Prelims
a) R1C12 = {19/28/37/46}, no 5
b) R1C34 = {39/48/57}, no 1,2,6
c) R34C9 = {15/24}
d) R45C3 = {18/27/36/45}, no 9
e) R56C7 = {18/27/36/45}, no 9
f) R67C1 = {29/38/47/56}, no 1
g) R9C45 = {17/26/35}, no 4,8,9
h) R9C67 = {15/24}
i) R9C89 = {16/25/34}, no 7,8,9
j) 19(3) cage at R7C6 = {289/379/469/478/568}, no 1
k) 20(3) cage at R8C4 = {389/479/569/578}, no 1,2
l) 29(7) cage at R4C6 = {1234568}, no 7,9
m) 37(8) cage at R6C2 = {12345679}, no 8
1a. 45 rule on R9 3 innies R9C123 = 24 = {789}, locked for N7, 7 also locked for R9, clean-up: no 2,3,4 in R6C1, no 1 in R9C45
1b. Killer pair 2,5 in R9C45 and R9C67, locked for R9
1c. 20(3) cage at R8C4 = {389/479/578} (cannot be {569} which clashes with R9C45), no 6
2a. 45 rule on N3 2 outies R4C89 = 7 = [25/34/52/61] -> R4C8 = {2356}
2b. 45 rule on N36 2 innies R4C7 + R6C9 = 12 = [39/48/57/84] -> R4C7 = {3458}, R6C9 = {4789}
2c. 29(7) cage at R4C6 = {1234568}, CPE no 1,2,6 in R6C56
2d. 8 in N5 only in R45C456 + R6C4, CPE no 8 in R4C7, clean-up: no 4 in R6C9
3a. 45 rule on N9 3(1+2) outies R6C9 + R79C6 = 13
3b. Min R6C9 = 7 -> max R79C6 = 6 -> max R7C6 = 5, max R9C6 = 4 (because no 1 in R7C6), clean-up: no 1 in R9C7
3c. 19(3) cage at R7C6 = {289/379/469/478/568}
3d. R7C6 = {2345} -> no 2,3,4,5 in R7C78
3e. 8 in N8 only in 20(3) cage at R8C4, locked for R8
3f. 20(3) cage (step 1c) = {389/578}, no 4
3g. R9C45 = {26} (cannot be {35} which clashes with 20(3) cage), locked for R9 and N8, clean-up: no 1 in R9C89
3h. Naked pair {34} in R9C89, locked for R9 and N9 -> R9C78 = [15], clean-up: no 4 in R56C7, no 7 in R6C9 (step 2b)
3i. R9C6 = 1 -> R6C9 + R7C6 = 12 = [84/93], no 5 in R7C6
3j. 4 in N8 only in R7C456, locked for R7, clean-up: no 7 in R6C1
[The first key step, a wellbeback-style one.]
4a. 45 rule on C789 1 outie R7C6 = 1 remaining innie R4C7
4b. R4C7 ‘sees’ all of N5 except for R6C56 and indirectly ‘sees’ R6C6 because R4C7 = R7C6 -> R4C7 = R6C5 = {34}
4c. R6C3 ‘sees’ all of N5 except for R4C45 -> at least one of {1234568} must be in R4C45
4d. 7,9 in N5 can only be in R4C45 and R6C6, R4C45 cannot contain both of 7,9 -> R6C6 = {79}
[The next key step but, as Ed said, many more are needed.]
5. 45 rule on N8 3 remaining innies R7C456 = 16 contains 4 = {349/457}
5a. 19(3) cage at R7C6 = {469/478} (cannot be {379} = 3{79} which clashes with R7C456 = {49}3, CCC) -> R7C6 = 4
5b. R7C6 = 4 -> R4C7 = 4 (step 4a) -> R6C9 = 8 (step 2b), clean-up: no 2 in R3C9, no 3 in R4C8 (step 2a), no 5 in R5C3, no 1 in R56C7, no 3 in R7C1
5c. 8 in N9 only in 19(3) cage at R7C6 = 4{78}, 7 locked for R7 and N9
5d. 7 in N8 only in 20(3) cage at R8C4 (step 3h) = {578}, 5 locked for R8 and N8
5e. Naked pair {39} in R7C45, locked for R7 and 37(8) cage at R6C2 -> R6C56 = [47], clean-up: no 2 in R5C7
5f. 9 in N5 only in R4C45, locked for R4 and 24(4) cage at R3C5
5g. 9 in C6 only in R12C6, locked for N2, clean-up: no 3 in R1C3
5h. 29(7) cage at R4C6 = {1234568}, 8 locked for N5
5i. R8C13 = {34} (hidden pair in N7)
5j. 1 in N7 only in R7C23 + R8C2, locked for 37(8) cage at R6C2, no 1 in R6C2
5k. Killer pair 2,6 in R4C78 and R56C7, locked for N6
5l. 7 in N6 only in R5C789, locked for R5, clean-up: no 2 in R4C3
6a. 45 rule on N78 R6C56 = [47] = 11 -> 2 remaining outies R6C12 = 11 = [56/65/92]
6b. 45 rule on N4 2 other innies R4C1 + R6C3 = 9 = {36}/[72/81] -> R4C1 = {3678}, no 5 in R6C3
6c. 29(7) cage at R4C6 = {1234568}, 5 locked for N5
6d. 16(3) cage at R4C2 = {178/349/358/367/457} (cannot be {169/259/268} which clash with R6C12), no 2
6e. 1 in N4 only in R4C1 + R6C3 = [81] or R45C3 = {18} or 16(3) cage = {178} -> 16(3) cage = {178/349/367/457} (cannot be {358}, locking-out cages)
6f. 7 of {178/367/457} must be in R4C2, 3 of {349} must be in R4C2 -> R4C2 = {37}
7a. 45 rule on R56789 3 remaining outies R4C236 = 16 = {178/358/367} (cannot be {268} which clashes with R4C89), no 2
[At this stage I saw
{358} can only be [385] (cannot be [358] because R45C3 = [54] clashes with 16(3) cage at R4C2 = {349/358} while 7 of 16(3) cage at R4C2 = {367} must be in R4C2) -> no 5 in R4C3, clean-up: no 4 in R5C3
However I then saw the more powerful short forcing chain.]
7b. Consider combinations for R4C89 (step 2a) = {25}/[61]
R4C89 = {25}, 5 locked for R4 => R4C236 = {178/367}
or R4C89 = [61], R6C7 = 2 (hidden single in N6), no 7 in R4C1 (step 6b) => 7 in R4 only in R4C236 = {178/367}
-> R4C236 = {178/367}, no 5, 7 locked for R4, clean-up: no 4 in R5C3, no 2 in R6C3 (step 6b)
7c. 1 of {178} must be in R4C3 -> no 8 in R4C3, clean-up: no 1 in R5C3
7d. 4 in N4 only in 16(3) cage at R4C2 = {349/457} -> R5C12 = {45/49}
7e. 5 in R4 only in R4C89 (step 2b) = {25}, locked for N6, 2 also locked for R4, clean-up: no 5 in R3C9, no 7 in R5C7
7f. Naked pair {36} in R56C7, locked for C7, 3 also locked for N6
7g. 1 in C1 only in R123C1, locked for N1, clean-up: no 9 in R1C1
7h. 1 in N4 only in R46C3, locked for C3
8. 45 rule for R6789 4 innies R6C3478 = 15 = {1239/1356}
8a. 2,5 only in R6C4 -> R6C4 = {25}
8b. Consider combinations for R6C3478
R6C3478 = {1239}, 2,9 locked for R6 => R5C3 = 2 (hidden single in N4), R4C3 = 7
or R6C3478 = {1356} => R6C8 = 1, R4C3 = 1 (hidden single in N4), R5C3 = 8
-> R45C3 = [18/72], no 3,6
8c. R4C236 (step 7b) = {178/367}
8d. 6,8 only in R4C6 -> R4C6 = {68}
9. 24(4) cage at R3C5 contains 9 = {1689/2679/3579} (cannot be {2589} because 2,5,8 only in R3C56)
9a. 7 of {2679/3579} must be in R3C5 -> no 2,3,5 in R3C5
9b. 5 of {3579} must be in R3C6 -> no 3 in R3C6
9c. Consider combinations for R4C236 (step 7b) = {178/367}
R4C236 = {178} => R8C6 = 5 => 24(4) cage = {1689/2679}
or R4C236 = {367}, 3 locked for R4 => 24(4) cage = {1689/2679}
-> 24(4) cage = {1689/2679}, no 3,5
9d. 3 in R4 only in R4C12, locked for N4, clean-up: no 6 in R4C1 (step 6b)
9e. R6C7 = 3 (hidden single in R6) -> R5C7 = 6
9f. 6 in R4 only in R4C456, CPE no 6 in R3C6
9g. 24(4) cage = {1689/2679}
9h. R3C6 = {28} -> no 8 in R3C5
9i. 24(4) cage = {1689} must be [68]{19} (cannot be [18]{69} which clashes with R34C6 = [86]) -> no 1 in R3C5
9j. Consider combinations for 24(4) cage = {1689/2679}
24(4) cage = {1689} => R3C6 = 8
or 24(4) cage = {2679} => R3C56 = [72], R4C45 = {69}, locked for R4 => R4C6 = 8
-> 8 in R34C6, locked for C6 -> R8C6 = 5
10. Consider placements for R4C1 = {38}
R4C1 = 3 => R4C6 = 8 (hidden single in R4) => R35C6 = [23]
or R4C1 = 8, R6C3 = 1 (step 6b), R4C3 = 7 => R5C3 = 2 => R5C6 = 3
-> R5C6 = 3
10a. 2 in C6 only in R123C6, locked for N2
10b. 12(3) cage at R2C4 = {156/237/246} (cannot be {129} because 2,9 only in R2C6, cannot be {138/147/345} because R2C6 only contains 2,6,9), no 8,9
[Now it’s getting a lot easier. Ed thought this comment should have been after step 11; the reason for it here was that it was consistent with the comment posted before my walkthrough, I had to go back and find a way to make progress after step 9.]
10c. R1C6 = 9 (hidden single in C6), clean-up: no 1 in R1C1, no 3 in R1C4
10d. 24(4) cage at R3C5 (step 9c) = {1689/2679} -> R3C56 = [68/72]
10e. 12(3) cage at R2C4 = {156/237} (cannot be {246} which clash with R3C56), no 4
10f. 6 of {156} must be in R2C6 -> no 6 in R2C45
10g. Killer pair 6,7 in 12(3) cage and R3C56, locked for N2, clean-up: no 5 in R1C3
11a. 12(3) cage at R2C4 (step 10e) = {156/237}, R3C56 (step 10d) = [68/72] -> combined cage 12(3) + R3C56 = {156}[72]/{237}[68]
11b. R1C34 = [48/75/84], R1C5 = {1358} -> combined cage R1C345 = [75/84]1/[48/75/84]3/[84]5 (cannot be [48]1/[48/57]5/[75]8) which clash with combined cage 12(3) + R3C56) -> no 8 in R1C5
11c. Combined cage R1C345 = [75/84]1/[75]3/[84]5 (cannot be [48/84]3 which clash with R1C12) -> R1C34 = [75/84]
11d. Killer pair 7,8 in R1C3 and R45C3, locked for C3 -> R9C3 = 9
11e. 8 in N2 only in R3C46, locked for R3
12. 15(3) cage at R3C2 = {159/249/348/357/456} (cannot be {168} because 1,8 only in R3C4, cannot be {258} which clashes with R3C6, cannot be {267} which clashes with R3C5)
12a. 12(3) cage at R2C4 (step 10e) = {156/237}
12b. Consider combinations for R1C12 = {28/37/46}
R1C12 = {28}, locked for R1 => R1C34 = [75], 12(3) cage = {237}, 2 locked for N2 => R3C6 = 8
or R1C12 = {37/46} => 15(3) cage cannot be {348} = {34}8 which clashes with R1C12
-> no 8 in R3C4
[Cracked, at last.]
12c. R3C6 = 8 (hidden single in N2), R4C6 = 6, R4C45 = {19}, 1 locked for R4 and N5, R3C5 = 6 (cage sum), R6C3 = 1 (hidden single in N4), R6C8 = 9, clean-up: no 2 in R7C1
12d. R2C6 = 2 -> 12(3) cage = {237}, locked for R2, 3 locked for N2
12e. R4C3 = 7 -> R5C3 = 2, R1C3 = 8 -> R1C4 = 4, clean-up: no 2,6 in R1C12
12f. Naked pair {37} in R1C12, locked for R1 and N1
12g. 15(3) cage = {159} (only remaining combination, cannot be {249} because 2,9 only in R3C2) -> R3C234 = [951]
12h. R3C9 = 4 -> R4C9 = 2, R3C1 = 2, R3C78 = [73], R4C8 = 5 -> R2C8 = 8 (cage sum)
and the rest is naked singles.