Prelims
a) R23C5 = {59/68}
b) R3C23 = {15/24}
c) R3C78 = {18/27/36/45}, no 9
d) R4C67 = {18/27/36/45}, no 9
e) R6C34 = {29/38/47/56}, no 1
f) R6C78 = {49/58/67}, no 1,2,3
g) R7C23 = {13}
h) R78C5 = {18/27/36/45}, no 9
i) R7C78 = {49/58/67}, no 1,2,3
j) 21(3) cage at R1C6 = {489/579/678}, no 1,2,3
k) 20(3) cage at R7C4 = {389/479/569/578}, no 1,2
l) 14(4) cage at R1C3 = {1238/1247/1256/1346/2345}, no 9
m) 13(4) cage at R8C7 = {1237/1246/1345}, no 8,9
n) 26(4) cage at R8C3 = {2789/3689/4589/4679/5678}, no 1
Steps Resulting From Prelims and Immediate Placements
1a. Naked pair {13} in R7C23, locked for R7 and N7, clean-up: no 6,8 in R8C5
1b. R3C78 = {18/27/36} (cannot be {45} which clashes with R3C23), no 4,5
1c. 13(4) cage at R8C7 = {1237/1246/1345}, 1 locked for N9
1d. 45 rule on N3 1 outie R4C9 = 1 -> R2C89 + R3C9 = 20 = {389/479/569/578}, no 2, clean-up: no 8 in R4C67
1e. 45 rule on N7 1 outie R6C1 = 1
2. 45 rule on N8 1 outie R9C7 = 1 innie R7C6 + 6 -> R7C6 = 2, R9C7 = 8, clean-up: no 1 in R3C8, no 7 in R4C7, no 5 in R6C8, no 7 in R78C5, no 5 in R7C78
2a. R7C6 = 2 -> R6C56 = 11 = {38/47/56}, no 9
2b. 45 rule on N9 2 remaining innies R78C9 = 11 = [56/65/92] (cannot be {47} which clashes with R7C78), no 4,7, no 3,9 in R8C9
2c. 13(4) cage at R8C7 = {1237/1345} (cannot be {1246} which clashes with R78C9), no 6
2d. 9 in N9 only in R7C789, locked for R7
2e. 45 rule on R6 2 remaining innies R6C29 = 9 = {27/36/45}, no 8,9
2f. 9 in R6 only in R6C34 = {29} or R6C78 = {49} -> R6C34 = {29/38/56} (cannot be {47} locking-out cages), no 4,7
[2 in R6 only in R6C29 and R6C34 would give the same elimination.]
3a. 45 rule on N2 4 innies R1C45 + R23C4 = 10 = {1234}, 4 locked for N2
3b. 45 rule on N2 1 outie R1C3 = 1 innie R3C4 + 4 -> R1C3 = {5678}
3c. 7 in N2 only in 21(3) cage at R1C6, locked for C6, clean-up: no 2 in R4C7, no 4 in R6C5 (step 2a)
3d. 17(3) cage at R3C4 can only contain one of 1,2,3,4 -> no 2,3,4 in R4C45
4. 45 rule on R1 3 innies R1C126 = 2 outies R2C47 + 15
4a. Max R1C126 = 24 -> max R2C47 = 9, no 9 in R2C7
5. 16(4) cage at R1C7 = {1249/1258/1348/1456/2347/2356} (cannot be {1267/1357} which clash with R3C78)
5a. 7 on {2347} must be in R1C789 (R1C789 cannot be {234} which clash with R1C45, ALS block), no 7 in R2C7
6. 26(4) cage at R8C3 = {2789/4589/4679/5678}
6a. 8 of {2789/4589/5678} must be in R8C3 -> no 2,5 in R8C3
7. R23C5 = {59/68}, R78C5 = {45}/[63/81] -> combined cage = {59}[63]/{59}[81]/{68}{45}, 5 locked for C5, clean-up: no 6 in R6C6 (step 2a)
7a. 22(4) cage at R8C6 = {1489/1678/3478/3568}
7b. 7 of {3478} must be in R9C5, 6 of {3568} must be in R9C5 (R89C6 cannot be {56} which clashes with 21(3) cage at R1C6), no 3 in R9C5
7c. 22(4) cage = {1489/1678/3478} (cannot be {3568} because R89C6 = {35} clashes with 21(3) cage at R1C6 + R46C6, killer ALS block), no 5 in R89C6
7d. Consider combinations for R23C5 = {59/68}
R23C5 = {59} => 21(3) cage at R1C6 = {678}, 6,8 locked for C6, R4C6 = {345}, R6C6 = {345} => R89C6 cannot be {34}, ALS block
or R23C5 = {68}, locked for C5 => R78C5 = {45}, 4 locked for N8
-> 22(4) cage = {1489/1678} (cannot be {3478}), no 3 in R89C6, 1 locked for N8, clean-up: no 8 in R7C5
[My original step 7d was flawed. Thanks Ed for suggesting an alternative way, which I’ve rewritten in my solving style.]
7e. 3 in C6 only in R456C6, locked for N5, clean-up: no 8 in R6C3, no 8 in R6C6 (step 2a)
7f. Killer pair 5,6 in R23C5 and R78C5, locked for C5, clean-up: no 5 in R6C6 (step 2a)
7g. 8 in N8 only in 20(3) cage at R7C4, locked for C4, clean-up: no 3 in R6C3
7h. 20(3) cage = {389/578}, no 4,6
7i. Killer triple 7,8,9 in R23C5, R4C5 and R6C5, locked for C5
7j. 22(4) cage = {1489} (only remaining combination), 4,9 locked for N8, clean-up: no 5 in R78C5
7k. R78C6 = [63], clean-up: no 8 in R23C5, no 7 in R7C78, no 5 in R8C9 (step 2b)
7l. Naked pair {59} in R23C5, locked for C5 and N2
7m. Naked triple {678} in 21(3) cage at R1C6, 6,8 locked for C6, clean-up: no 3 in R4C7
7n. Naked triple {578} in 20(3) cage at R7C4, 5,7 locked for C4, clean-up: no 6 in R6C3
7o. 9 in C4 only in R456C4, locked for N5
7p. Naked pair {49} in R7C78, locked for R7 and N9, clean-up: no 2 in R8C9 (step 2b)
7q. R78C9 = [56], clean-up: no 3,4 in R6C2 (step 2e)
7r. 16(4) cage at R6C1 = {1258} (only possible combination, cannot be {1249} because R7C1 doesn’t contain 4,9) -> R7C1 = 8, R8C12 = {25}, locked for R8 and N7
7s. Naked pair {17} in R8C78, locked for R8 and N9
7t. R2C89 + R3C9 (step 1d) = {389/479/578} (cannot be {569} because 5,6 only in R2C8), no 6
[Note. There is now the Unique Rectangle elimination R6C78 cannot be {49} because R7C78 = {49} but I don’t use that type of step since it doesn’t fully solve a puzzle.]
8. 17(3) cage at R3C4 = [197/368/467] (cannot be {269} because no 7,8 in R4C5, cannot be {278} because no 7,8 in R4C4), no 2 in R3C4, clean-up: no 6 in R1C3 (step 3b)
8a. Consider permutations for 17(3) cage
17(3) cage = [179] => R6C5 = 8, R6C6 = 3 (cage sum) => R4C67 = {45} (only remaining combination)
or 17(3) cage = [368/467] => R4C67 = {45} (only remaining combination)
-> R4C67 = {45}, locked for R4
[This proved to be a key step.]
8b. 4 in N4 only in R5C123, locked for R5
8c. 4 in N5 only in R46C6, locked for C6 -> R89C6 = [91], R9C5 = 4, R8C3 = 4, clean-up: no 2 in R3C2
8d. 4 in N4 only in R5C12, locked for 31(6) cage at R2C1, no 4 in R23C1
9. 45 rule on N1 3 innies R1C3 + R23C1 = 17 = {278/359/368} (cannot be {269} because R1C3 only contains 5,7,8)
9a. 5 of {359} must be in R1C3 -> no 5 in R23C1
9b. 8 of {278} must be in R1C3 -> no 7 in R1C3, clean-up: no 3 in R3C4 (step 3b)
9c. 17(3) cage at R3C4 (step 8) = [197/467] -> R4C5 = 7, R6C5 = 8, R6C6 = 3 (cage sum), R5C6 = 5, R4C67 = [45], clean-up: no 6 in R6C2 (step 2e)
9d. Killer pair 1,4 in R3C23 and R3C4, locked for R3, clean-up: no 8 in R3C8
9e. Killer pair 3,7 in R2C89 + R3C9 (step 7t) and R3C78, locked for N3
9f. 1 in N3 only in 16(4) cage at R1C7 (step 5) = {1249/1456} (cannot be {1258} which clashes with R1C3), no 8, 4 locked for N3
9g. 5 of {1456} must be in R1C8 -> no 6 in R1C8
9h. R2C89 + R3C9 (step 7t) = {389/578}
9i. 5 of {578} must be in R2C8 -> no 7 in R2C8
10. 1,5 in R5 only in 22(5) cage at R5C3 = {12568/13567}, no 9, 6 locked for R5
10a. 8 of {12568} only in R5C3 -> no 2 in R5C3
11. R1C3 + R23C1 (step 9) = {278/359/368}
11a. 4 in N4 only in 31(6) cage at R2C1 = {234589/234679}
11b. {234589} only has one of 2,7, {234679} must have one of 2,7 in R6C2 so cannot have both in R23C1 -> R1C3 + R23C1 = {359/368} (cannot be {278} because cannot have both of 2,7 in R23C1), no 2,7, 3 locked for N1 and 31(6) cage, no 3 in R45C1 + R5C2
[Cracked.]
11c. 31(6) cage = {234589/234679}, 2 locked for N4, clean-up: no 9 in R6C4
11d. R4C4 = 9 (hidden single in N5) -> R3C4 = 1 (cage sum), R1C5 = 2, R1C3 = 5 (step 3b), R3C23 = [42], R6C3 = 9 -> R6C4 = 2, clean-up: no 7 in R3C78, no 4 in R6C78
[Removing any need for the Unique Rectangle.]
11e. Naked pair {67} in R6C78, locked for R6 and N6 -> R16C9 = [94]
11f. Naked pair {36} in R3C78, locked for R3 and N3
11g. Naked pair {78} in R23C9, 8 locked for C9 and N3 -> R2C8 = 5
11h. Naked pair {14} in R1C78, locked for R1 and N3 -> R2C7 = 2
11i. R5C4567 = [6153] -> R5C3 = 7 (cage sum)
and the rest is naked singles.