SudokuSolver Forum

Prelims

a) 28(4) cage at R1C6 = {4789/5689}, no 1,2,3
b) 14(4) cage at R8C1 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R5 1 innie R5C5 = 5, placed for NR3C5

2. 45 rule on R12 2 innies R12C1 = 6 = {15/24}

3. 45 rule on R34 2 innies R34C9 = 5 = {14/23}

4. 45 rule on R67 2 innies R67C1 = 10 = {19/28/37/46}, no 5

5. 45 rule on R89 2 innies R89C9 = 16 = {79}, locked for C9 and NR5C7

6. 28(4) cage at R1C6 = {4789/5689}, CPE no 8,9 in R1C345 + R34C7

7. 45 rule on NR6C1 using R67C1 = 10 (step 4), 3 remaining innies R67C2 + R9C3 = 21 = {489/579/678}, no 1,2,3
7a. 14(4) cage at R8C1 = {1238/1256/1346/2345} (cannot be {1247} which clashes with R67C2 + R9C3), no 7

8. Law of Leftovers (LoL) for R6789 three outies R5C789 must exactly equal three innies R6C5 + R7C45, no 7,9 in R5C789 -> no 7,9 in R6C5 + R7C45
8a. R5C789 = R6C5 + R7C45 -> R6C4 = R5C6 + 8 (difference between 23(4) and 15(4) cages) -> R6C4 = 9, R5C6 = 1, 1 placed for NR3C5, 9 placed for NR6C3, clean-up: no 1 in R7C1 (step 4)
[Is there a technically simpler way to do this step?]
8b. 15(4) cage at R5C6 = {1248} (only remaining combination), 2,4,8 locked for R5 and NR5C7
8c. R5C789 = {248} -> R6C5 + R7C45 = {248} (LoL for R6789)
8d. Naked triple {248} in R6C5 + R7C45, locked for NR3C5

9. R5C9 + R6C89 = {1356} -> 15(4) cage at R6C8 = {1356} (only combination containing at least three of 1,3,5,6)
9a. 45 rule on NR5C7 using R5C789 = {248} = 14 and R89C9 = {79} = 16, 1 outie R7C7 = 1 innie R6C8 -> R7C7

10. LoL for R1234 three outies R5C123 must exactly equal three innies R3C56 + R4C5 -> R5C4 = R4C6 + 2 (difference between 25(4) and 23(4) cages) -> R5C4 = {67}, R4C6 = {45}

11. 25(5) cage at R5C1 = {3679}, 3,9 locked for NR1C1

12. 8,9 in NR1C3 only in R1C6 + R2C67, locked for 28(4) cage at R1C6, no 8,9 in R1C7
12a. 9 in NR1C7 only in R1234C8, locked for C8

13. 45 rule on NR1C7 using R34C9 = 5 (step 3), 3 remaining innies R1C7 + R34C8 = 18 = {279/369/378/459/468/567} (cannot be {189} because no 1,8,9 in R1C7), no 1

14. LoL for C89 three outies R157C7 must exactly equal three innies R689C8, R6C8 = R7C7 (step 9a) -> R15C7 must exactly equal R89C8, no 1,3 in R15C7 -> no 1,3 in R89C8

15. 28(4) cage at R1C6 = {4789/5689} contains one of 6,7
15a. Hidden killer pair 6,7 in R1C345 + R34C7 and R1C6 + R2C67 for NR1C3, R1C345 + R34C7 must contain one of 6,7 (cannot contain both because R1C345 + R34C7 “see” all cells of the 28(4) cage) -> R1C6 + R2C67 must contain one of 6,7 -> no 6,7 in R1C7
15b. R1C7 = {45} -> no 4,5 in R1C6 + R2C67

16. LoL for C89 (step 14), no 6,7 in R15C7 -> no 6,7 in R89C8

17. R1C7 + R34C8 (step 13) = {459/468/567} (cannot be {279/369/378} because R1C7 only contains 4,5), no 2,3 in R34C8

18. Hidden killer pair 2,3 in 22(4) cage at R1C8 and R34C9 for NR1C7, R34C9 must contain both or neither of 2,3 -> 22(4) cage must contain both or neither of 2,3
18a. 22(4) cage at R1C8 = {1489/1579/1678/2389} (cannot be {2479/2569/2578/3469/3478/3568} which only contain one of 2,3, cannot be {4567} which clashes with R1C7)
18b. 22(4) cage = {1489/1678/2389} (cannot be {1579} = {79}{15} which clashes with R34C9 + R67C9, killer ALS block), no 5, 8 locked for NR1C7
18c. 3 of {2389} must be in R12C9 (R12C9 cannot be {28} which clashes with R34C9 + R5C9, killer ALS block), no 3 in R12C8

19. 5 in C9 only in R67C9, locked for 15(4) cage at R6C8 and NR5C7

20. 3 in C8 only in R67C8, locked for 15(4) cage at R6C8
20a. 3 in NR5C7 only in R7C78, locked for R7, clean-up: no 7 in R6C1 (step 4)

21. 28(4) cage at R1C6 = {4789/5689}, 22(4) cage at R1C8 (step 18b) = {1489/1678/2389}
21a. Caged X-Wing for 8 in 28(4) cage and 22(4) cage, no other 8 in R12

22. 8 in R34 only in 20(4) cage at R3C1 and 20(4) cage at R3C3, which must each contain 8

23. R1C7 + R34C8 (step 17) = {459/567}, CPE no 5 in R34C7
23a. 45 rule on NR1C7 using R34C9 = 5 (step 3), 2 outies R34C7 = 1 innie R1C7 + 4
23b. R1C7 = {45} -> R34C9 = 8,9 -> no 4 in R34C9

24. Hidden killer pair 4,5 in R1C345 and R4C6 for NR1C3, R4C6 = {45} -> R1C345 must contain one of 4,5
24a. Killer pair 4,5 in R1C345 and R1C7, locked for R1, clean-up: no 1,2 in R2C1 (step 2)

25. Hidden killer pair 1,2 in R1C345 and R34C7 for NR1C3, R34C7 cannot contain both of 1,2 (because of 22(4) cage at R3C7), R1C345 cannot contain both of 1,2 (which would clash with R1C1) -> R1C345 contains one of 1,2 and R34C7 contains one of 1,2
25a. Killer pair 1,2 in R1C1 and R1C345, locked for R1

26. R1C7 + R34C8 (step 17) = {459/567}
26a. R34C7 = R1C7 + 4 (step 23a)
26b. Consider placements for R1C7
R1C7 = 4 => R1C7 + R34C8 = {459}
or R1C7 = 5 => R34C7 = 9 = {27/36} => R1C7 + R34C8 = {459} (cannot be {567} which clashes with R34C7)
-> R1C7 + R34C8 = {459}, locked for NR1C7, clean-up: no 1 in R34C9 (step 3)
26c. Naked pair {23} in R34C9, locked for C9 and NR1C7
26d. R5C9 = 4 (hidden single in C9)
26e. 8 in C9 only in R12C9, locked for NR1C7
26f. 1 in NR1C7 only in R2C89, locked for R2
26g. 2,3 in R2 only in R2C2345, locked for NR1C2, no 2,3 in R1C2 + R3C4 + R4C234

27. 23(4) cage at R3C5 = {3479/3569}, R34C9 = {23}
27a. Caged X-Wing for 3 in 23(4) cage and R34C9, no other 3 in R34

28. 23(4) cage at R3C5 = {3479/3569}, R1C7 + R34C8 = {459} with 9 in R34C8
28a. Caged X-Wing for 9 in 23(4) cage and R1C7 + R34C8, no other 9 in R34

29. Naked quad {6789} in R1C2689, locked for R1

30. 15(4) cage at R1C4 = {1239/1257/1347/1356/2346}
30a. Killer quad 6,7,8,9 in 15(4) cage, R2C67 and R2C89, locked for R2
30b. Max R1C3 + R2C23 = 12 -> min R1C2 = 7
30c. 19(4) cage at R1C2 = {2359/3457} (cannot be {1459} which clashes with R2C1), no 1
30d. 3 in R12 only in 19(4) cage and 15(4) cage -> 15(4) cage = {1239/1347/1356/2346}
[3 in R12 is an alternative way to eliminate {1459} from 19(4) cage.]

31. R67C2 + R9C3 (step 7) = {489/579/678}
31a. 5 of {579} must be in R67C2 (R67C2 cannot be {79} which clashes with R1C2), no 5 in R9C3

32. 9 in C1 only in R57C1, 9 in C3 only in R59C3 -> grouped X-Wing for 9, no other 9 in NR1C1 and NR6C1

33. R1C2 = 9 (hidden single in C2), placed for NR1C2
33a.19(4) cage at R1C2 (step 30c) = {2359} (only remaining combination), no 4

34. 25(4) cage at R6C2 = {4678} (only remaining combination)
34a. Caged X-Wing for 4,8 in 25(4) cage and R6C5 + R7C45, no other 4,8 in R67, clean-up: no 2,6 in R67C1 (step 4)
34b. Naked quad {4678} in 25(4) cage, CPE no 4,6,7,8 in R9C3 using NR6C1 -> R9C3 = 9, placed for NR6C1, R89C9 = [97]
34c. R67C1 = 10 (step 4) -> R6C1 = 3, R7C1 = 7, both placed for NR6C1
34d. Naked quad {2468} in R7C2345, locked for R7
34e. Naked pair {13} in R7C78, locked for R7 and NR5C7 -> R67C9 = [65], R67C8 = [13], 1 placed for NR6C6, R7C6 = 9, placed for NR6C6
34f. R7C67 = [91] = 10 -> R6C67 = 7 = {25}, locked for R6 and NR6C6
34g. R2C6 = 9 (hidden single in NR1C3)
34h. R5C1 = 9 (hidden single in C1)
34h. R6C3 = 7 (hidden single in R6), placed for NR6C3

35. Naked pair {48} in R89C8, locked for C8, NR6C6 and 22(4) cage at R8C7 -> R5C78 = [82]
35a. R1C7 = 4 (hidden single in NR1C7) -> 28(4) cage at R1C6 = {4789}, 7,8 locked for C6 and NR1C3
35b. Naked pair {26} in R34C7, locked for C7 and NR1C3 -> R6C67 = [25], R9C7 = 3, placed for NR6C3, R8C7 = 7

36. R1C1 = 2 (hidden single in R1), R2C1 = 4 (step 2), both placed for NR1C1

37. R12C9 = [81], R12C6 = [78], R12C8 = [67]
37a. 19(4) cage at R1C2 (step 33a) = {2359}, 2 locked for R2

38. 2 in NR6C1 only in R89C2, locked for C2
38a. R2C3 = 2 (hidden single in R2)

39. R9C3 = 9 -> 16(4) cage at R8C3 = {1249}, locked for NR6C3, 2 locked for C4

40. R9C3 = 9 -> R78C2 = 12 (step 7) = {48}, locked for C2, NR6C1 and 25(4) cage at R6C2 -> R7C3 = 6, placed for NR6C3, R9C56 = [85], R5C3 = 3, R1C3 = 5, R2C2 = 3, R2C45 = [56], R4C6 = 4

41. Naked pair {18} in R34C3, locked for C3 and 20(4) cage at R3C3 -> R8C3 = 4

42. Naked pair {12} in R89C4, locked for C4 -> R1C45 = [31]

43. R34C3 = {18} = 9 -> R34C4 = 11 = [47]

44. 8 in C1 only in R34C1, locked for NR1C1 -> R3C3 = 1, placed for NR1C1

and the rest is naked singles, without using the nonets.