Prelims
a) R12C1 = {17/26/35}, no 4,8,9
b) R1C45 = {18/27/36/45}, no 9
c) R89C9 = {18/27/36/45}, no 9
d) R9C56 = {19/28/37/46}, no 5
e) 20(3) cage at R1C2 = {389/479/569/578}, no 1,2
f) 8(3) cage at R2C3 = {125/134}
g) 22(3) cage at R3C4 = {589/679}
h) 13(4) cage at R4C2 = {1237/1246/1345}, no 8,9
Steps resulting from Prelims
1a. 8(3) cage at R2C3 = {125/134}, CPE no 1 in R2C1, clean-up: no 7 in R1C1
1b. 13(4) cage at R4C2 = {1237/1246/1345}, 1 locked for N4
[The first key 45.]
2. 45 rule on R12 1 innie R2C9 = 2 outies R3C38 + 5
2a. Min R3C38 = 3 -> min R2C9 = 8
2b. Max R3C38 = 4 -> R3C38 = {12/13}, 1 locked for R3
[I overlooked no 1 in R8C8, CPE using D\]
2c. R2C9 = {89} -> 15(4) cage at R2C9 = {1239/1248}, no 5,6,7, 1,2 locked for C9, 1 also locked for N6,clean-up: no 7,8 in R89C9
2d. R2C9 = {89} -> no 8,9 in R345C9
2e. Killer pair 3,4 in 15(4) cage and R89C9, locked for C9
2f. 16(3) cage at R6C9 must contain one of 1,2,3,4 -> R7C8 = {1234}
3. 45 rule on C12 3 outies R148C3 = 19 = {289/379/469/478/568}, no 1
3a. 2 of {289} must be in R4C3 -> no 2 in R8C3
3b. 13(4) cage at R4C2 = {1237/1246/1345}, 1 locked for C2
4. 45 rule on N5 2 innies R4C4 + R6C6 = 10 = [64/73/82/91]
4a. 18(3) cage at R6C6 = {189/279/369/378/459/468} (cannot be {567} because R6C6 only contains 1,2,3,4)
4b. R6C6 = {1234} -> no 1,2,3,4 in R7C56
5a. 45 rule on N1 2(1+1) outies R2C4 + R4C1 = 7 = [16/25/34/43/52]
5b. 45 rule on N9 2(1+1) outies R6C9 + R8C6 = 12 = [57/66/75/84/93]
5c. 45 rule on C1 1 outie R3C2 = 1 innie R9C1 + 4, no 2,3,4 in R3C2, no 6,7,8,9 in R9C1
5d. 45 rule on C9 1 innie R1C9 = 1 outie R8C8 + 5, no 5 in R1C9
5e. 16(3) cage at R6C9 = {178/259/268/367/457} (cannot be {349} because 3,4 only in R7C8, cannot be {358} which clashes with R89C9, cannot be {169} which clashes with R1C9 + R7C8 = [61])
5f. 16(3) cage at R3C1 = {259/268/349/358/457} (cannot be {367} which clashes with R12C1)
[At this stage I can’t see any direct possibilities for the combined cages 8(2) at R1C1 and 16(3) at R3C1, or 16(3) at R6C9 and 9(2) at R8C9.]
6. 45 rule on R89 2 outies R7C27 = 1 innie R8C1 + 7, no 7 in R7C7 (IOU)
[Taking step 2 further …]
7. 15(4) cage at R2C9 (step 2c) = {1239/1248}
7a. R2C9 + R3C389 = 9{13}2/8{12}4, 2 locked for R3, no 3 in R3C9
7b. 4 of {1248} must be in R3C9 -> no 4 in R45C9
7c. 16(3) cage at R3C1 (step 5f) = {259/268/349/358/457}
7d. {268} must be {68}2 -> no 6 in R4C1, clean-up: no 1 in R2C4 (step 5a)
7e. 8(3) cage at R2C3 = {125/134}, 1 locked for C3 and N1, clean-up: no 7 in R2C1
7f. 16(3) cage at R3C1 = {268/349/358/457} (cannot be {259} which clashes with R12C1)
7g. 9 of {349} must be in R3C2 -> no 9 in R3C1
7h. R12C1 and 16(3) cage ‘see’ each other -> combined cage 16(3) cage + R12C1 = {268}{35}/{349}{26}/{358}{26}/{457}{26}, 2 locked for C1, clean-up: no 6 in R3C2 (step 5c)
8. 45 rule on N3 4 innies R13C7 + R23C9 = 20, R23C9 (step 7a) = [84/92] = 11,12 -> R13C7 = 8,9 = [17/26]/{35}/[18]{36/45}, no 7,8,9 in R1C7, no 9 in R3C7
[Another key 45 but a harder one to spot.]
9. 45 rule on N1236 2 outies R4C14 = 1 innie R6C9 + 1
9a. Min R4C14 = 8 -> min R6C9 = 7, clean-up: no 6,7 in R8C6 (step 5b)
9b. Max R4C14 = 10 -> no 5 in R4C1, no 9 in R4C4, clean-up: 2 in R2C4 (step 5a), no 1 in R6C6 (step 4)
9c. 22(3) cage at R3C4 = {589/679}, 9 locked for R3 and N2, clean-up: no 5 in R9C1 (step 5c)
9d. 8 of {589} must be in R4C4 -> no 8 in R3C45
9e. 8(3) cage at R2C3 = {125/134}
9f. 5 of {125} must be in R2C4 -> no 5 in R2C3
9g. 16(3) cage at R3C1 (step 7f) = {268/358/457}
9h. R4C1 = {234} -> no 3,4 in R3C1
9i. 1,2 in N2 only in R1C123 + R2C23, CPE no 1,2 in R1C7
9j. R13C7 (step 8) = 8,9 = {35/36/45}, no 7,8
[And a related 45.]
10. 45 rule on N1236 5(4+1) innies R3C1245 + R6C9 = 37
10a. Min R3C1245 = 28 must contain 8 -> 8 in R3C12, locked for R3 and N1
10b. 16(3) cage at R3C1 (step 9g) must contain 8 = {268/358}, no 4,7, clean-up: no 3 in R2C4 (step 5a)
10c. Killer pair 2,3 in R12C1 and R4C1, locked for C1
10d. 7 in N1 only in 20(3) cage at R1C2 = {479}, 4 locked for N1
10e. R148C3 (step 3) = {289/379/469/478} (cannot be {568} because R1C3 only contains 4,7,9), no 5
11. 7 in R3 only in R3C456, locked for N2, clean-up: no 2 in R1C45
11a. R1C45 and 15(4) cage at R1C6 ‘see’ each other -> combined cage R1C45 + 15(4) cage at R1C6 = 24(6) = {123468} (only remaining combination without 7,9), no 5, clean-up: no 4 in R1C45
11b. Caged X-Wing for 4 in 20(3) cage at R1C2 and 15(4) cage at R1C6, no other 4 in R12
[Cracked. The rest is straightforward.]
11c. R2C4 = 5 -> R23C3 = 3 = {12}, locked for C3 and N1, R12C1 = [53], 5 placed for D\, R3C12 = [68], R4C1 = 2, R9C1 = 4 (step 5c), placed for D/, clean-up: no 4,5 in R8C9, no 6 in R9C56
11d. Naked pair {36} in R89C9, locked for C9 and N9 -> R345C9 = [412], R2C9 = 8 (cage sum), clean-up: no 1 in R7C8 (step 5d), no 4 in R8C6 (step 5b)
11e. R3C45 = {79}, locked for R3, R4C4 = 6 (cage sum), R6C4 = 4 (step 4), both placed for D\ -> R89C9 = [63], 3 placed for D\, clean-up: no 3 in R1C5, no 7 in R9C56
11f. R2C5 = 4 (hidden single in N2), R3C6 = 3, clean-up: no 6 in R1C5
11g. Naked pair {18} in R1C45, locked for R1 and N2
11h. Naked pair {26} in R12C6, locked for C6 and 15(4) cage at R1C6, R1C7 = 3, R3C7 = 5, placed for D/, clean-up: no 8 in R9C5
11i. R8C4 = 5, R6C9 = 7 (step 5b), R1C9 = 9, placed for D/, R7C9 = 5, R7C8 = 4 (cage sum)
11j. R3C67 = [35] = 8 -> R45C7 = 12 = {48}, locked for C7 and N6
11k. Naked pair {47} in R1C23, locked for R1 and N1 -> R2C2 = 9, placed for D\
11l. Naked pair {12} in R3C3 + R7C7, locked for D\
11m. Naked pair {78} on D\, CPE no 7,8 in R8C5
11n. 13(4) cage at R4C2 = {1345}, locked for N4, 5 also locked for C2
11o. 7 in R4 only in R4C56, locked for N5 -> R5C5 = 8, placed for both diagonals, R4C6 = 7, placed for D/, R8C8 = 7, R1C45 = [81], clean-up: no 9 in R9C6
11p. R6C6 = 4 -> R7C56 = 14 = [68], R9C6 = 1 -> R9C5 = 9, R9C78 = [28], R7C7 = 1, placed for D\, R7C3 + R8C2 = [32], placed for D/
and the rest is naked singles, without using the diagonals.