Prelims
a) R34C1 = {17/26/35}, no 4,8,9
b) R34C4 = {12}
c) R3C89 = {19/28/37/46}, no 5
d) R67C6 = {89}
e) R67C9 = {39/48/57}, no 1,2,6
f) R7C12 = {19/28/37/46}, no 5
g) 19(3) cage at R2C2 = {289/379/469/478/568}, no 1
h) 24(3) cage at R3C2 = {789}
i) 19(3) cage at R4C2 = {289/379/469/478/568}, no 1
j) 11(3) cage at R5C8 = {128/137/146/236/245}, no 9
k) 6(3) cage at R6C7 = {123}
l) 11(3) cage at R8C6 = {128/137/146/236/245}, no 9
Steps resulting from Prelims
1a. Naked triple {789} in 24(3) cage at R3C2, CPE no 7,8,9 in R12C3
1b. Naked pair {12} in R34C4, locked for C4
1c. Naked pair {89} in R67C6, locked for C6
1d. Naked triple {123} in 6(3) cage at R6C7, CPE no 1,2,3 in R89C7
2a. 45 rule on N4 2 innies R4C13 = 11 = [29/38] -> R34C1 = [53/62]
2b. 24(3) cage at R3C2 = {789}, 7 locked for R3 and N1, clean-up: no 3 in R3C89
2c. 19(3) cage at R4C2 = {469/478/568} (cannot be {289} which clashes with R4C3, cannot be {379} which clashes with R4C13), no 2,3
2d. Killer pair 8,9 in 19(3) cage and R4C3, locked for N4
2e. 45 rule on N6 2 innies R6C79 = 9 = [18/27] -> R67C9 = [75/84]
2f. 6(3) cage at R6C7 = {123}, 3 locked for R7 and N9, clean-up: no 7 in R7C12
2g. 9 in R6 only in R6C456, locked for N5
2h. 11(3) cage at R5C8 = {146/236/245} (cannot be {128} which clashes with R6C7, cannot be {137} which clashes with R6C79), no 7,8
2i. Killer pair 1,2 in 11(3) cage and R6C7, locked for N6
2j. 1 in R4 only in R4C456, locked for N5
3. 45 rule on R34567 2 innies R3C7 + R7C3 = 11 = [29/38/47/56/65/92], no 1,8 in R3C7, no 1,4 in R7C3
3a. 45 rule on N5 4 innies R46C46 = 21 cannot contain both of 1,2, R4C4 = {12} -> no 1,2 in R4C6
4. 45 rule on N4 3 outies R3C123 = 21 = {579/678}
4a. Combined cages R3C123 + R3C89 = 5{79}{28}/5{79}{46}/6{78}{19}, 9 locked for R3, clean-up: no 2 in R7C3 (step 3)
4b. 45 rule on N6 3 outies R7C789 = 9 = {135/234}
4c. Combined cages R7C12 + R7C789 = {19}{23}4/{28}{13}5/{46}{13}5, 1 locked for R7
[Taking steps 4a and 4c a bit further …]
4d. R3C123 + R3C89 (step 4a) = 5{79}{28}/5{79}{46}/6{78}{19}
R3C123 + R3C89 = 5{79}{28} => R3C4 = 1, R3C56 = {34} (cannot be {36} because 12(3) cage cannot be {36}3, cannot be {46} because no 2 in R4C6), R3C7 = 6
or R3C123 + R3C89 = 5{79}{46} => R3C5 = 8 (hidden single in R3), R34C6 = [13], R3C4 = 2, R3C7 = 3
or R3C123 + R3C89 = 6{78}{19} => R3C4 = 2, R3C567 = {345}
-> 12(3) cage at R3C5 = {345}/[813], R3C7 = {3456}, clean-up: no 9 in R7C3 (step 3)
4e. R46C46 = 21 (step 3a) = {1389/1479/1569/1578/2379/2469/2478/2568} can only contain one of 3,4,5, R4C6 = {345} -> no 3,4,5 in R6C4
4f. R7C12 + R7C789 (step 4c) = {19}{23}4/{28}{13}5/{46}{13}5
R7C12 + R7C789 = {19}{23}4 => R7C6 = 8, R7C45 = {567} (5 must be in R7C45 because no 5 in R6C4), R7C3 = {67}
or R7C12 + R7C789 = {28}{13}5 => R7C6 = 9, R7C45 = {46} (cannot be {47} because 18(3) cage cannot be 7{47}, R7C3 = 7
or R7C12 + R7C789 = {46}{13}5 => R7C5 = 2 (hidden single in R7), R67C4 = {79}, R7C3 = {78}
-> R7C3 = {678}, 18(3) cage at R6C4 = {279/468/567}, clean-up: no 6 in R3C7 (step 3)
4g. Hidden killer pair 6,9 in R3C123 and R3C89 for R3, R3C123 contains one of 6,9 -> R3C89 must contain one of 6,9 -> R3C89 = {19/46}, no 2,8
4h. R3C4 = 2 (hidden single in R3) -> R4C4 = 1
4i. R46C46 = {1389/1479/1569} (cannot be {1578} which clashes with R6C9), 9 locked for N5
4j. Consider placement for 9 in N5
R6C4 = 9 => R7C45 = 9 = [72] (cannot be {45} which clashes with R7C9)
or R6C6 = 9 => R7C6 = 8
-> R7C12 = {19/46}, no 2,8
4k. Killer pair 1,4 in R7C12 and R7C789, locked for R7
4l. 18(3) cage at R6C4 = {279/567}, no 8
4m. 2 of {279} must be in R7C5 -> no 9 in R7C5
5. R3C123 (step 4) = {579/678}, R3C89 = {19/46}
5a. Consider placements for R3C7 = {345}
R3C7 = 3 => R7C3 = 8 (step 3) => R4C3 = 9 => R3C23 = {78}
or R3C7 = 4 => R3C89 = {19} (locked for R3) => R3C23 = {78}
or R3C7 = 5, R3C1 = 6 => R3C23 = {78}
-> R3C1 = 6, R3C23 = {78}, locked for R3, N1 and 24(3) cage at R3C2, R4C1 = 2, R4C3 = 9, clean-up: no 4 in R3C89, no 4 in R7C2
5b. Naked pair {19} in R3C89, locked for R3 and N3
5c. 12(3) cage at R3C5 = {345}, CPE no 3,4,5 in R12C6
5d. R5C7 = 9 (hidden single in N6)
5e. 19(3) cage at R2C2 = {289/379/469} (cannot be {478/568} because 6,7,8 only in R4C4), no 5
5f. 6,7,8 only in R2C4 -> R2C4 = {678} -> R2C2 = 9, clean-up: no 1 in R7C1
5g. 19(3) cage at R4C2 = {478/568}
5h. 4 of {478} must be in R45C2 (R45C2 cannot be {78} which clashes with R3C2), no 4 in R5C1
6. 9 in N2 only in R1C45 -> 15(3) cage at R1C3 = {159/249}
6a. 2 of {249} must be in R1C3 -> no 4 in R1C3
6b. 8 in N2 only in R2C45, locked for R2
6c. 8 in N2 only in 19(3) cage R2C2 = [928] or 13(3) cage at R2C5 = [814] => 19(3) cage cannot be [946] (locking-out cages), no 4 in R2C3, no 6 in R2C4, also no 2 in R2C7
6d. Consider combinations for 19(3) cage at R2C2 = {289/379}
19(3) cage = [928] => 15(3) cage at R1C2 = {159}
or 19(3) cage = [937] => R2C5 = 8 (hidden single in N2) => 13(3) cage at R2C5 = [814] => R3C7 = {35}, 4 in R3 only in R3C56, locked for N2 => 15(3) cage at R1C3 = {159}
-> 15(3) cage at R1C3 = {159}, locked for R1
6e. R1C3 + R2C1 = {15} (hidden pair in N1)
6f. 4 in N1 only in R1C12, locked for R1
6g. Hidden killer triple 6,7,8 in R1C6, R2C4 and 13(3) cage at R2C5, R1C6 = {67}, R2C4 = {78} for N2 -> 13(3) cage must contains one of 6,7,8 in N2
6h. 13(3) cage can only contain one of 6,7,8 -> no 6,7 in R2C7
6i. 13(3) cage = {148/346} (cannot be {157} which clashes with R2C1), no 5,7
6j. 8 of {148} must be in R2C5, 6 of {346} must be in R2C6 -> no 1,6 in R2C5
6k. Killer pair 3,8 in 19(3) cage and 13(3) cage, locked for R2
6l. 6 in N2 only in R12C6, locked for C6
7. 18(3) cage at R6C4 (step 4l) = {279/567}, R46C46 (step 4i) = {1389/1479/1569}
7a. Consider placements for R1C4 = {59}
R1C4 = 5 => R3C56 = {34}, locked for 12(3) cage at R3C5 => R4C6 = 5, R6C4 = 6 => 18(3) cage = {567}
or R1C4 = 9 => 18(3) cage = {567}
-> 18(3) cage at R6C4 = {567}
7b. 18(3) cage = {567}, 5 locked for R7 and N8 -> R7C9 = 4, R6C9 = 8, R67C6 = [98], R7C1 = 9 -> R7C2 = 1
7c. Naked pair {23} in R7C78, locked for N9 and 6(3) cage at R6C7 -> R6C7 = 1
7d. R5C3 = 1 (hidden single in N4), 15(3) cage at R1C3 = [591], R12C6 = [76], R2C4 = 8 -> R2C3 = 2 (cage sum)
7e. 5 in N2 only in R3C56, locked for R3 and 12(3) cage at R3C5, no 5 in R4C6
7f. R3C7 + R7C3 = 11 (step 3) = [47] (only remaining combination), R2C57 = [43], R3C23 = [78], R7C78 = [23]
7g. R3C56 = {35} -> R4C6 = 4 (cage sum)
7h. Naked pair {56} in R7C45, locked for N8 and 18(4) cage at R6C4 -> R6C4 = 7
7i. Naked pair {34} in R89C4, locked for C4 and N8
7j. Naked pair {12} in R89C6, locked for C6 and N8
7k. R5C1 = 7 (hidden single in C1) -> R45C2 = 12 = [84]
7l. R6C8 = 4 (hidden single in N6) -> R5C89 = 7 = {25}, locked for R5 and N6
[When I analysed cages in R12, I hadn’t expected that they’d provide the key to unlocking R7, but I couldn’t see any direct way to solve R7.]
8. 17(3) cage at R8C3 = {467} (only remaining combination) = [647]
8a. R9C5 = 9 -> R9C67 = 6 = [15]
and the rest is naked singles.