R4C4 = 5 -> min R45C3 = 12, no 2 in R5C3
7a. 9(3) cage = {126/234}, 2 locked for R1
7b. R18C4 (step 3a) = {16/34} (cannot be [25] which clashes with R4C4), no 2,5
7c. R189C5 (step 3c) = 18 = {279/369/459/567}, no 1
7d. 9(3) cage = {126/234} = [162]/{234}, no 6 in R1C4, clean-up: no 1 in R8C4
7e. Consider combinations for 9(3) cage
9(3) cage = [162] => R8C6 = 4, R23C5 = {38} (hidden pair in N2) => R4C5 = 2 (cage sum) => 4 in C5 only in R56C5
or 9(3) cage = {234} => R23C5 = {18} (hidden pair in N2) => R4C5 = 4 (cage sum)
-> 4 in R456C5, locked for C5 and N5
8a. 45 rule on N14 3 innies R345C3 = 19 = {379/469/478} (cannot be {568} because R3C3 only contains 4,7), no 5
8b. 45 rule on N36 3 innies R345C7 = 12 = {129/138/237} (cannot be {147/156} because R3C7 only contains 2,3, cannot be {345} because 4,5 only in R5C7, cannot be {246} = [264] which clashes with R3C7 + R4C8 = [24], combining steps 2a and 2d), no 4,5,6
[Only just spotted this, although I’d seen the 45s a lot earlier but couldn’t see any use for them at that time.]
9a. 45 rule on R6 3 innies R6C456 = 15
9b. 45 rule on R6789 3 outies R5C456 = 15
9c. 1,6,7 in N5 only in R56C456, neither of R5C456 and R6C456 can contain both of 1,7, nor both of 6,7 (because 2 in R4C45) -> one of R5C456 and R6C456 must be {168}, locked for N5
9d. R4C6 = 9 -> R4C45 = [24] (step 6j), R23C5 = 9 = {18}, 1 locked for C5 and N2, R4C8 = 5, R4C2 = 3 (step 4e), R3C3 = 4 (step 4i), R3C7 = 3 (step 2a) -> R23C6 = 11 = {56}, 6 locked for C6 and N2, clean-up: no 6 in R8C4 (step 3a)
9e. Naked triple {234} in 9(3) cage at R1C4, 3,4 locked for R1
9f. Naked pair {34} in R18C4, locked for C4
9g. Naked pair {79} in R23C4, 7 locked for C4
9h. R9C4 = R7C3 + 1 (step 1b) -> R7C3 = {57}, R9C4 = {68}
9i. 19(4) cage at R5C4 = {1567} (only remaining combination) -> R7C3 = 7, R9C4 = 8
9j. R4C4 = 2 -> R45C3 = 15 = [69]
9k. R9C4 = 8 -> R89C3 = 5 = {23}, locked for C3 and N7
9l. Naked triple {234} in R8C346, locked for R8 -> R8C9 = 5, R8C2 = 1, R8C1 = 6 (step 1c)
9m. Naked pair {89} in R7C12, locked for R7
9n. Naked pair {45} in R9C12, locked for R9
10a. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 + 1 -> R7C7 = {24}, R9C6 = {13}
10b. Naked pair {13} in R79C6, locked for C6 and N8 -> R18C4 = [34], R1C5 = 2, R18C6 = [42], R89C3 = [32]
10c. R8C46 = [42] = 6 -> R89C5 = 16 = {79}, 7 locked for C5
10d. R4C8 = 5 -> R23C8 = 7 = {16}, locked for C8 and N3
10e. 21(3) cage at R1C9 = {489} (only remaining combination) -> R2C9 = 4, R13C9 = {89}, locked for C9 and N3, R1C78 = [57], R2C7 = 2
10f. R4C6 = 9 -> R45C7 = 9 = {18}, locked for C7 and N6
10g. R4C9 = 7 -> R5C89 = 8 = [26], R6C9 = 3, R9C9 = 1 -> R89C8 = 17 = [89]
10h. 4 in R5 only in 13(3) cage at R4C1 = {148}, locked for N4, 1 locked for C1
10i. R4C2 = 3 -> R23C2 = 12 = {57}, locked for C2 and N1
and the rest is naked singles.