Prelims
a) R12C7 = {13}
b) R5C67 = {18/27/36/45}, no 9
c) R8C45 = {18/27/36/45}, no 9
d) R89C6 = {39/48/57}, no 1,2,6
e) R8C89 = {39/48/57}, no 1,2,6
f) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
g) 19(3) cage at R1C8 = {289/379/469/478/568}, no 1
h) 8(3) cage at R2C2 = {125/134}
i) 19(3) cage at R8C1 = {289/379/469/478/568}, no 1
j) 27(4) cage at R2C5 = {3789/4689/5679}, no 1,2
1a. 8(3) cage at R2C2 = {125} (cannot be {134} which clashes with R2C7), locked for R2 -> R12C7 = [13]
1b. 45 rule on R1 2 remaining outies R2C69 = 17 = {89}, locked for R2
1c. 45 rule on C789 (or N36) 3 outies R345C6 = 6 = {123}, locked for C6, R5C7 = {678}, clean-up: no 9 in R89C6
1d. R8C45 = {18/27/36} (cannot be {45} which clashes with R89C6), no 4,5
1e. 45 rule on R9 2 innies R9C16 = 13 -> R9C1 = {5689}
1f. 45 rule on R89 2 innies R8C37 = 6 = [15/24/42]
1g. 45 rule on N78 1 outie R6C6 = 2 innies R7C12, no 9 in R7C12
1h. 27(4) cage at R2C5 = {3789/4689/5679}, CPE no 9 in R1C5
1i. 3,4 of {3789/4689} must be in R2C5 + R3C56 (R3C56 cannot be {89} which clashes with R2C6), no 3,4 in R4C5
2a. 45 rule on R9 3 outies R8C126 = 18 = {279/468/567} (cannot be {369} because no 3,6,9 in R8C6, cannot be {378} which clashes with R8C89, cannot be {459} which clashes with R8C37), no 3
2b. 7 of {279} must be in R8C6, 7 of {567} must be in R8C6 (R8C12 cannot contain both of 6,7) -> no 7 in R8C12
2c. {279} must be {29}7 with R9C1 = 8 (cage total), 6 of {468/567} must be in R8C12 -> no 6 in R9C1, clean-up: no 7 in R9C6 (step 1e), no 5 in R8C6
2d. 9 in R8 only in R8C126 = {279} = {29}7
or in R8C89 = {39}
-> no 7 in R8C89 (locking-out cages), clean-up: no 5 in R8C89
2e. 7 in R8 only in R8C456, locked for N8
2f. 45 rule on N78 7 innies R7C123456 + R8C3 = 32 form 32(7) cage because they ‘see’ each other = {1234589/1234679/1235678}, 3 locked for R7
2g. R7C123456 + R8C3 = {1234589/1234679} (cannot be {1235678} because 4,9 in R7C789 clashes with R8C89), 9 locked for R7 and 32(6) cage
2h. Max R6C6 = 8 -> max R7C12 (step 1g) = 8, no 8 in R7C12
2i. R7C789 must contain 5,8 or 6,7 and one of 1,2,4
2j. 14(3) cage at R9C7 = {158/167/239} (cannot be {149/347} which clash with R8C89, cannot be {257/356} which clash with R7C789, cannot be {248} which clashes with R9C16), no 4
3a. 45 rule on N6 2 innies R45C7 = 1 outie R3C9 + 6
3b. R45C7 cannot be [26/28] which eliminate 2 from C6 and cannot be [27] because no 3 in R3C9 -> no 2 in R4C7
3c. Min R45C7 = 10 -> min R3C9 = 4
4a. Consider combinations for R8C37 (step 1f) [15/24/42]
R8C37 = [15] => 1 in R7 must be in R7C89 => 14(3) cage at R9C7 (step 2j) = {239}, locked for N9 => R8C89 = {48}
or R8C37 = {24}
-> 4 in R8C3789, locked for R8
4b. R8C126 (step 2a) = {279/567} -> R8C6 = 7, R9C6 = 5, R9C1 = 8 (step 1e), placed for D/, clean-up: no 2 in R8C45
5a. R6C6 = R7C12 (step 1g)
5b. R7C123456 + R8C3 (step 2g) contains 4
5c. Either R7C12 contains 4 => R6C6 greater than 4
or R6C6 ‘sees’ 4 in R7C3456+ R8C3
-> no 4 in R6C6
5d. R6C6 = {68} -> R7C12 = 6,8 = {15/24/17/35} (cannot be {26} which clashes with R8C12), no 6
5e. Whichever of 6,8 is in R6C6 must also be in R7C89, R7C789 cannot contain both of 6,8 (which clashes with R7C3456, there’s no 6,8 in R7C12 + R8C3) -> 32(6) cage at R6C6 must contain both of 6,8 = {135689/234689} (cannot be {245678} because R7C3456 must contain 9 for R7), no 7, 3 locked for R7
5f. R7C789 cannot contain both of 6,8 -> no 6,8 in R7C7
6a. 45 rule on N2356 2 innies R2C4 + R6C6 = 1 outie R5C3 + 1
6b. Min R2C4 + R6C6 = 7 -> min R5C3 = 6
6c. Max R2C4 + R6C6 = 10 -> no 5 in R2C4
6d. 5 in R2 only in R2C23, locked for N1
7a. R8C37 (step 1f) = [15/24/42]
7b. R7C12 (step 5d) = {15/17} (cannot be {24} because 2 in R7C12 + 5 in R8C35 = [15] clash with R8C12), no 2,4, 1 locked for R7 and N7, clean-up: no 5 in R8C7
[And the rest is fairly straightforward.]
7c. Naked pair {24} in R8C37, locked for R8, clean-up: no 8 in R8C89
7d. Naked pair {39} in R8C89, locked for R8 and N9, clean-up: no 6 in R8C45
7e. Naked pair {56} in R8C12, locked for N7
7f. Naked pair {17} in R7C12, 7 locked for R7 and N7
7g. R7C12 = {17} = 8 -> R6C6 (step 1g) = 8, placed for D\, R2C69 = [98]
7h. R7C8 = 8 (hidden single in N9) -> R7C79 + R8C7 = 11 = {245}, no 6, 2 locked for N9
7i. Naked triple {167} in 14(3) cage at R9C7, 1,6 locked for R9
7j. 1 on D/ only in R4C6 + R5C5 + R6C4, locked for N5, clean-up: no 8 in R5C7
7k. Naked pair {67} in R59C7, locked for C7
7l. R4C7 = 8 (hidden single in N6)
7m. R2C4 + R6C6 = R5C3 + 1 (step 6a), R6C6 = 8 -> R2C4 + R5C3 = [18/29]
7n. R2C6 = 9 -> R1C456 = 14 = {248/347/356} (cannot be {257} because R1C6 only contains 4,6)
7o. R1C6 = {46} -> no 4,6 in R1C45
7p. Killer triple 1,2,3 in R1C456, R2C4 and R3C6, locked for N2
7q. 27(4) cage at R2C5 = {5679} (cannot be {4689} which clashes with R1C6) -> R3C5 = 9, R2C5 + R3C45 = {567}, locked for N2, 5 locked for R3
7r. R1C6 = 4 -> R1C45 = {28}, locked for R1, 2 locked for N2
7s. R2C4 = 1 -> R5C3 = 8 (step 7m), R8C45 = [81], R1C45 = [28], R3C6 = 3, R5C6 = 2 -> R5C7 = 7
7t. R2C9 = 8 -> R1C89 = 11 = {56}, 6 locked for R1 and N3
7u. Naked triple {379} in 19(3) cage at R1C1, 7,9 locked for N1
7v. R3C6 + R4C67 = [318] = 12 -> R2C8 + R3C78 = 13 = {247}, 4,7 locked for N3, 2 locked for R3, 7 locked for C8
7w. R6C7 = 9 (hidden single in C7) -> R56C8 = 8 = [35/53/62]
7x. Killer pair 5,6 in R1C8 and R56C8, locked for C8
7y. Naked pair {56} in R1C9 + R8C2, locked for D/
8a. R3C7 + R8C3 = [29] (hidden pair on D/)
8b. 3 on D/ only in R5C5 + R6C4, locked for N5
8c. R8C7 = 4, R7C7 = 5, placed for D\
8d. R3C2 = 8 (hidden single in N1), R2C1 + R3C13 = {146} = 11 -> R4C12 = 12 = {57}, locked for R4 and N4
8e. R1C3 = 7 (hidden single in C3)
8f. R38C9 = [93], R8C8 = 9, placed for D\, R1C1 = 3, placed for D\, R5C5 = 4, placed for both diagonals, R4C4 = 6, placed for D\, R2C8 = 7, R3C138 = [614], R8C1 = 5 -> R8C2 = 6, placed for D/
and the rest is naked singles, without using the diagonals.