Prelims
a) R12C1 = {19/28/37/46}, no 5
b) R2C78 = {16/25/34}, no 7,8,9
c) R34C8 = {19/28/37/46}, no 5
d) R67C2 = {79}
e) R8C23 = {29/38/47/56}, no 1
f) R89C9 = {19/28/37/46}, no 5
g) 11(3) cage at R5C7 = {128/137/146/236/245}, no 9
h) 7(3) cage at R6C3 = {124}
i) 23(3) cage at R6C6 = {689}
j) 26(4) cage at R1C3 = {2789/3689/4589/4679/5678}, no 1
1a. Naked pair {79} in R67C2, locked for C2, clean-up: no 2,4 in R8C3
1b. Naked triple {689} in 23(3) cage at R6C6, CPE no 6,8,9 in R89C6
1c. Caged X-Wing for 9 in R67C2 and 23(3) cage at R6C6, no other 9 in R67
1d. 45 rule on C1 1 outie R9C2 = 1 innie R3C1, no 7,9 in R9C2 -> no 7,9 in R3C1
1e. 45 rule on N5 4 innies R46C46 = 25 = {1789/2689/4579/4678} (cannot be {3589/3679} because R6C4 only contains 1,2,4), no 3
1f. R6C4 = {124} -> no 1,2,4 in R4C46
1g. Min R4C4 = 5 -> max R3C45 = 9, no 9 in R3C45
1h. Min R4C6 = 5 -> max R34C7 = 8, no 8,9 in R34C7
1i. 45 rule on N12 3(1+2) outies R1C7 + R4C24 = 21
1j. Max R4C24 = 17 -> min R1C7 = 4
1k. Max R1C7 + R4C4 = 18 -> min R4C2 = 3
2a. 45 rule on C9 1 outie R1C8 = 1 innie R7C9 + 4 -> R1C8 = {56789}, R7C9 = {12345}
2b. 22(4) cage at R6C8 = {2578/3478/3568/4567} (cannot be {1678} because R7C78 = {67/68/78/678 clash with R7C2 + R7C56, killer ALS block), no 1, clean-up: no 5 in R1C8
2c. R7C78 cannot be {67/68/78} -> one of 6,7,8 must be in R6C8 = {678}
2d. Killer quad 6,7,8,9 in R7C2 + R7C56 + R7C78, locked for R7
2e. 1 in R7 only in R7C134, CPE no 1 in R9C3
2f. 39(7) cage at R1C8 must contain 3, locked for C9, clean-up: no 7 in R1C8, no 7 in R89C9
3. 45 rule on N36 2(1+1) innies R1C7 + R6C8 = 1 outie R4C6 + 10
3a. Max R1C7 + R6C8 = 17 -> max R4C6 = 7
3b. Min R4C6 = 5 -> min R1C7 + R6C8 = 15, no 4,5,6 in R1C7
3c. 26(4) cage at R1C3 = {2789/3689/4589/4679/5678}
3d. 2 of {2789} must be in R1C345 (R1C345 cannot be {789} which clashes with R1C7) -> no 2 in R2C5
3e. 3 of {3689} must be in R1C345 (R1C345 cannot be {689} which clashes with R1C8) -> no 3 in R2C5
4. 45 rule on C789 2 innies R19C7 = 2 outies R48C6 + 2
4a. Max R48C6 = 13 -> max R19C7 = 15, min R1C7 = 7 -> max R9C7 = 8
5. 45 rule on N47 2(1+1) innies R4C2 + R9C3 = 1 outie R6C4 + 7
5a. Max R6C4 = 4 -> max R4C2 + R9C3 = 11, min R4C2 = 3 -> max R9C3 = 8
5b. R789C4 cannot be {124} = 7 (which clashes with R6C4) -> no 8 in R9C3
6. 45 rule on N89 3(2+1) outies R6C68 + R9C3 = 21
6a. Max R6C68 = 17 -> min R9C3 = 4
6b. R6C68 cannot be [97] = 16 (which clashes with R6C2) -> no 5 in R9C3
6c. Min R9C3 = 4 -> max R789C4 = 11, no 9 in R89C4
7. R4C2 + R9C3 = R6C4 + 7 (step 5)
7a. Max R6C4 = 4 -> max R4C2 + R9C3 = 11, min R9C3 = 4 -> no 8 in R4C2
7b. R1C7 + R4C24 = 21 (step 1i), max R1C7 + R4C2 = 15 -> min R4C4 = 7 (since R4C24 cannot be [66])
7c. Min R4C4 = 7 -> max R3C45 = 7, no 7,8 in R3C45
7d. R4C2 + R9C3 = R6C4 + 7, R6C4 + R4C2 + R9C3 = [236/447/456] (cannot be [144] which clashes with R67C3 = {24}, cannot be [254] which clashes with R67C3 = {14}), no 1 in R6C4, no 6 in R4C2, no 4 in R9C3
7e. Naked triple {124} in 7(3) cage at R6C3, 1 locked for C3
7f. 15(4) cage at R7C4 = {1257/1347/1356/2346} (cannot be {1248} because R9C3 only contains 6,7), no 8
7g. R9C3 = {67} -> no 6,7 in R789C4
7h. 15(4) cage = {1257/1347/1356} (cannot be {2346} which clashes with R6C4), 1 locked for C4 and N8
7i. 15(4) cage = {1257/1356} (cannot be {1347} which clashes with R6C4 + R4C2 + R9C3 = [447]), no 4, 5 locked for C4 and N8
7j. Min R4C4 = 7 -> max R3C45 = 7, min R3C4 = 2 -> max R3C5 = 5
8. 45 rule on N1 2(1+1) outies R2C4 + R4C2 = 1 innie R1C3 + 7
8a. Max R2C4 + R4C2 = 14 -> max R1C3 = 7
8b. Min R1C3 = 2 -> min R2C4 + R4C2 = 9, max R4C2 = 5 -> min R2C4 = 4
9. 45 rule on N9 2(1+1) outies R6C8 + R8C6 = 1 innie R9C7 + 8
9a. Max R6C8 + R8C6 = 15 -> max R9C7 = 7
10. R46C46 (step 1e) = {2689/4579/4678}, R1C7 + R4C24 = 21 (step 1i)
10a. Consider candidates for R4C2 = {345}
R4C2 = {34} => R1C7 + R4C4 = 17,18 = [89/98/99], no 7 in R4C4 => R46C46 = {2689/4678} (cannot be {4579} = [7549]), no 5
or R4C2 = 5
-> no 5 in R4C6
10b. 13(3) cage at R3C7 cannot contain both of 6,7, R4C6 = {67} -> no 6,7 in R34C7
10c. R46C46 = {2689/4678}, 6 locked for C6 and N5
10d. 6 in N8 only in R789C5, locked for C5
11. R1C7 + R6C8 = R4C6 + 10 (step 3)
11a. R4C6 = {67} -> R1C7 + R6C8 = 16,17 = [88/97/98], no 7 in R1C7, no 6 in R6C8
11b. 19(4) cage at R1C6 cannot contain both of 8,9, R1C7 = {89} -> no 8,9 in R123C6
11c. 26(4) cage at R1C3 = {2789/4589/4679/5678} (cannot be {3689} which clashes with R1C78, ALS block), no 3
11d. 4 of {4589/4679} must be in R1C345 (R1C345 cannot be {589} which clashes with R1C7, cannot be {679} which clashes with R1C78, ALS block) -> no 4 in R2C5
11e. 5 of {4589/5678} must be in R1C345 (R1C345 cannot be {489} which clashes with R1C7, cannot be {678} which clashes with R1C78, ALS block) -> no 5 in R2C5
12. R1C7 + R4C24 = 21 (step 1i), R4C2 + R9C3 = R6C4 + 7 (step 5)
12a. R4C6 = {67} -> R1C7 + R6C8 = [88/97/98] (step 11a)
12b. Consider combinations for R46C46 (step 10c) = {2689/4678}
R46C46 = {2689}, R6C4 = 2 => R4C2 + R9C3 = [36] => R1C7 + R4C4 = 18 = [99]
or R46C46 = {4678} with R4C6 = 6 = [7648] => R1C7 + R4C2 = 14 = [95]
or R46C46 = {4678} with R4C6 = 7 => R1C7 + R6C8 = [98]
-> R1C7 = 9, clean-up: no 1 in R2C1, no 1 in R4C8
[Ed pointed out that after this placement, R3C3 becomes a hidden single for R3; don’t know how much difference this would make if it had been spotted.]
13. R1C7 = 9 -> R123C6 = 10 = {127/145/235}
13a. Consider combinations for R123C6
R123C6 = {127}, locked for C6
or R123C6 = {145} => R89C6 = {27/37} (cannot be {23} which clashes with R789C4), 7 locked for C6
or R123C6 = {235} => R89C6 = {47}, locked for C6
-> R4C6 = 6, clean-up: no 4 in R3C8
13b. R1C7 = 9, R4C6 = 6 -> R6C8 = 7 (step 12a), R67C2 = [97], R67C6 = [89], R7C5 = 6, clean-up: no 3 in R34C8, no 4 in R8C2
13c. R9C3 = 6 -> R789C4 = 9 = {135} (only remaining combination), locked for C4 and N8, clean-up: no 5 in R8C23, no 4 in R8C9
13d. R1C7 + R4C24 = 21 (step 1i), R1C7 = 9 -> R4C24 = 12 = [39/57], no 4
13e. 8 in R7 only in R7C78, locked for N9, clean-up: no 2 in R89C9
14. R123C6 (step 13) = {145/235} (cannot be {127} which clashes with R89C6, ALS block), no 7, 5 locked for C6 and N2
14a. Hidden killer pair 1,3 in R123C6 and R5C6 for C6, R123C6 contains one of 1,3 -> R5C6 = {13}
14b. Hidden killer pair 1,3 in R123C6 and R3C5 for N2, R123C6 contains one of 1,3 -> R3C5 = {13}
14c. 45 rule on N8 using R789C4 = 9, 1 remaining innie R8C6 = 1 outie R9C7 + 1 -> R8C6 + R9C7 = [21/43]
14d. R9C6 = 7 (hidden single in C6)
14e. 8 in N8 only in R89C5, locked for C5
15. 45 rule on N2 using R123C6 = 10, 2(1+1) outies R1C3 + R4C4 = 1 innie R2C4 + 5 -> no 5 in R1C3 (IOU)
15a. 26(4) cage at R1C3 = {2789/4679} -> R1C4 = {68}, R2C5 = 9, 7 locked for R1, clean-up: no 1 in R1C1, no 3 in R2C1
15b. Naked pair {68} in R1C48, locked for R1, clean-up: no 2,4 in R2C1
15c. R1C3 + R4C4 cannot total 12,13 -> no 7,8 in R2C4
15d. R1C45 = [87] (hidden pair in N2) -> R1C3 = 2, R1C8 = 6, clean-up: no 8 in R2C1, no 1 in R2C78, no 4 in R4C8
15e. Naked pair {14} in R67C3, locked for C3 and 7(3) cage at R6C4 -> R6C4 = 2
15f. R46C46 (step 10) = {2689} (only remaining combination) -> R4C4 = 9 -> R4C2 = 3 (step 13d), clean-up: no 1 in R3C8, no 8 in R8C3
15g. R4C4 = 9 -> R3C45 = 5 = [41], R2C4 = 6, R5C4 = 7
15h. R2C1 = 7 -> R1C1 = 3, R1C6 = 5
15i. R8C6 =4 (hidden single in C6), R9C7 = 3 (step 14c)
15j. R4C6 = 6 -> R34C7 = 7 = {25} (only remaining combination), locked for C7 -> R2C7 = 4, R2C8 = 3, R7C7 = 8, R1C9 = 1, clean-up: no 9 in R89C9
15k. R89C9 = [64]
15l. Naked pair {28} in R34C8, locked for C8 -> R7C89 = [52]
15m. R1C2 + R2C4 = [46] = 10 -> R2C23 = 9 = [18], R45C3 = [75], R5C2 = 2 (cage sum)
and the rest is naked singles.