Prelims
a) R12C1 = {29/38/47/56}, no 1
b) R1C23 = {79}
c) R1C56 = {49/58/67}, no 1,2,3
d) R34C4 = {79}
e) R34C5 = {14/23}
f) R3C67 = {89}
g) R56C6 = {19/28/37/46}, no 5
h) R6C12 = {29/38/47/56}, no 1
i) R78C9 = {49/58/67}, no 1,2,3
j) R9C12 = {19/28/37/46}, no 5
k) 7(3) cage at R1C4 = {124}
1a. Naked pair {79} in R1C23, locked for R1 and N1, clean-up: no 2,4 in R12C1, no 4,6 in R1C56
1b. Naked pair {58} in R1C56, locked for R1 and N2, clean-up: no 3,6 in R2C1
1c. R3C67 = [98] -> R34C3 = [79], clean-up: no 1 in R56C6
1d. 7(3) cage at R1C4 = {124}, CPE no 1,2,4 in R2C56
1e. Naked pair {36} in R2C56, locked for R2 and 21(5) cage at R1C7, 3 locked for N2, clean-up: no 2 in R4C5
1f. R2C56 = 9 -> R1C78 + R2C7 = 12 = {129/147} -> R2C7 = {79}, R1C78 = {12/14}, 1 locked for R1 and N3
1g. Naked triple {124} in R1C478, 2,4 locked for R1
1h. 7(3) cage at R1C4 = {124}, 1 locked for R2
2a. 45 rule on N36 using R1C78 + R2C7 = 12, 1 remaining outie R4C6 = 7, clean-up: no 3 in R56C6
2b. 7 in N6 only R5C89, 7 locked for R5
2c. 1 in C6 only in R789C6, locked for N8
2d. 45 rule on N14 3 innies R245C3 = 11 = {128/146/236/245}, no 9
3a. 45 rule on R123456 1 outie R7C5 = 1 innie R6C4 + 1, no 8 in R7C5
3b. 26(6) cage at R4C3 = {123569/123578/124568/134567} (cannot be {123479} because 7,9 only in R7C5
3c. 7,9 of {123569/123578/134567} must be in R7C5 -> no 3 in R7C5, clean-up: no 2 in R6C4
3d. R6C4 + R7C5 = [12/45/56/67/89] (cannot be [34] which clashes with R34C5 = [41]), no 3 in R6C4, no 4 in R7C5
3e. R6C4 + R7C5 = [45/56/67/89] (cannot be [12] which clashes with R34C5 = [23] because 3 in N5 must be in R4C5 when 26(6) cage = {124568}), no 1 in R6C4, no 2 in R7C5
3f. R6C4 + R7C5 = [56/67/89] (cannot be [45] because R56C6 = {28} and 26(6) cage = {124568} would require 2,4,8 in R45C3), no 4 in R6C4, no 5 in R7C5
3g. R6C4 + R7C5 = [67/89] (cannot be [56] because R56C6 = {46}, only other place for 6 in N5 and R7C5 clash with R2C56), no 5 in R6C4, no 6 in R7C5
[Some of those can be considered to be equivalent to forcing chains.]
3h. 26(6) cage at R4C3 = {123569/123578/134567}
3i. 5 in N5 only in R5C45 + R6C5, locked for 26(6) cage, no 5 in R45C3
3j. Killer pair 6,8 in R56C6 and R6C4, locked for N5
4a. 45 rule on N3 using R1C78 + R2C7 = 12, 1 innie R3C8 = 1 outie R4C9 + 2 -> R3C8 = {3456}, R4C9 = {1234}
4b. 23(5) cage at R1C9 = {12569/13469/14567/23459/23567} (cannot be {12479} because R1C9 must contain one of 3,6)
4c. 23(5) cage = {12569/14567/23459/23567} (cannot be {13469} which clashes with R3C8 + R4C9 = [31]), 5 locked for N3, clean-up: no 3 in R4C9
5a. 45 rule on N124 using R2C56 = 9, 2 innies R45C3 = 1 outie R4C5 + 6, IOU no 6 in R5C3
5b. R245C3 (step 2d) = {128/146/236}
5c. 6 of {146/236} must be in R4C3 -> no 3,4 in R4C3
[I ought to have spotted this obvious step earlier but hadn’t checked and found that 26(6) cage without 7 or 9 has one combination; I ought to have realised that since it’s missing 7 and 9 it must also be missing 3 to make a total of 19 missing. I’d also used combinations in step 3 for the other 26(6) cage.]
6a. 26(6) cage at R2C2 = {124568}, no 3
6b. R1C1 = 3 (hidden single in N1) -> R2C1 = 8, R1C9 = 6, clean-up: no 3,8 in R6C2, no 7 in R78C9, no 2,7 in R9C2, no 4 in R4C9 (step 4a)
6c. 26(6) cage = {124568} -> R4C2 = 8, clean-up: no 2 in R9C1
6d. 5,6 in N1 only in R2C2 + R3C123, locked for 26(6) cage, no 5,6 in R4C1
6e. 26(6) cage at R4C3 (step 3h) = {123569/134567} -> R4C3 = 6, clean-up: no 5 in R6C12
6f. 5 in N4 only in 15(3) cage at R5C1 = {159/357}, no 2,4
6g. 7 of {357} must be in R6C3 -> no 3 in R6C3
6h. 45 rule on N4 2 remaining innies R4C1 + R5C3 = 5 = [14/23/41], no 2 in R5C3
6i. 18(3) cage at R9C7 = {279/369/378/567} (cannot be {189/459/468} which clash with R78C9), no 1,4
6j. R9C12 = {19/46} (cannot be [73] which clashes with 18(3) cage), no 3,7
6k. 18(3) cage = {279/378/567} (cannot be {369} which clash with R9C12), 7 locked for R9 and N9
6l. 14(3) cage at R7C1 = {239/257/347/356} (cannot be {149/167} which clash with R9C12), no 1
6m. 3 of {239/347/356} must be in R8C2 -> no 4,6,9 in R8C2
6n. 45 rule on N9 4 innies R78C78 = 14 = {1238/1256/1346} (cannot be {2345} which clashes with R78C9), no 9
7a. 5 in R4 only in R4C78, locked for N6
7b. 7 in R5 only in 23(4) cage at R3C8 = [35]{78}/{34}{79}, 3 locked for C8
7c. 45 rule on R1234 2 outies R5C89 = 1 remaining innie R4C7 + 11
7d. R5C89 = {78/79} = 15/16 -> R4C7 = {45}
[Something else I ought to have spotted earlier.]
8a. 3 in N4 only in R5C23, locked for R5
8b. 3 in N5 only in R46C5, locked for C5 -> R2C56 = [63]
9a. 15(3) cage at R5C1 (step 6f) = {159/357}, 14(3) cage at R7C1 (step 6l) = {239/257/347/356}
9b. Consider combinations for R4C1 + R5C3 (step 6h) = [14/23/41]
R4C1 + R5C3 = {14} => 15(3) cage = [537] => 14(3) cage = {27}5
or R4C1 + R5C3 = [23] => 15(3) cage = {159} => R6C12 = {47}=> 14(3) cage = {257/356} (cannot be {29}3 which clashes with R4C1, cannot be {47}3 which clashes with R6C1)
-> 14(3) cage = {257/356}, no 4,9, 5 locked for N7
10a. 1 in C9 only in R46C9, locked for N6
10b. 31(6) cage at R4C6 = {145678/234679/235678} (cannot be {124789} which clashes with R5C89, cannot be {135679} which clashes with 23(4) cage at R3C8, step 7b)
10c. 31(6) cage = {234679/235678} (cannot be {145678} = [75{46}81] which clashes with R6C12 + R6C46, killer ALS block), no 1, 3 locked for R6 and N6
10d. Naked pair {45} in R4C78, 4 locked for R4 and N6
10e. R4C9 = 1 (hidden single in N6) -> R3C8 = 3 (step 4a), R4C1 = 2, R4C5 = 3 -> R3C5 = 2, R12C4 = [41], R2C3 = 2, R5C3 = 3 (step 6h), clean-up: no 9 in R6C12
10f. Naked pair {47} in R6C12, 4 locked for R6, 7 locked for N4, clean-up: no 6 in R5C6
10g. R1C78 = {12}, R2C56 = [63] -> R2C7 = 9 (cage sum)
10h. R9C7 = 7 (hidden single in C7) -> R9C89 = 11 = {29}/[65/83], no 5 in R9C8, no 8 in R9C9
10i. 14(3) cage at R7C1 (step 9b) = {257/356} -> R8C2 = {23}, R78C1 = {56/57}, 5 locked for C1
10j. R78C2 = {23} (hidden pair in C2)
11a. Consider combinations for 21(4) cage at R7C2 = 2{478}/3{189}
21(4) cage = 2{478}, 7 locked for C3 => R1C3 = 9 => R6C3 = {15}
or 21(4) cage = 3{189}, 1,9 locked for C3 => R6C3 = 5
-> R6C3 = {15}
[Ed pointed out 21(4) cage at R7C2 = 2{478}/3{189} -> killer pair 7,9 in R1C3 and R789C3, locked for C3.
I also realised that R6C35 = {15} (hidden pair for R6) was simpler.
So I’d seen this step as a forcing chain when both killer pair and hidden pair were simpler.]
11b. 9 in N4 only in R5C12, locked for R5
11c. R5C89 = {78}, 8 locked for R5 and N6, R3C8 = 3 -> R4C8 = 5 (cage total), clean-up: no 2 in R6C6
11d. R4C7 = 4, R5C7 = 6 (hidden single in R5)
11e. 5 in N3 only in R23C9, locked for C9, clean-up: no 8 in R78C9, no 6 in R9C8 (step 10h)
11f. Naked pair {49} in R78C9, locked for C9 and N9 -> R23C9 = [75], R2C28 = [54], clean-up: no 2 in R9C89 (step 10h)
11g. R6C3 = 5 (hidden single in N4) -> R6C5 = 1
11h. R9C89 = [83]
11i. 2,5 in R9 only in R9C456, locked for N8
11j. 12(3) cage at R8C6 = {156/246}, no 8, 6 locked for R8
11k. 12(3) cage = {156/246} = [156/651/426], no 1 in R8C7 = {25}, no 2 in R8C8
11l. R59C4 = {25} (hidden pair in C4)
[I’ve now reached Ed’s “sting in the tail”, how to eliminate 12(3) cage at R8C6 = [156]. I refuse to use Unique Rectangle, R8C6 cannot be 1 => R1C78 and R7C78 both {12} whose order cannot be resolved, as that relies on the solution being unique but walkthroughs are meant to reach a unique solution.]
[Or maybe that’s not initially the right approach.]
12a. R56C6 = [28/46], R7C5 = R6C4 + 1 (step 3a), 14(3) cage at R7C1 (step 9b) = {257/356}
12b. Consider permutations for 12(3) cage at R8C6 (step 11k) = [156/651/426]
12(3) cage = [156] => 14(3) cage = [572], 6 in N7 only in R9C12 = {46}, R9C35 = [19] (hidden pair in R9) => R7C5 = 7, R6C4 = 6 => R56C6 = [28]
or 12(3) cage = [426/651] => R8C6 = {46} => R56C6 = [28] (cannot be [46] which clashes with R8C6)
-> R56C6 = [28], R5C45 = [54], R6C4 = 6, R7C5 = 7, R1C56 = [85], R89C5 = [95], R78C9 = [94]
12c. 12(3) cage = [156/651] -> R8C7 = 5, 1 locked for R8
12d. R8C1 = 7 -> R7C1 + R8C2 = 7 = [52], R6C1 = 4
12e. 6 in N7 only in R9C12 = [64]
and the rest is naked singles.