Prelims
a) R12C1 = {49/58/67}, no 1,2,3
b) R3C12 = {39/48/57}, no 1,2,6
c) R56C6 = {29/38/47/56}, no 1
d) R5C89 = {16/25/34}, no 7,8,9
e) R7C12 = {12}
f) R78C3 = {89}
g) 21(3) cage at R4C3 = {489/579/678}, no 1,2,3
h) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
i) 37(8) disjoint cage at R5C4 = {12345679}, no 8
Steps Resulting From Prelims and Initial Placements
1a. Naked pair {12} in R7C12, locked for R7 and N7
1b. Naked pair {89} in R78C3, locked for C3 and N7
1c. 45 rule on N1 1 innie R3C3 = 2
1d. 45 rule on N4 1 outie R6C4 = 9, clean-up: no 2 in R56C6
1e. 45 rule on N7 1 innie R9C3 = 3
1f. R6C4 = 9 -> R6C123 = 10 = {127/136/145/235}, no 8
2. 45 rule on N78 3 innies R7C456 = 14 = {347/356}, no 9, 3 locked for R7, N8 and 37(8) disjoint cage at R5C4
2a. 45 rule on N69 2 innies R9C78 = 10, must contain 9 for 37(8) disjoint cage = {19}, locked for R9 and N7, 1 also locked for 37(8) cage
2b. 2 in 37(8) cage only in R5C45 + R6C5, locked for N5
2c. R7C3 = 9 (hidden single in R7) -> R8C3 = 8
2d. 1 in N5 only in 12(3) cage at R4C4, locked for R4
2e. 19(3) cage at R8C6 = {289/478/568} (cannot be {469} which clashes with R7C456), 8 locked for R9
2f. 9 of {289} must be in R8C6 -> no 2 in R8C6
3a. 21(3) cage at R4C3 = {579/678} (cannot be {489} because 8,9 only in R5C2), no 4
3b. 8,9 of {579/678} only in R5C2 -> R5C2 = {89}
3c. 21(3) cage = {579/678}, 7 locked for C3 and N4
3d. 45 rule on R4 1 innie R4C3 = 1 outie R5C1 + 3, R4C3 = {567} -> R5C1 = {234}
3e. Hidden killer pair 8,9 in 14(3) cage at R4C1 and R5C2 for N4, R5C2 = {89} -> 14(3) cage must contain one of 8,9 = {239/248}, no 5,6, 2 locked for N4
4. 45 rule on C6789 using R9C78 = 10, 1 outie R9C5 = 2 innies R47C6 + 1
4a. Min R47C6 = 4 -> min R9C5 = 5
4b. Max R9C5 = 8 -> max R47C6 = 7, no 7 in R7C6
4c. Max R47C6 = 7, min R7C6 = 3 -> max R4C6 = 4
4d. Deleted
5a. 17(3) cage at R5C7 = {269/278/359/368/458/467} (cannot be {179} which clashes with R9C7), no 1
5b. 1 in R5 only in R5C89 = {16}, locked for N6, 6 also locked for R5 clean-up: no 5 in R6C6
5c. 45 rule on C9 2 innies R45C9 = 10 = [46/91]
5d. 16(3) cage at R4C7 = {259/349/457} (cannot be {358} because R4C9 only contains 4,9), no 8
5e. Consider placement for 2 in N4
2 in R4C12 => 16(3) cage = {349/457}
or 2 in R5C1 => R4C3 = 5 (step 3d) => 16(3) cage = {349}
-> 16(3) cage = {349/457}, no 2, 4 locked for R4 and N6
5f. 1 in N5 only in 12(3) cage at R4C4 = {138/156}, no 7
5g. 2 in R4 only in R4C12, locked for N4, clean-up: no 5 in R4C3 (step 3d)
[After this step I missed Ed’s 4,7 in N5 only in R5C456 + R6C56, CPE no 4,7 which would have led to an early placement R4C6 = 1, as in his walkthrough.]
6. R9C3 = 3 -> R8C45 + R9C4 = 12 and must contain 1 for N8 = {129/147/156}
6a. 2 of {129} must be in R9C4 -> no 2 in R8C45
6b. 2 in N8 only in R9C46, locked for R9
7. 1 in N1 only in 18(4) cage at R1C2 = {1359/1368/1458/1467}
7a. Consider combinations for R6C123 = 10 (step 1f) = {136/145}
R6C123 = {136} => 4 in C3 only in R12C3 => 18(4) cage = {1458/1467}
or R6C123 = {145} => R45C3 = [67] => R126C3 = {145} => 18(4) cage = {1359/1458/1467} (cannot be {1368} which requires 1,6 in R12C3)
-> 18(4) cage = {1359/1458/1467}
7b. R3C12 = {39/48} (cannot be {57} which clashes with 18(4) cage), no 5,7
7c. R12C1 = {58/67} (cannot be {49} which clashes with R3C12), no 4,9
8. Hidden killer pair 2,3 in 22(4) cage at R6C8 (in R8C78) and R8C9 for R8, 22(4) cannot contain both of 2,3 (because it doesn’t contain 9) -> R8C9 = {23}, R8C78 must contain one of 2,3
8a. 22(4) cage can only contain one of 2,3 in R8C78 -> no 2,3 in R6C8
8b. 22(4) contains one of 2,3 = {2578/3478/3568}, 8 locked for C8
8c. 18(4) cage at R6C9 contains at least one of 2,3 = {2358/2367/2457/3456}
8d. 8 of {2358} must be in R7C9 -> no 8 in R6C9
9. 45 rule on N3 1 outie R1C6 = 1 innie R3C7 + 1, no 1,3 in R1C6, no 9 in R3C7
9a. 45 rule on N3 3 innies R1C78 + R3C7 = 12 = {138/147/156/237/246/345} (cannot be {129} = {29}1 because 13(3) cage at R1C6 cannot be 2{29}), no 9
10. 45 rule on N12 3 innies R123C6 = 16
10a. Hidden killer pair 2,9 in R123C6 and R89C6 for C6, R89C6 must contain both or neither of 2,9 -> R123C6 must contain both or neither = {178/259/358/457} (cannot be {169/268/349} which only contain one of 2,9, cannot be {367} which clashes with R56C6), no 6, clean-up: no 5 in R3C7 (step 9)
[Before I spotted the hidden killer pair, I’d eliminated {169/349} which clash with R4567C6 and R567C6 respectively, killer ALS blocks.]
10b. Consider combinations for R123C6
R123C6 = {178} => R89C6 = [92] (hidden pair in C6)
or R123C6 = {259/358/457}, 5 locked for C6
-> no 5 in R89C6
10c. 19(3) cage at R8C6 (step 2e) = {289/478/568}
10d. 6 of {568} must be in R8C6 -> no 6 in R9C6
10e. 17(4) cage at R2C5 contains 2 = {1268/2348/2456} (cannot be {1259/2357} which clash with R123C6), no 7,9
10f. 14(3) cage at R1C4 = {167/239/347} (cannot be {149/158/248/356} which clash with 17(4) cage, cannot be {257} which clashes with R123C6), no 5,8
11a. Hidden killer pair 1,9 in 17(3) cage at R1C9 and R45C9 for C9, R45C9 contains both or neither of 1,9 -> 17(3) cage must contain both or neither of 1,9 = {179/278/368/458/467} (cannot be {269/359} which contain 9 but not 1)
11b. 9 in N3 only in 17(3) cage at R1C9 and 16(3) cage at R2C7, 17(3) cage only contain 9 in {179}, 16(3) cage cannot contain both of 7,9 so cannot contain 7 = {169/259/268/349/358}, no 7
[I had to think hard to find this forcing chain.]
12. R9C5 = R47C6 + 1 (step 4), R123C6 (step 10a) = {178/259/358/457}
12a. Consider combinations for R7C456 (step 2) = {347/356}
R7C456 = {347} => R5C45 + R6C5 = {256} => R56C6 = {38/47} => R123C6 (step 10a) = {259/358/457} (cannot be {178} which clashes with R56C6)
or R7C456 = {356}, locked for N8, R9C5 = {78} => R47C6 = 6,7 = [15/16] (cannot be [34] because no 4 in R7C6 in this path)
-> R123C6 = {259/358/457}, no 1, 5 locked for C6 and N2, clean-up: no 6 in R6C6
[Amended because step 4d deleted.]
12b. R4C6 = 1 (hidden single in C6)
[Extra sub-step, now that I know about Ed’s CPE.
4 in N5 only in R5C456 + R6C56, CPE no 4 in R7C6.]
12c. R9C5 = {58} -> R47C6 = 4,7 -> R7C6 = {36}
12d. 6 in C6 only in R78C6, locked for N8
12e. 15(4) cage at R8C4 contains 1 for N8 = {1239/1347}, no 5
12f. 17(4) cage at R2C5 (step 10e) = {1268/2348}, 8 locked for N2, clean-up: no 3 in R123C6, no 7 in R3C7 (step 9)
12g. 15(3) cage at R2C6 cannot be {249} = [294] which clashes with R3C12 -> no 4 in R3C7, clean-up: no 5 in R1C6 (step 9)
13. 45 rule on R6789 4 outies R5C4567 = 20, must contain 2 for R5 = {2378/2459}
13a. {2378} = {27}[38] (cannot be {27}[83] because 16(3) cage at R4C7 (step 5e) must contain both or neither of 3,9), {2459} = {25}[49] -> R5C45 {25/27}, R5C6 = {34}, R5C7 = {89}
13b. R5C6 = {34} -> R6C6 = {78}
13c. Hidden killer pair 4,6 in R6C123 and R6C5 for R6, R6C123 (step 1f) contains one of 4,6 -> R6C5 = {46}
13d. 4 in 37(8) disjoint cage only in R6C5 + R7C45, CPE no 4 in R8C5
13e. 17(3) cage at R5C7 (step 5a) = {269/278/359/368/458} (cannot be {467} because R5C7 only contains 8,9)
13f. R5C7 = {89} -> no 8 in R67C7
13g. 2 of {278} must be in R6C7 -> no 7 in R6C7
13h. Consider placement for 8 in N6
R5C7 = 8 => 17(3) cage = {278/368/458}
or R6C8 = 8 => R7C9 = 8 (hidden single in N9) => R689C9 = 10 = {23}5 => 17(3) cage = {269}
-> 17(3) cage = {269/278/368/458}
13i. 17(3) cage {458} = [854] -> no 5 in R7C7
13j. Consider combinations for R7C456 (step 2) = {347/356}
R7C456 = {347}, locked for R7 => R7C789 = [685] (cannot be [658] which clashes with 22(4) cage at R6C9 = [2835/3825]) => R5C7 = 8 (hidden single in N6)
or R7C456 = {356}, locked for 37(8) disjoint cage => R6C5 = 4, R5C6 = 3 => R5C4567 = {27}[38]
-> R5C7 = 8, clean-up: no 9 in R1C6 (step 9)
[Cracked. The rest is fairly straightforward.]
14a. R5C2 = 9 -> R45C3 = 12 = [75]
14b. R6C123 (step 1f) = {136} (only remaining combination), locked for R6, 3 also locked for N4
14c. R6C5 = 4, R5C6 = 3 -> R6C6 = 8
14d. 16(3) cage at R4C7 = {349} (hidden triple in R4)
14e. R7C6 = 6, R9C5 = 8 (hidden single in N8) -> R7C45 = {35} (hidden pair in N8), 5 locked for R7
14f. R3C4 = 8 (hidden single in N2) -> R3C12 = [93]
14g. R5C1 = 4, R467C1 = [231] (hidden triple in C1) -> R4C2 = 8
14h. 8 in N1 only in R12C1 = {58}, 5 locked for C1 and N1
14i. R89C2 = {45} (hidden pair in N7), 4 locked for C2
14j. R3C34 = [28] = 10 -> R23C5 = 7 = {16}, locked for C5 and N2
14k. R4C45 = [65], R7C45 = [53]
14l. Naked pair {16} in R3C57, locked for R3
14m. R3C7 = {16} -> R1C6 = {27} (step 9)
14n. R7C8 = 8 (hidden single in C8)
15. 8 in C9 only in 17(3) cage at R1C9 = {278} (only possible combination, cannot be {368} because R3C9 only contains 4,5,7, cannot be {458} which clashes with R3C8) -> R3C9 = 7, R12C9 = {28}, 2 locked for C9 and N3
15a. R45C9 = [91] (hidden pair in C9) -> R5C8 = 6
15b. 9 in N3 only in 16(3) cage at R2C7 = {349} (only possible combination, cannot be {169} which clashes with R3C7) -> R3C8 = 4, R2C78 = {39}, locked for R2, 3 also locked for N3
15c. R123C6 (step 12a) = {457} (only remaining combination) = [745], R3C7 = 6 (step 9)
and the rest is naked singles.