Prelims
a) R12C9 = {39/48/57}, no 1,2,6
b) R23C6 = {17/26/35}, no 4,8,9
c) R78C4 = {15/24}
d) R89C1 = {18/27/36/45}, no 9
e) R89C9 = {29/38/47/56}, no 1
f) 20(3) cage at R4C1 = {389/479/569/578}, no 1,2
g) 9(3) cage at R5C2 = {126/135/234}, no 7,8,9
h) 22(3) cage at R5C6 = {589/679}
i) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9
j) 24(3) cage at R9C2 = {789}
k) 10(3) cage at R9C6 = {127/136/145/235}, no 8,9
l) 14(4) cage at R3C1 = {1238/1247/1256/1346/2345}, no 9
1a. Naked triple {789} in 24(3) cage at R9C2, locked for R9, clean-up: no 1,2 in R8C1, no 2,3,4 in R8C9
1b. 22(3) cage at R5C6 = {589/679}, 9 locked for R5
1c. 25(4) cage at R7C6 = {2689/3589/3679/4678} (cannot be {1789} which clashes with R9C4, cannot be {4579} which clashes with R78C4), no 1
1d. Killer triple 7,8,9 in 25(4) cage and R9C4, locked for N8
1e. 45 rule on C9 2 innies R37C9 = 10 = {19/28/37/46}, no 5
2a. 45 rule on C456789 3 innies R159C4 = 19 = {289/379/469/478/568}, no 1
2b. Max R5C4 = 6 -> min R19C4 = 13, no 2,3 in R1C4
2c. 1 in N5 only in R4C456 + R5C5 + R6C456, locked for 45(9) cage at R3C5
2d. 45 rule on C789 3 outies R159C6 = 12
2e. Min R59C6 = 6 -> max R1C6 = 6
3a. 45 rule on N9 3(2+1) outies R6C78 + R9C6 = 8
3b. Min R6C78 = 3 -> max R9C6 = 5
3c. Max R6C78 = 7, no 7,8,9 in R6C78
3d. 45 rule on N69 4(2+2) outies R3C89 + R59C6 = 13
3e. Min R59C6 = 6 -> max R3C89 = 7, no 7,8,9 in R3C89, clean-up: no 1,2,3 in R7C9 (step 1e)
3f. Min R7C9 = 4 -> max R6C78 + R7C8 = 11, no 9 in R7C8
4a. 45 rule on N3 3(2+1) outies R1C6 + R4C78 = 15
4b. Max R4C78 = 13 -> min R1C6 = 2
4c. R159C6 = 12 (step 2d) = {129/138/147/246/345} (cannot be {156/237} which clash with R23C6)
4d. R5C6 = {56789} -> no 5,6 in R19C6
4e. Max R1C6 = 4 -> min R1C78 = 11, no 1 in R1C78
4f. 45 rule on N3 2 innies R3C89 = 1 outie R1C6 + 1
4g. Max R1C6 = 4 -> max R3C89 = 5, no 5,6 in R3C89, clean-up: no 4 in R7C9 (step 1e)
4h. Max R3C89 = 5 -> min R4C78 = 11, no 1 in R4C78
4i. 10(3) cage at R9C6 = {136/145/235}
4j. 15(4) cage at R6C7 = {1239/1248/1257/1347/1356/2346}
4k. R7C9 = {6789} -> no 6,7,8 in R6C78 + R7C8
5a. 45 rule on N2 3 innies R1C46 + R3C5 = 17 = {269/278/359/368/458/467}
5b. R1C6 = {234} -> no 2,3,4 in R1C4 + R3C5
6a. R89C9 = {29/38/47} (cannot be {56} which clashes with 10(3) cage), no 5,6
6b. 12(3) cage at R4C9 = {129/138/147/156/246} (cannot be {237} which clashes with R89C9, cannot be {345} which clashes with R12C9)
6e. 6 in C9 only in R37C9 = [46] (step 1e) or 12(3) cage = {156/246} -> 12(3) cage = {129/138/156/246} (cannot be {147}, locking-out cages), no 7
[This result can also be reached using 5 in C9.]
7a. 45 rule on N9 2 innies R7C89 = 1 outie R9C6 + 7
7b. Max R9C6 = 4 -> max R7C89 = 11 but cannot be [56] which clashes with 10(3) cage at R9C6 -> no 5 in R7C8
8a. 12(3) cage at R4C9 (step 6e) = {129/138/156/246}
8b. Consider combinations for 17(3) cage at R7C7 = {179/269/359/368/458/467} (cannot be {278} which clashes with R89C9)
17(3) cage = {179/269/359/467} => 8 in R78C9, locked for C9
or 17(3) cage = {368/458} => R78C9 = {79} (hidden pair in N9), locked for C9 => R12C9 = {48}, locked for C9
-> 12(3) cage = {129/156/246}, no 3,8
9a. Consider permutations for R89C9 = [74/83/92]
R89C9 = [74], no 8 in R12C9 => R7C9 = 8 (hidden single in C9), R3C9 = 2 (step 1e), naked triple {156} in 12(3) cage at R4C9, 1 locked for N6, R6C78 + R7C8 = 7 = {124} => R7C8 = 1, 1 in N3 only in 17(3) cage at R2C7 = {179}, 9 locked for N3
or R89C9 = [83/92] => R12C9 = {48/57} (cannot be {39} which clash with R89C9
-> R12C9 = {48/57}
9b. R89C9 = [83/92] (cannot be [74] which clashes with R12C9), no 4,7
9c. 45 rule on R9 3 innies R9C159 = 11 = {236/245} (cannot be {146} because R9C9 only contains 2,3), no 1, 2 locked for R9, clean-up: no 8 in R8C1
9d. 17(3) cage at R2C7 = {179/269/359/368} (cannot be {278/458/467} which clash with R12C9), no 4
9e. R6C78 + R9C6 = 8 (step 3a)
9f. R9C6 = {134} -> R6C78 = 4,5,7 = {13/23/34} (cannot be {14/25} which clash with 12(3) cage at R4C9), no 5, 3 locked for R6, N6 and 15(4) cage at R6C7
9g. 15(4) cage at R6C7 contains 3 = {1239/1347/2346}, no 8, clean-up: no 2 in R3C9 (step 1e)
9h. Consider placement for 3 in C9
R3C9 = 3 => R7C9 = 7 (step 1e) => 15(4) cage = {1347}, no 2 => R6C78 cannot total 5 => no 3 in R9C6
or R9C9 = 3
-> R9C6 = {14}, R6C78 = {13/34}, no 2
9i. Killer pair 1,4 in 12(3) cage and R6C78, locked for N6
9j. Killer pair 1,4 in R78C4 and R9C6, locked for N8
9k. R6C78 + R9C6 = {13}4/{34}1, CPE no 1,4 in R6C6
9l. R9C159 = {236/245}
9m. 4 of {245} must be in R9C1 -> no 5 in R9C1, clean-up: no 4 in R8C1
9n. 4 in 45(9) cage at R3C5 only in R4C456 + R5C5 + R6C45, locked for N5
10a. 45 rule on N8 3 innies R7C5 + R9C46 = 14 = {158/167/248/347} (cannot be {149} because 1,4 only in R9C6, cannot be {239/257/356} because R9C6 only contains 1,4), no 9
10b. 9 in R9 only in R9C23, locked for N7
10c. R159C4 (step 2a) = {289/379/568}, R9C4 = {78} -> no 7,8 in R1C4
11a. R3C89 = R1C6 + 1 (step 4f), max R1C6 = 4 -> max R3C89 = 5
11b. Consider placement 2 in N6
2 in R4C78 => 16(4) cage at R3C8 can only be {14}{29} (with max R3C89 = 5), no 3
or 2 in 12(3) cage at R4C9, locked for C9 => R9C9 = 3
-> no 3 in R3C9, clean-up: no 7 in R7C9 (step 1e)
[It gets easier from here]
11c. 7 in C9 only in R12C9 = {57}, locked for N3, 5 locked for C9
11d. 12(3) cage at R4C9 (step 8b) = {129/246}, 2 locked for C9 and N6, R9C9 = 3 -> R8C9 = 8, clean-up: no 6 in R8C1
11e. 10(3) cage at R9C6 = {145} (only remaining combination), no 6, 4,5 locked for R9, 5 locked for N9, clean-up: no 5 in R8C1
11f. 15(4) cage at R6C7 (step 9g) = {1239/2346} -> R7C8 = 2, clean-up: no 4 in R8C4
11g. 2 in 45(9) cage at R3C5 only in R4C456 + R5C5 + R6C456, locked for N5
11h. 16(4) cage at R3C8 = {1357/1456} (cannot be {1348} because 1,3,4 only in R3C89), no 8,9, 5 locked for R4 and N6
11i. 8 in N6 only in 22(3) cage at R5C6 = {589} -> R5C6 = 5, R5C78 = {89}, 8 locked for R5, 9 locked for N6
11j. R159C6 (step 2d) = 12, R5C6 = 5 -> R19C6 = 7 = [34], clean-up: no 2 in R8C4
11k. Naked pair {15} in R78C4, locked for C4, 5 locked for N8
11l. 17(3) cage at R7C7 contains 4,7 for N9 = {467}, no 1,9
11m. R7C9 = 9 (hidden single in N9) -> R3C9 = 1 (step 1e), clean-up: no 7 in R2C6
11n. Naked triple {246} in 12(3) cage at R4C9, 4,6 locked for N6
11o. R4C78 = {57}, R3C9 = 1 -> R3C8 = 3, R6C78 = [31], R9C78 = [15], R4C78 [57]
11p. R1C6 = 3 -> R1C78 = 12 = {48}, locked for R1, 8 locked for N3
11q. R159C4 (step 10c) = {379} = [937]
11r. R1C46 + R3C5 = 17 (step 5a), R1C46 = [93] -> R3C5 = 5
11s. R7C5 + R9C46 = 14 (step 10a), R9C46 = [74] -> R7C5 = 3
11t. R7C6 = 8 (hidden single in N8)
12a. 45 rule on N7 2 innies R7C12 = R9C4 + 1
12b. R9C4 = 7 -> R7C12 = 8 = {17}, locked for R7 and N7 -> R78C4 = [51], R7C3 = 4
12c. R8C1 = 3 -> R9C1 = 6, R9C5 = 2
12d. 2 in C1 only in R123C1, locked for N1
12e. R1C4 = 9 -> R1C23 = 8 = {17}, locked for R1 and N1 -> R1C159 = [265], clean-up: no 2 in R23C6
12f. R7C1 = 1 (hidden single in C1) -> R7C2 = 7, R1C23 = [17]
12g. 3 in N4 only in R4C23 -> 14(4) cage at R3C1 = {1346} (only remaining combination, cannot be {1238} because 1,2,3 only in R4C23) -> R3C12 = [46], R4C23 = [31]
12h. R5C4 = 3 -> R5C23 = 6 = [42]
12i. 20(3) cage at R4C1 = {578} = [875]
and the rest is naked singles.
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