Prelims
a) R1C23 = {39/48/57}, no 1,2,6
b) R12C4 = {17/26/35}, no 4,8,9
c) R12C6 = {49/58/67}, no 1,2,3
d) R1C89 = {15/24}
e) R23C3 = {14/23}
f) R3C12 = {29/38/47/56}, no 1
g) R34C4 = {19/28/37/46}, no 5
h) R3C67 = {18/27/36/45}, no 9
i) R5C12 = {19/28/37/46}, no 5
j) R5C34 = {69/78}
k) R6C12 = {19/28/37/46}, no 5
l) R7C45 = {39/48/57}, no 1,2,6
m) R89C9 = {29/38/47/56}, no 1
n) 10(3) cage at R4C1 = {127/136/145/235}, no 8,9
o) 22(3) cage at R8C8 = {589/679}
p) 12(4) cage at R7C7 = {1236/1245}, no 7,8,9
1a. 22(3) cage at R8C8 = {589/679}, 9 locked for N9
1b. 12(4) cage at R7C7 = {1236/1245}, 2 locked for N9
1c. Killer pair 5,6 in 22(3) cage and 12(4) cage, locked for N9
2a. 45 rule on N4 2 innies R56C3 = 15 = {69/78}
2b. 5 in N4 only in 10(3) cage at R4C1, locked for R4
2c. 10(3) cage = {145/235}, no 6,7
2d. 45 rule on N78 2 outies R6C34 = 11 = [65/74/83/92]
2e. 45 rule on N78 2 innies R7C23 = 12 = {39/48/57}, no 1,2,6
2f. 45 rule on N8 1 outie R8C3 = 1 innie R9C4 + 4 -> R8C3 = {56789}, R9C4 = {12345}
2g. 45 rule on C1234 3(2+1) innies R78C4 + R8C3 = 20, max R7C4 + R8C3 = 18 -> min R8C4 = 2
2h. 45 rule on N2 2 innies R3C46 = 8 = {17/26/}/[35], no 4,8,9, no 3 in R3C6, clean-up: no 1,5,6 in R3C7, no 1,2,6 in R4C4
2i. 45 rule on C789 3 innies R356C7 = 10 = {127/136/145/235}, no 8,9, clean-up: no 1 in R3C6, no 7 in R3C4, no 3 in R4C4
2j. 4 of {145} must be in R3C7 -> no 4 in R56C7
3. Hidden killer pair 8,9 in 16(3) cage at R1C5 and R12C6 for N2, each can only contain one of 8,9 -> R12C6 = {49/58}, no 6,7
3a. Hidden killer pair 4,8 in 16(3) cage at R1C5 and R12C6 for N2, R12C6 contains one of 4,8 -> 16(3) cage must contain one of 4,8 = {178/268/349/358/457} (cannot be {169/259/367} which don’t contain 4 or 8)
3b. 16(3) cage contains of 8,9 = {178/268/349/358} (cannot be {457} which clashes with R12C6
4. R3C3467 cannot be {14}[263] which clashes with R3C12 -> no 2 in R3C4, no 6 in R3C6, no 3 in R3C7, clean-up: no 8 in R4C4
[One of my most important steps, which I found while doing the Practice version.]
4a. R356C7 (step 2i) = {127/145/235} (cannot be {136} because R3C7 only contains 2,4,7), no 6 in R56C7
4b. R3C12 = {29/38/56}, (cannot be {47} which clashes with R3C67), no 4,7
5. 45 rule on C6 4 innies R3456C6 = 18 = {1278/1467/2349/2358/2367} (cannot be {1359/1458/2457/3456} which clash with R12C6, cannot be {1368} because R3C6 only contains 2,5,7, cannot be {1269} = 2{169} because 19(5) cage at R4C6 cannot contain both of 6,9)
5a. 45 rule on N2 1 outie R4C4 = 1 innie R3C6 + 2
5b. R3456C6 = {1278/1467/2358/2367} (cannot be {2349} = 2{349} which clashes with R3C6 + R4C4 = [24]), no 9
6a. 45 rule on C5 3 innies R789C5 = 15
6b. 45 rule on N8 using R789C5 = 15, 3 innies R789C4 = 16 = {178/259/358/457} (cannot be {169} which clashes with R12C4 + R3C4 (killer ALS block), cannot be {349} which clashes with R34C4, cannot be {367} which clashes with R12C4, cannot be {268} = [862] which clashes with R8C3 + R9C4 = [62], step 2f), no 6
6c. Consider combinations for R789C4
R789C4 = {178}, locked for C4 => R34C4 = [64], R12C4 = {35}, locked for C4
or R789C4 = {259/358/457}, 5 locked for C4
-> no 5 in R6C4, clean-up: no 6 in R6C3 (step 2d), no 9 in R5C3 (step 2a), no 6 in R5C4
6d. 6 in C4 only in R123C4, locked for N2
6e. 16(3) cage at R1C5 (step 3b) = {178/349/358}, no 2
7. 45 rule on N478 2 outies R56C4 = 11 = [74/83/92]
7a. Consider placements for 6 in C4
R12C4 = {26}, locked for C4 => R56C4 = [74/83]
or R34C4 = [64] => 10(3) cage at R4C1 = {235} (only remaining combination), locked for N4, no 7,8 in R56C12 => R56C3 = {78} (hidden pair in N4) => R5C34 = {78}=> R56C4 = [83]
-> R56C4 = [74/83], no 9 in R5C4, no 2 in R6C4, clean-up: no 6 in R5C3, no 9 in R6C3 (step 2a)
7b. Naked pair {78} in R56C3, locked for C3 and N4, clean-up: no 4,5 in R1C2, no 2,3 in R56C12, no 4,5 in R7C2 (step 2e), no 3,4 in R9C4 (step 2f)
7c. 10(3) cage at R4C1 = {235} (hidden triple in N4), locked for R4
7d. R789C4 (step 6b) = {259/358/457} (cannot be {178} which clashes with R5C4), no 1, 5 locked for C4, clean-up: no 3 in R12C4, no 5 in R8C3 (step 2f)
[5 locked for N8 is used in step 8.]
7e. 1 in C4 only in R123C4, locked for N2
7f. 16(3) cage at R1C5 (step 6e) = {349/358}, no 7, 3 locked for C5 and N2, clean-up: no 5 in R3C6 (step 2h), no 4 in R3C7, no 7 in R4C4, no 9 in R7C4
7g. Naked pair {27} in R3C67, locked for R3, clean-up: no 3 in R2C3, no 9 in R3C12
7h. R356C7 (step 4a) = {127/235}, 2 locked for C7
7i. 12(4) cage at R7C7 = {1236/1245}, 2 locked for R7
7j. 6 of {1236} must be in R78C7 (R78C7 cannot be {13} which clashes with R356C7), no 6 in R7C89
8. 5 in C4 only in R789C4, locked for N8, clean-up: no 7 in R7C4
8a. R6C34 = 11 (step 2d), R7C23 = 12 (step 2e), R789C4 (step 7d) = {259/358/457}
8b. Consider placements for R6C4 = {34}
R6C4 = 3 => R6C3 = 8, R7C23 = [75], no 5 in R7C4 => R789C4 = {358/457}
or R6C4 = 4, R4C4 = 9 => R789C4 = {358}
-> R789C4 = {358/457}, no 2,9
[Cracked. I spent quite a lot of time earlier trying to eliminate {259} before the puzzle was ready for me to make this important elimination. The rest is straightforward.]
8c. R9C4 = 5, R8C3 = 9 (step 2f), clean-up: no 3 in R1C2, no 3 in R7C23 (step 2e), no 7 in R7C5
8d. 2 in C4 only in R12C6 = {26}, locked for C4 and N2 -> R3C46 = [17], R3C7 = 2, R4C4 = 9, clean-up: no 4 in R1C89, no 4 in R2C3
8e. Naked pair {15} in R1C89, locked for R1 and N3, clean-up: no 7 in R1C2, no 8 in R2C6
8f. Naked pair {34} in R13C3, locked for C3 and N1, R7C3 = 5, R7C2 = 7 (step 2e), R4C3 = 2, R2C3 = 1 -> R3C3 = 4, R1C3 = 3 -> R1C2 = 9, R6C3 = 8, R6C4 = 3 (cage sum), R5C34 = [78], R7C4 = 4 -> R7C5 = 8, R9C3 = 6, R9C2 = 2 (cage sum), clean-up: no 4 in R2C6, no 8 in R3C12, no 1 in R56C1
8g. Naked pair {56} in R3C12, locked for R3 and N1 -> R2C2 = 8
8h. R8C34 = [97], R9C5 = 1 -> R8C5 = 6 (cage sum)
8i. Naked pair {39} in R79C6, locked for C6 -> R2C6 = 5, R1C6 = 8, R8C6 = 2
8j. Naked triple {146} in R456C6, locked for N5 and 19(5) cage at R4C6 -> R4C5 = 7
8k. R456C6 = {146} = 11 -> R56C7 = 8 = [35]
8l. 12(4) cage at R7C7 = {1236} (only remaining combination) -> R78C7 = [61], R7C89 = {23}, locked for R7 and N9, clean-up: no 8 in R89C9
8m. R89C9 = [47]
8n. Naked pair {89} in R9C89, locked for R9 and N9 -> R8C8 = 5, R1C8 = 1
9. 45 rule on N3 1 remaining innie R3C8 = 1 outie R4C9 + 2 -> R3C8 = {38}, R4C9 = {16}
9a. R3C9 = 8 (hidden single in C9), R3C8 = 3, R4C9 = 1
9b. R3C8 = 3 -> R4C78 + R5C8 = 21 = {489} (only remaining combination) -> R4C78 = {48}, R5C8 = 9, clean-up: no 1 in R5C2
and the rest is naked singles.