Prelims
a) R12C1 = {89}
b) R12C9 = {18/27/36/45}, no 9
c) R2C78 = {18/27/36/45}, no 9
d) R3C46 = {18/27/36/45}, no 9
e) R78C1 = {17/26/35}, no 4,8,9
f) R9C12 = {39/48/57}, no 1,2,6
g) R9C45 = {13}
h) 10(3) cage at R5C5 = {127/136/145/235}, no 8,9
Steps Resulting From Prelims
1a. Naked pair {89} in R12C1, locked for C1 and N1, clean-up: no 3,4 in R9C2
1b. Naked pair {13} in R9C45, locked for R9 and N8, clean-up: no 9 in R9C2
2. 45 rule on N1 2 outies R2C4 + R4C3 = 17 = {89}
2a. Naked pair {89} in R2C14, locked for R2, clean-up: no 1 in R1C9, no 1 in R2C78
2b. 45 rule on N4 1 outie R7C2 = 1 innie R4C3 -> R7C2 = {89}
2c. 45 rule on N7 2 innies R7C23 = 10 = [82/91]
2d. 45 rule on C123 3 outies R267C4 = 24 = {789}, locked for C4, clean-up: no 1,2 in R3C6
3. 45 rule on N2 3 innies R1C6 + R2C4 + R3C5 = 21 = {489/579/678}, no 1,2,3
4. 45 rule on C89 3 innies R123C8 = 19 = {289/379/469/478/568}, no 1
4a. 2 of {289} must be in R2C8 -> no 2 in R13C8
4b. Min R1C6 + R1C8 = 7 -> max R1C7 = 7
5. R78C1 = {17/26/35}, R7C23 = [82/91] -> combined cage R7C23 + R78C1 = [82]{35}/[91]{26}/[91]{35} (cannot be [82]{17} which clashes with R9C12) -> R78C1 = {26/35}, no 1,7
5a. Hidden killer pair 4,7 in 15(3) cage at R8C2 or R9C12 for N7, R9C12 contains one of 4,7 -> 15(3) cage must contain one of 4,7 = {249/267/348/357} (cannot be {159/168/258} which don’t contain 4 or 7, cannot be {456} which clashes with R78C1), no 1
5b. R7C3 = 1 (hidden single in N7), placed for D/, R7C2 = 9, R4C3 = 9 (hidden single in N4) -> R2C14 = [98], R1C1 = 8, placed for D\, R7C4 = 7, R6C4 = 9, placed for D/, clean-up: no 1 in R2C9, no 1 in R3C4
5c. 9 on D\ only in R8C8 + R9C9, locked for N9 and 30(6) cage at R5C8
5d. R5C7 = 9 (hidden single in C7)
5e. R2C4 = 8 -> R1C6 + R3C5 (step 3) = {49/67}, no 5
5f. R1C6 + R3C5 = {49/67}, R3C46 = [27]/{36/45} -> combined cage R1C6 + R3C345 = {49}[27]/{49}{36}/{67}{45}, 4 locked for N2
5g. 45 rule on N5 1 outie R3C5 = 1 innie R6C6 + 2, R3C5 = {4679} -> R6C6 = {2457}
5h. 8 in N5 only in R4C56, locked for R4
5i. 45 rule on C9 1 innie R9C9 = 1 outie R4C8 + 4 -> R9C9 = {5679}, R4C8 = {1235}
6. 45 rule on R12 1 innie R2C2 = 1 remaining outie R3C3 + 3 -> R2C2 = {567}, R3C3 = {234}
6a. 1 on D\ only in R4C4 + R8C8, CPE no 1 in R4C8, clean-up: no 5 in R9C9
6b. 13(3) cage at R2C2 = {157/256} (cannot be {247/346} = 7{24}/6{34} which clash with R3C3 + R3C46, ALS block), no 3,4, 5 locked for N1
[Alternatively variable hidden killer triple 2,3,4 in R3C12, R3C3 and R3C46 for R3, R3C4 = {234}, R3C46 contains one of 2,3,4 -> R3C12 cannot contain more than one of 2,3,4 …]
6c. 13(3) cage at R2C2 = {157} (only remaining combination, cannot be {256} because 5{26} clashes with R2C2 + R3C3 = [52] while R2C2 + R3C123 = 6{25}3 clashes with R3C46), locked for N1, 1 also locked for R3, clean-up: no 3 in R3C3
6d. R1C7 = 1 (hidden single in N3) -> R1C68 = 13 = {49/67}, no 3,5 in R1C7
7. R2C2 = R3C3 + 3 (step 6), R3C5 = R6C6 + 2 (step 5g)
7a. Consider placements for R2C2 = {57}
R2C2 = 5
or R2C2 = 7, R3C3 = 4, no 4 in R3C5 => no 2 in R6C6 => R6C6 = {457} => naked triple [745] in R2C2 + R3C3 + R6C6
-> 5 in R2C2 + R6C6, locked for D\
5 on D\ in R2C2 + R6C6, CPE no 5 in R2C6 + R6C2
8. R1C6 + R3C5 (step 5f) = {49/67}, R3C46 = [27]/{36/45}
8a. Consider permutations for 13(3) cage at R2C2 = {157}
R2C2 = 5 => R3C12 = {17}, locked for R3
or R2C2 = 7, R3C12 = {15} => R3C46 = [27]/{36} => R1C6 + R3C5 = {49) (cannot be {67} which clashes with R3C46)
-> no 7 in R3C5, clean-up: no 6 in R1C6, no 7 in R1C8 (step 6d), no 5 in R6C6 (step 5g)
[Spotted later, an alternative which I prefer because it is more balanced and starts at a distance from both remaining 5s on D\
Consider permutations for R1C6 + R3C5
R1C6 + R3C5 = {49} => R6C6 = {27} (step 5g)
or R1C6 + R3C5 = {67}, locked for N2 => R3C46 = {45}, locked for R3 => R3C12 = {17}, locked for N1 => R2C2 = 5, placed for D\
-> no 5 in R6C6
Then 7 is eliminated from R3C5 by step 9, with the associated clean-ups.]
9. R2C2 = 5 (hidden single on D\), R3C3 = 2 (step 6), placed for D\, naked pair {17} in R3C12, locked for R3, clean-up: no 4 in R1C9, no 4 in R2C78, no 4 in R3C5 (step 5g), no 9 in R1C6 (step 5f), no 4 in R1C8 (step 6d), no 7 in R9C1
9a. R1C23 + R2C3 must be {34}6/{36}4 (cannot be {46}3 which clashes with R1C68), 3 locked for R1 and N1, clean-up: no 6 in R2C9
9b. Killer pair 4,6 in R1C23 and R1C68, locked for R1, clean-up: no 3 in R2C9
10. 8 in N5 only in 28(5) cage at R3C5 = {14689/23689/24589} (cannot be {13789} which clashes with 10(3) cage at R5C5, cannot be {34678} which clashes with R6C6, cannot be {25678} because R3C5 + R4C4 don’t contain 2,5,7,8) -> R3C5 = 9, R1C6 = 4 (step 5g), R1C8 = 9 (cage sum), R6C6 = 7, placed for D\, clean-up: no 5 in R3C46
[Cracked.]
10a. Naked pair {36} in R3C46, locked for R3 and N2
10b. 5 in N2 only in R1C45, locked for R1, clean-up: no 4 in R2C9
10c. Naked pair {27} in R12C9, locked for C9 and N3
10d. R9C9 = 9 (hidden single in C9) -> R4C8 = 5 (step 5i)
10e. R3C7 = 5 (hidden single in N3), placed for D/, R9C1 = 4, placed for D/, R9C2 = 8
10f. Naked pair {36} in R2C8 + R5C5, locked for D/, CPE no 3,6 in R5C8 + R8C8 (using D\)
10g. R4C6 = 8 (hidden single on D/)
11. R123C8 = 19 (step 4), R1C8 = 9 -> R23C8 = 10 = [64], 6 placed for D/ -> R5C5 = 3, placed for D\, R8C8 = 1, placed for D\, R2C7 = 3, R9C45 = [31], R3C46 = [63], R4C4 = 4, placed for D\
11a. R5C4 = 1 (hidden single in C4), R3C5 + R4C46 = [948] -> R4C5 = 6 (cage sum)
11b. R6C6 = 7, R7C7 = 6 -> R6C7 + R7C6 = 9 = [45]
11c. R8C4 = 2, R8C2 = 7, placed for D/, R89C3 = 8 = [35]
11d. R3C9 = 8, R4C8 = 5, R5C9 = 6 -> R4C9 = 1 (cage sum)
and the rest is naked singles, without using diagonals.