Prelims
a) R1C45 = {39/48/57}, no 1,2,6
b) R1C67 = {49/58/67}, no 1,2,3
c) R2C56 = {14/23}
d) R45C1 = {18/27/36/45}, no 9
e) R45C8 = {19/28/37/46}, no 5
f) R67C1 = {89}
g) R8C45 = {18/27/36/45}, no 9
h) R9C56 = {49/58/67}, no 1,2,3
1. 45 rule on R12 1 innie R2C7 = 6, R3C78 = 8 = {17/35}
1a. 45 rule on N3 2 innies R1C7 + R3C9 = 6 = [42] (cannot be [51] which clashes with R3C78) -> R1C6 = 9, clean-up: no 3,8 in R1C45, no 4 in R9C5
2. Naked pair {57} in R1C45, locked for R1 and N2
2a R1C89 = {18/38} (cannot be {13} which clashes with R3C78), 8 locked for R1 and N3
2b. Killer pair 1,3 in R1C89 and R3C78, locked for N3
2c. 9 in N3 only in R2C89, locked for R2
2d. 2,6 in R1 only in R1C123, locked for N1
2e. 9 in N1 only in R3C123, locked for 35(6) cage at R3C1, no 9 in R4C23
3. R3C9 = 2 -> R456C9 = 21 = {489/678} (cannot be {579} which clashes with R2C9), no 1,3,5, 8 locked for C9 and N6, clean-up: no 2 in R45C8
3a. R1C8 = 8 (hidden single in R1)
3b. R45C8 = {19/37} (cannot be {46} which clashes with R456C9), no 4,6
3c. Killer pair 7,9 in R45C8 and R456C9, locked for N6
3d. Hidden killer pair 4,6 in R6C8 and R456C9 for N6, R456C9 contains one of 4,6 -> R6C8 = {46}
3e. 2,5 in N6 only in R456C7, locked for C7, clean-up: no 3 in R3C8 (step 1)
3f. 9 in C7 only in R789C7, locked for N9
3g. 3 in N3 only in R1C9 + R3C7, locked for D/
4. 45 rule on R89 4 outies R56C2 + R7C23 = 11 = {1235}, locked for 24(6) cage at R5C2, 3 also locked for C2
4a. R56C2 + R7C23 = 11 -> R8C23 = 13 = {49/67}
4b. Naked quad {1235} in R56C2 + R7C23, CPE no 1,2,5 in R9C2
5. Naked pair {89} in R67C1, locked for C1, clean-up: no 1 in R45C1
6. 45 rule on N5 2 innies R4C6 + R6C4 = 15 = [69/78/87]
7. 26(5) cage at R3C5 = {23678/24578/34568} (cannot be {14678} because R45C7 only contain 1,2,3,5), no 1
7a. R45C7 = {235} -> no 3 in R3C56
8. 9 in N4 only in R56C3 + R6C1, CPE no 9 in R6C4, clean-up: no 6 in R3C5 (step 5)
8a. Naked pair {78} in R4C6 + R6C4, locked for N5 and D/, clean-up: no 1 in R3C8 (step 1), no 6 in R8C3 (step 4a)
8b. Naked pair {13} in R1C9 + R3C7, locked for D/
8c. Naked quad {1235} in R56C2 + R7C23, 1 locked for C2
9. 15(4) cage at R1C1 = {1248/1356} (cannot be {1347} because R1C12 must contain one of 2,6, cannot be {1257} which clashes with R2C89, ALS block, cannot be {2346} which clashes with R2C56), no 7, 1 locked for C1 and N1
9a. R1C2 = {26} -> no 2,6 in R1C1
9b. 8 of {1248} must be in R2C2, 5 of {1356} must be in R2C2 -> R2C2 = {58}, no 5 in R2C1
9c. R1C23 = {26} (hidden pair in R1)
9d. Killer pair 2,5 in R12C2 and R567C2, locked for C2
9e. Naked pair {13} in R1C19, CPE no 1,3 in R9C9 using D\
10. Killer triple 5,7,9 in R23C8 and R45C8, locked for C8, 5 also locked for N3
11. 14(3) cage at R1C3 = {167/248/356} (cannot be {158/347} because R1C3 only contains 2,6, cannot be {257} because 5,7 only in R2C3)
11a. R1C3 = {26} -> no 2 in R2C4
11b. 2 in R2 only in R2C56 = {23}, locked for R2 and N2
11c. 14(3) cage = {167/248}, no 5
11d. 1 in N2 only in R23C4, locked for C4, clean-up: no 8 in R8C5
12. 45 rule on N1 4(2+2) outies R23C4 + R4C23 = 19
12a. Max R23C4 = 9 (because 1 must be in R23C4) -> min R4C23 = 10, no 1 in R4C3
13. 45 rule on N1 5 innies R12C3 + R3C123 = 30 = {24789/35679} (cannot be {25689/34689/45678} which clash with 15(4) cage at R1C1)
13a. R12C3 = [24/28/67] -> R3C123 = {359/479/789}
14. 26(5) cage at R3C5 (step 7) = {23678/24578/34568}, R3C7 = {13} -> 26(5) cage + R3C7 = {23678}1/{24578}{13}/{34568}1 [1 and/or 4 in R3]
14a. R3C123 = {359/479/789} = 17,20,24 -> R3C4 + R4C23 = 11,15,18 = {146/168/468} (cannot be {236/245/258} because 2,3,5 only in R4C3, other combinations clash with R3C123), CPE no 6 in R4C4
14b. 35(6) cage at R3C1 = {345689} (only remaining combination, cannot be {146789} = {479}1{68} which clashes with 26(5) cage + R3C7, cannot be {146789} = {789}1{46} which clashes with R45C1 because 7 of {789} must be in R3C1) -> R3C123 = {359}, R3C4 + R4C23 = {468}
[Cracked. The rest is fairly straightforward. Routine clean-ups omitted from here.]
15. R3C2 = 9, R3C13 = {35}, locked for R3 and N1 -> R1C1 = 1, placed for D\, R2C1 = 4, R2C2 = 8, placed for D\, R1C2 = 2 (cage sum), R12C3 = [67], R3C78 = [17], R2C9 = 9, R2C8 = 5, placed for D/, R7C3 = 2, R9C1 = 6, R8C2 = 4, R8C3 = 9, R9C2 = 7, R67C1 = [98]
15a. R45C1 = {27} (hidden pair in C1)
16. R7C7 = 7 (hidden single on D\)
16a. Naked triple {235} in R456C7, locked for C7 and N6 -> R89C7 = [89]
16b. R9C56 = {58} (only remaining combination), locked for R9 and N8 -> R9C9 = 4, placed for D\
16c. R78C9 = {15} (hidden pair in C9), locked for N9
17. R9C3 = 1 (hidden single in R9)
17a. R9C123 = [671] = 14 -> R8C1 + R9C4 = 7 = [52]
17b. R8C45 = {36} (only remaining combination), locked for R8 and N8 -> R8C8 = 2, placed for D\, R8C9 = 1, R8C6 = 7, R4C6 = 8, R6C4 = 7
18. R3C1 = 3, R3C3 = 5, placed for D\
18a. R5C5 = 9 (hidden single on D/)
and the rest is naked singles, without using the diagonals.
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