Prelims
a) R1C23 = {69/78}
b) R23C9 = {19/28/37/46}, no 5
c) R3C45 = {49/58/67}, no 1,2,3
d) R5C12 = {39/48/57}, no 1,2,6
e) R5C34 = {17/26/35}, no 4,8,9
f) R5C89 = {59/68}
g) R7C45 = {29/38/47/56}, no 1
h) R78C9 = {59/68}
i) R9C23 = {14/23}
j) 7(3) cage at R6C8 = {124}
k) 29(7) cage at R1C5 = {1234568}, no 7,9
l) 42(7) cage at R8C7 = {3456789}, no 1,2
1. Naked triple {124} in 7(3) cage at R6C8, CPE no 1,2,4 in R4C8
1a. R7C78 = {12} (hidden pair in N9), locked for R7, clean-up: no 9 in R7C45
1b. Naked triple {124} in 7(3) cage at R6C8, 4 locked for R6 and N6
1c. 45 rule on N9 2 outies R9C56 = 14 = {59/68}
1d. R7C45 = {38/47} (cannot be {56} which clashes with R9C56), no 5,6 in R7C45
2. R5C89 + R78C9 form naked quad {5689} (because R5C89 and R78C9 cannot have the same combinations), CPE no 5,6,8,9 in R4C9
3. 45 rule on R123 1 innie R3C8 = 1 outie R4C1 + 6, R3C8 = {789}, R4C1 = {123}
3a. 45 rule on R123 3 outies R4C189 = 10 = {127/136/235}, no 8,9
3b. 1,2 of {127} must be in R4C19 -> no 7 in R4C8
3c. 5,6,7 only in R4C8 -> R4C8 = {567}
4. 45 rule on R1234 3 innies R4C567 = 19 = {289/379/469/478/568}, no 1
5. 45 rule on R5 3 innies R5C567 = 11 = {128/137/146/236/245}, no 9
6. 45 rule on R789 1 outie R6C1 = 1 innie R7C8 + 5, R7C8 = {12} -> R6C1 = {67}
7. 45 rule on N1 2 innies R23C3 = 1 outie R4C1 + 5
7a. Max R4C1 = 3 -> max R23C3 = 8, no 8,9 in R23C3
8. 45 rule on N7 2 innies R78C3 = 1 outie R6C1 + 6
8a. Min R6C1 = 6 -> min R78C3, no 1,2 in R8C3
9. 45 rule on N7 4(1+3) outies R6C1 + R8C45 + R9C4 = 16
9a. Min R6C1 = 6 -> max R8C45 + R9C4 = 10, no 8,9 in R8C45 + R9C4
10. 12(3) cage at R7C6 = {129/138/147/246} (cannot be {156} which clashes with R9C56, cannot be {237} which clashes with R7C45, cannot be {345} because R7C7 only contains 1,2), no 5
10a. 9 of {129} must be in R7C6 -> no 9 in R8C6
[Step 11 may not have been necessary, if I’d spotted steps 13 and 14 earlier, but it contains obvious steps so I’ve kept it in.]
11. 9 in R5 only in R5C12 = {39} or in R5C89 = {59} -> R5C12 = {39/48} (cannot be {57}, locking-out cages), no 5,7 in R5C12
11a. Killer pair 8,9 in R5C12 and R5C89, locked for R5
11b. 9 in R5 only in R5C12 = {39} or in R5C89 = {59} -> R5C34 = {17/26} (cannot be {35}, locking-out cages), no 3,5 in R5C34
11c. R5C567 (step 5) = {137/236/245} (cannot be {146} which clashes with R5C34)
12. 45 rule on C6789 2 outies R19C5 = 9 = [18/36/45], clean-up: no 5 in R9C6 (step 1c)
12a. 9 in N8 only in R78C6, locked for C6
[With hindsight I ought to have looked at the combinations for the large cages in C12 earlier.]
13. 25(6) cage at R1C1 = {123469/123478} (cannot be {123568/124567} which clash with R1C23), no 5, 4 locked for N1
13a. Killer pair 6,7 in 25(5) cage and R1C23, locked for N1
13b. 5 in N1 only in R23C3, locked for C3 and 21(5) cage at R4C1, no 5 in R1C4 + R2C45
14. 34(6) cage at R6C1 = {145789/235789} (cannot be {136789/245689/345679} which clash with R9C23), no 6 -> R6C1 = 7, R7C8 = 2 (step 6), R7C7 = 1, clean-up: no 1 in R5C4
14a. R78C3 = {67} (hidden pair in N7), locked for C3 and 22(5) cage at R7C3, no 6,7 in R8C45 + R9C4, clean-up: no 8,9 in R1C2, no 2 in R5C4
14b. R78C3 = {67} = 13 -> R8C45 + R9C4 = 9 = {135/234}, 3 locked for N8, clean-up: no 8 in R7C45
14c. Naked pair {47} in R7C45, locked for R7 and N8 -> R78C3 = [67], R8C45 + R9C4 = {135}, locked for N8, clean-up: no 4 in R1C5 (step 12), no 9 in R9C6 (step 1c)
14d. Naked pair {68} in R9C56, locked for N8 and 42(7) cage at R8C7 -> R78C6 = [92]
15. R78C9 = {68} (hidden pair in N9) -> R7C9 = 8, R8C9 = 6, clean-up: no 2,4 in R23C9, no 6,8 in R5C8
15a. Naked pair {59} in R5C89, locked for R5 and N6, clean-up: no 3 in R5C12
15b. Naked pair {48} in R5C12, locked for R5 and N4
15c. Naked pair {14} in R6C89, locked for R6 and N6
16. Naked pair {35} in R7C12, locked for N7, clean-up: no 2 in R9C23
16a. Naked pair {14} in R9C23, locked for R9 and N7
16b. Naked pair {89} in R8C12, locked for R8 and N7 -> R9C1 = 2
17. R1C3 = 8 (hidden single in C3), R1C2 = 7
17a. 25(6) cage at R1C1 (step 13) = {123469} (only remaining combination), 2 locked for C2 and N1
18. R4C189 (step 3a) = {127/136} -> R4C1 = 1, R3C8 = 7 (step 3), R4C8 = 6, R4C9 = 3, R5C3 = 2, R5C7 = 7, R5C4 = 6, R4C3 = 9, R4C2 = 5, R4C4 = 2 (cage sum), R6C23 = [63], R6C4 = 8 (cage sum)
18a. R4C67 = [48], R6C67 = [52], R46C5 = [79], R5C5 = 1 (cage sum), R5C6 = 3, R1C5 = 3, R9C5 = 6 (step 12), R9C6 = 8, R8C5 = 5, R89C4 = [13], R7C45 = [74], R3C5 = 8, R3C4 = 5
19. Naked pair {19} in R23C9, locked for C9 and N3 -> R5C89 = [95], NS R6C89 = [14]
and the rest is naked singles.
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