Prelims
a) R45C9 = {14/23}
b) R6C56 = {39/48/57}, no 1,2,6
c) R78C6 = {19/28/37/46}, no 5
d) R7C78 = {17/26/35}, no 4,8,9
e) R8C78 = {13}
f) 20(3) cage at R3C8 = {389/479/569/578}, no 1,2
g) 8(3) cage at R5C4 = {125/134}
h) 41(7) cage at R4C2 = {2456789}, no 1,3
Steps resulting from Prelims
1a. Naked pair {13} in R8C78, locked for R8 and N9, clean-up: no 7,9 in R7C6, no 5,7 in R7C78
1b. Naked pair {26} in R7C78, locked for R7 and N9, clean-up: no 4,8 in R8C6
1c. 8(3) cage at R5C4 = {125/134}, 1 locked for C4
2. 45 rule on C9 2 outies R26C8 = 15 = {69/78}
3. 45 rule on C89 4 innies R1789C8 = 10 = {1234} -> R7C8 = 2, R7C7 = 6, placed for D\, R9C8 = 4, naked pair {13} in R18C8, locked for C8
3a. R9C8 = 4 -> R9C67 = 13 = [58/67/85]
3b. 9 in N9 only in R789C9, locked for C9 and 31(5) cage at R6C8, no 9 in R6C8, clean-up: no 6 in R2C8 (step 2)
3c. Min R6C8 + R789C9 = 6{579} = 27 -> max R6C9 = 4
3d. 6 in C9 only in R123C9, locked for N3
4. Naked pair {13} in R18C8, CPE no 1,3 in R1C1 using D/ R1C1
4a. 14(3) cage at R1C6 cannot contain both of 1,3, R1C8 = {13} -> no 1,3 in R1C67
5. 41(7) cage at R4C2 = {2456789}, 6 locked for N4
5a. 41(7) cage at R4C2 = {2456789}, CPE no 2 in R4C1
6. 45 rule on C1 2 innies R19C1 = 1 outie R5C2 + 10
6a. Max R19C1 = 17 -> max R5C2 = 7
6b.Min R19C1 = 11, no 1 in R9C1
7. 45 rule on C6789 2 outies R56C5 = 1 innie R3C6 + 10
7a. Max R56C5 = 17 -> max R3C6 = 7
7b. Min R56C5 = 11, no 1 in R5C5
8. 45 rule on N8 2 innies R7C4 + R9C6 = 1 outie R9C3 + 7
8a. Max R7C4 + R9C6 = 13 -> max R9C3 = 6
9. 45 rule on N1 4(1+3) outies R2C4 + R4C1 + R5C12 = 11
9a. Min R4C1 + R5C12 = 6 -> max R2C4 = 5
9b. Min R2C4 = 2 -> max R4C1 + R5C12 = 9, no 7,8,9 in R4C1 + R5C12
9c. Max R4C1 + R5C12 = 9 -> min R23C1 = 12, no 1,2 in R23C1
10. 45 rule on C123 2 outies R24C4 = 1 innie R9C3 + 5
10a. Min R24C4 = 7 (cannot be {24} which clashes with 8(3) cage at R5C4) -> min R9C3 = 2
11. 28(6) cage at R7C5 = {123589/123679/124579/124678/134569/134578} (cannot be {234568} which clashes with R9C6), 1 locked for C5 and N8, clean-up: no 9 in R8C6
11a. 9 in N8 only in 28(6) cage = {123589/123679/124579/134569}
11b. Variable hidden killer pair 2,7 in 28(6) cage and R8C6 for N8, R8C6 cannot contain both of 2,7 -> 28(6) cage must contain at least one of 2,7 = {123589/123679/124579}
11c. 3 of {123589} must be in R9C3 (otherwise combination clashes with R78C6 = [37]), 2 of {124579} must be in R9C3 (otherwise clashes with R78C6 = [82]), no 5 in {123679} -> no 5 in R9C3
12. 8(3) cage at R5C4 = {125/134}, 1 locked for N5
12a. 5 of {125} must be in R7C4 -> no 5 in R56C4
12b. 1 in N2 only in R23C6, CPE no 1 in R3C7
12c. 1 on D/ only in R1C9 + R6C4, CPE no 1 in R6C9
13. 45 rule on N7 2 innies R79C3 = 1 outie R6C1 + 2
13a. Min R79C3 = 6 -> min R6C1 = 4
14. 1,3 in N4 only in R4C1 + R5C12, locked for 21(5) cage at R2C1, no 3 in R23C1
14a. R2C4 + R4C1 + R5C12 = 11 (step 9)
14b. R4C1 + R5C12 = {123/134/135} = 6,8,9 -> R2C4 = {235}
15. R24C4 = R9C3 + 5 (step 10)
15a. R9C3 = {236} -> R24C4 = 7,8,11 = {25/34/35/29/38}, no 7 in R4C4
16. 6 in C9 only in 24(4) at R1C9 = {1689/2679/3678/4569}
16a. 1 in C9 only in 24(4) cage or in R45C1 = {14} -> 24(4) cage = {1689/2679/3678} (cannot be {4569}, locking-out cages), no 4,5
16b. 4 in C9 only in R456C9, locked for N6
16c. 5 in C9 only in R789C9, locked for N9, clean-up: no 8 in R9C6 (step 3a)
17. 28(6) cage at R7C5 (step 11b) = {123589/123679/124579} contains 5 or 6
17a. All cells of 28(6) cage “see” R9C6 -> 28(6) cage + R9C6 must contain 5, locked for N8
18. 8(3) cage at R5C4 = {134} (only remaining combination), locked for C4
19. R2C4 = {25} -> R4C1 + R5C12 (step 14b) = {123/135}, no 4
19a. R4C1 + R5C12 = {123/135} = 6,9 -> R23C1 = 12,15 = {48/69/78} (cannot be {57} when R4C1 + R5C12 = {135}), no 5 in R23C1
20. R24C4 (step 15a) = {25/29} -> R9C3 = {26}
20a. R24C4 = {25/29}, 2 locked for C4
21. Hidden killer pair 2,8 in 28(6) cage at R7C5 and R78C6 for N8, R78C6 must contain both or neither of 2,8 -> 28(6) cage must contain both or neither of 2,8 in N8 -> 28(6) cage (step 11b) = {123679/124579} (cannot be {123589} which cannot contain both of 2,8 in N8 because it doesn’t contain 6), no 8
21a.R7C6 = 8 (hidden single in N8), R8C6 = 2, clean-up: no 4 in R6C5
21b. 28(6) cage = {123679/124579} -> R9C3 = 2
22. R24C4 = R9C3 + 5 (step 10)
22a. R9C3 = 2 -> R24C4 = 7 = {25}, locked for C4
22b. 8 in C4 only in R13C4, locked for N2
22c. 41(7) cage at R4C2 = {2456789}, 8 locked for N4
23. 1,3 in C3 only in R123C3, locked for N1
23a. R2C4 = {25} -> 15(4) cage at R1C3 = {1239/1356}, no 4,7,8
23b. 5 of {1356} must be in R2C4 -> no 5 in R123C3
24. R23C1 (step 19a) = {48/78} (cannot be {69} which clashes with 15(4) cage at R1C3), 8 locked for C1 and N1
25. 8 on D\ only in R5C5 + R9C9, CPE no 8 in R1C9 using D/
26. 45 rule on N7 3 remaining innies R7C13 + R8C1 = 14 = {149/167/347/356}
26a. 1,3 only in R7C1 -> R7C1 = {13}
26b. 6 of {356} must be in R8C1 -> no 5 in R8C1
26c. 14(3) cage at R6C1 = {149/167/356} (cannot be {347} which clashes with R23C1, ALS block)
26d. 6 of {167} must be in R8C1 -> no 7 in R8C1
27. R4C1 + R5C12 = {123/135} (step 19)
27a. Killer pair 1,3 in R45C1 and R7C1, locked for C1
27b. Hidden killer pair 1,3 in R45C1 and R7C1 for C1, R7C1 = {13} -> R45C1 cannot contain both of 1,3 -> R5C2 = {13}
28. R19C1 = R5C2 + 10 (step 6)
28a. R5C12 = {13} -> R19C1 = 11,13 = [29/47/56/49/76], no 9 in R1C1, no 5 in R9C1
29. 2 in C1 only in R15C1, CPE no 2 in R5C5 using D\
29a. 2 in N5 only in R4C45, locked for R4, clean-up: no 3 in R5C9
29b. 2 on D/ only in R1C9 + R3C7, locked for N3
30. R9C25 = {13} (hidden pair in R9)
30a. Naked pair {13} in R7C1 + R9C2, locked for N7
31. R9C67 = [58] (cannot be [67] which clashes with R9C14, ALS block), clean-up: no 7 in R6C5
31a. Naked triple {579} in R789C9, locked for 31(5) cage at R6C8, no 7 in R6C8, clean-up: no 8 in R2C8 (step 2)
31b. R789C9 = {579} = 21 -> R6C89 = 10 = [64/82]
31c. Killer pair 2,4 in R45C9 and R6C9, locked for C9 and N6
31d. 8 in C9 only in R123C9, locked for N3
32. R3C7 = 2 (hidden single in N3)
33. Hidden killer pair 1,3 in 24(4) cage at R1C9 and R45C9 for C9, R45C9 contains one of 1,3 -> 24(4) cage must contain one of 1,3
33a. Killer pair 1,3 in R1C8 and 24(4) cage, locked for N3
34. R5C5 = 8 (hidden single on D\), placed for D/, clean-up: no 4 in R6C6
34a. R8C3 = 8 (hidden single in N7)
[I’ve been a bit slow at spotting some things, I really ought to have spotted this earlier.]
35. 5 in C3 only in R4567C3, locked for 41(7) cage at R4C2, no 5 in R46C2 + R4C4
35a. R4C4 = 2, placed for D\
36. R2C4 = 5 -> R123C3 = 10 = {136}, locked for C3 and N1
37. 45 rule on N1 2 remaining innies R23C1 = 15 = {78}, locked for C1 and N1
37a. R23C1 = 15 -> R4C1 + R5C12 = 6 = {123} -> R5C1 = 2, R4C1 = {13}, clean-up: no 3 in R4C9
37b. Naked pair {14} in R45C9, locked for C9 and N6 -> R6C9 = 2, R6C8 = 8 (step 31b), R2C8 = 7 (step 2), placed for D/
38. R1C8 = 1 (hidden single in N3), R8C8 = 3, placed for D\, R8C7 = 1, R3C3 = 1
39. R2C6 = 1 (hidden single in N2), R3C7 = 2 -> R24C7 = 14 = [95], R1C7 = 4, R1C6 = 9 (cage sum), R6C6 = 7, placed for D\, R6C5 = 5, R56C7 = [73]
40. R9C9 = 9, R9C1 = 6, placed for D/, R1C9 = 3, placed for D/, R4C6 = 4, placed for D/
and the rest is naked singles, without using the diagonals.
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