SudokuSolver Forum

I agree with you that the SS score seems a touch high. However it might be based on step 5a. Something just below 1.0 seems reasonable. It definitely took me quite a lot longer than the "newspaper" hardest killers which I do another site; they would be rated at least 0.5 on Mike's original rating definitions. Those puzzles typically take me about 20 minutes; of course I don't write walkthroughs for them, that adds to the time taken.

I’ll attempt to limit myself to insertion solving, if not pure paper solving (if there is such a thing)

1. 45 rule on C1234 3 innies R345C4 = 24 = {789}

2. 45 rule on C1234 1 innie R5C4 = 1 outie R3C5 + 6, R3C5 = {123}

3. 45 rule on C6789 1 outie R7C5 = 1 innie R5C6 + 5, R5C6 = {1234}, R7C5 = {6789}

4. 45 rule on C6789 3 innies R567C6 = 11
4a. 6(2) cage at R2C6 + hidden 11(3) cage at R5C6 = 17(5) must contain 1,2,3 -> hidden 11(3) cage must contain 3 and one of 1,2 = {137/236}, no other 3 in C6, no 4 in R5C6 -> no 9 in R7C5 (step 3)

5. 45 rule on N1 3(2+1) outies R12C4 + R4C1 = 7 -> max R12C4 = 6
5a. Min R12C4 = 6, otherwise min R12C4 + R23C6 + R3C5 less than 15 -> R12C4 = 6, R4C1 = 1, R3C1 = 5
5b. R12C4 = 6 and R23C6 = 6 cannot contain 3 -> R3C5 = 3, R5C4 = 9 (step 2), R34C4 = {78}

6. R12C5 = {69} (cannot be {78} which clashes with R3C4) -> R1C6 = {78} (only other place in N2), R1C7 = {23}

7. Min R5C12 + R5C6 + R5C89 = 15, R5C12 + R5C89 = 13 -> min R5C6 = 2, min R7C5 = 7 (step 3)

8. R5C89 = {14} (cannot be {23} which clashes with R5C6)

9. Deleted. Thanks Simon for pointing out my careless error. Re-worked from here, starting with modifying the next two steps.

10. 45 rule on R1234 1 outie R5C7 = 1 innie R4C5 + 4 -> R5C7 = {68}, R4C5 = {24}

11. 45 rule on R6789 1 outie R5C3 = 1 innie R6C5 + 3 -> R5C3 = {578}, R6C5 = {245}

12. R89C5 = {15} (cannot be {24} which clashes with R4C5), R6C5 = {24}, R5C6 = 3, R7C5 = 8 (step 3)
12a. R78C4 = {34} (only remaining combination)

13. R5C5 = 7 (only remaining cell in C5), R34C4 = [78], R1C6 = 8, R1C7 = 2

14. R5C12 = {26} (only remaining combination) -> R5C7 = 8, R4C5 = 4 (step 10), R5C3 = 5, R6C5 = 2
[Now I’m back where I thought I was.]

15. R12C4 = 6 (step 5a) = {15} (only remaining combination) -> R23C6 = {24}

16. R67C6 = 8 = [17]

17. R4C6 = 5 (only remaining valid cell in N5), R4C7 = 6 (cage sum), R6C4 = 6, R6C3 = 4 (cage sum)

18. R9C4 = 2 (only remaining place in C4), R9C3 = 7

19. R89C6 = {69} (only remaining places in N8) = 15 -> R9C78 = 7 = {34} (only remaining combination)

20. 45 rule on N9 1 remaining innie R7C9 = 9, R6C9 = 3 -> R89C9 = {68} (only remaining combination), R78C8 = {25}, R78C7 = [17]

21. R6C78 = {59} (only remaining combination) = [59]

22. R6C12 = {78} (only remaining places in R6) = 15 -> R78C1 = 10

23. R4C23 = {39} (only remaining places in N4)

24. R4C89 = {27} (only remaining places in R4) = [72] = 9 -> R23C9 = 9

25. R3C78 = {69} (only remaining combination) = [96]

26. R7C23 = {23} (only remaining combination) -> R78C4 = [43], R78C8 = [52]

27. R6C12 = 15 (step 22), R7C1 = 6 (only remaining place in R7), R8C1 = 4 (cage sum), R5C12 = [26]

28. R8C23 = {58} (only remaining combination) = [58], R89C5 = [15], R89C9 = [68], R89C6 = [96]

29. R9C12 = [91] (only remaining places in R9)

30. R23C9 = 9 (step 24) = {45} (only remaining combination) = [54], R12C4 = [51], R23C6 = [42], R5C89 = [41], R9C78 = [43]

31. R2C7 = 3 (only remaining cell in C7), R2C8 = 8

32. R1C89 = [17] (only remaining places in N3)

33. R12C4 = 6 (step 5a) -> R1C23 = 10 = {46} (only remaining combination) = [46], R12C5 = [96]

34. R1C1 = 3 (only remaining cell in R1), R2C1 = 7, R6C12 = [87]

35. R23C2 = {28} (only valid combination) = [28], R7C23 = [32], R4C23 = [93]

36. R23C3 = [91] (only remaining cells in C3)

1:
Innie C1234: R345C4 = 24 = {789}
N25 18(3): R3C5 <> {456789} (or min sum = 4+7+8 = 19 > 18)
Outies N1: R12C4+R4C1 = 7 <> {6789}
N2 6(2) <> {6789}
Hidden quad N2: R1C56+R2C5+R3C4 = {6789}
N2 15(2): R1C6+R3C4 = 6+7+8+9-15 = 15
Innies N12: R1C6+R3C145 = 23
--> R3C15 = 23-15 = 8
--> C1 6(2) = [51], R3C5 = [3]

2a:
N56 19(3), N6 14(2), N69 12(2) <> [1]
--> [1] of N6 locked in N6 5(2) = {14}
--> R6C9 <> [4]
Outies N36: R14C6+R7C9 = 22
--> R7C9 <> [4] (or max total = 4+8+9 = 21 < 22)
--> C9 12(2) <> {48}, = {39/57} has [5/9]
--> N9 14(2) <> {59}, = {68}

2b:
Outies N47: R69C4 = 8 <> [4]
R9 9(2): R9C3 <> [5]
N7 5(2) & 10(2) <> [5]
--> [5] of N7 locked in N7 13(2) = {58}
--> N9 14(2) = [68]

3:
Outies R9: R8C56 = 10 = {19/37}
--> N8 6(2) = [15], R8C6 = [9]
N9 8(2) & 7(2) <> [9]
Hidden single N9: C9 12(2) = [39]
Outies N9: R9C6 = [6]
Outies N36: R14C6 = 13 = {58}
--> N23 10(2) = [82], R4C6 = [5]
Innies N12: N25 18(3) = [738]
Innies N3: R23C9 = 9 = [54]

Mop-up:
N2 6(2) = [42], N56 19(3) = [568], N6 5(2) = [41]
--> N3 8(2) = [17], N3 11(2) = [38], N3 15(2) = [96]
--> N36 18(4) = [5472], N6 14(2) = [59]
--> N9 8(2) = [17], N89 22(4) = [9643]
R9: N7 10(2) = [91], R9 9(2) = [72]
R3: C2 10(2) = [28], C3 10(2) = [91]
--> N2 15(2) = [96], N4 12(2) = [93], N7 13(2) = [58]
--> N4 8(2) = [26], N7 5(2) = [32], N9 7(2) = [52]
--> N12 16(4) = [4651], N47 25(4) = [8764], N8 7(2) = [43]
--> C1 10(2) = [37], N45 15(3) = [546], N58 16(2) = [187]
--> N5 25(5) = [49732]

Step 1 (three 45s used, 3 placements made):
Similar to Andrew's step 1+2+5 but slightly different logic. Basically first establish the hidden quad {6789} in N2 (I think hidden pairs/triples/quads like this are fairly straight forward to spot in pencilmarkless solving), and then establish R1C6+R3C4 = 15, and then using innies of N12 to establish R3C15 = 8 = [53]. All lines before the last one are to be followed mentally, i.e. no explicit marking needed at all.

Step 2a (one 45 used, 0 placement made):
Again to be followed mentally only: first nail the [1] of N6 into 5(2) = {14}, then using outies of N36 mentally narrow down the possibility of C9 12(2) to be {39/57}, thus narrow down N9 14(2) to be {68}. (If one finds it necessary, one can make a tiny note of this result outside the grid when "paper solving", personally I don't have this need since I can process all these logic mentally.)

Step 2b (one 45 used, 2 placements made):
Using outies of N47 nail the [5] of N7 into 13(2) = {58}, thus establish N9 14(2) to be [68].

Step 3 (five 45s used, 13 placements made):
Using outies of R9 establish R8C56 = 10 = [19], then via a series of innies & outies plus a hidden single establish a few more cell placements to approach the final mop-up stage.

63 placements made in mop-up

Total: ten 45s used, 81 placements made