SudokuSolver Forum

Preliminaries courtesy of SudokuSolver

Cage 4(2) n23 - cells ={13}
Cage 7(2) n36 - cells do not use 789
Cage 13(2) n89 - cells do not use 123
Cage 10(2) n69 - cells do not use 5
Cage 10(4) n1 - cells ={1234}
Cage 14(4) n36 - cells do not use 9
Cage 26(4) n12 - cells do not use 1
Cage 26(4) n3 - cells do not use 1
Cage 26(4) n56 - cells do not use 1

1. 4(2)r1c6 = {13} only: both locked for r1

2. "45" on r1: 3 innies r1c189 = 21 = [4]{89} only
2a. r1c89 = {89} only: both locked for r1 and n3

3. Naked triple {123} in n1: all locked for n1

4. "45" on c1: 4 outies r2378c2 = 11 = {1235} only: all locked for c2; 5 locked for n7

5. "45" on n4: 2 outies r37c1 = 9
= [81/72/63]
5a. r3c1 = (678), r7c1 = (123)

6. 9 in n1 only in c3: locked for c3

7. 5 in c1 only in n4: locked for n4
7a. r37c1 = 9 -> r456c1 = 16 (cage sum) and must have 5 = {259/358}(no 1,6,7)
[Andrew noticed that a more direct way is to do “45” on n4: 3 innies r456c1 = 16. Thanks Andrew. I never saw that "45" at this spot, was blinded by the outies n4]

8. 29(6) must have 1,4,6 & 7 for n4 = {124679/134678}

The key
9. from step 8, 29(6) must have both 6 & 7, but can't both be in c2 because r1c2 = (67) -> at least one of 6 or 7 must be in r456c3 -> min. r456c3 = {126} = 9
9a. "45" on c12: 2 innies r19c2 - 5 = 3 outies r456c3; min. 3 outies = 9 -> min. r19c2 = 14 -> r19c2 = [68/69/78/79]
9b. r9c2 = (89)

10. 4 in c2 only in n4: locked for n4
10a. 1 in n4 only in c3: locked for c3

11. 16(4)r9c2 must have 8 or 9 for r9c2 = {1249/1258/1348}(no 6,7)
11a. must have 1 which is only in n8 -> 1 locked for r9 and n8

12. 16(4)r8c3 = {2347/2356}(no 8,9)
12a. must have 2 & 3, both locked for r8

13. "45" on r89: 1 innie r8c9 = 1 outie r7c2 = {15} only common candidates

14. Naked pair {15} in r78c2: 1 locked for c2 and n7
14a. Hidden single 1 in n1 -> r2c1 = 1
14b. no 8 in r3c1 (h9(2)r37c1)
14c. r78c2 = {15} = 6 -> r89c1 = 15 = {69/78}(no 2,3) = [6/7..]

15. 8 in n1 only in c3: locked for c3 (I see this is now actually hidden pair {89} in r23c3 -> r23c3 = {89})

16. Naked pair {67} in n1: both locked for n1
16a. r1c3 = 5
16b. 2 in r1 only in n2: 2 locked for n2

17. Naked pair {15} in r8c29: both locked for r8

18. 16(4)r8c3 = {2347} only combination: 4 & 7 locked for r8
18a. no 8 in r9c1 (h15(2)r89c1)

19. 16(4)r9c2 = {1249/1258/1358}: ie, can't have both of 2 & 3
19a. -> r9c89 must have one of 2 or 3 for r9 (Hidden killer pair)
19b. 24(4)r8c7 must have 2 or 3 for r9 and must also have two of {689} for r8c78 = {2589/2679/3489/3678}
19c. = exactly two of 6,8,9 which must go in r8c78 -> no 6,8,9 in r9c89

20. 6 in r9 in 13(2) = {67} or in r9c1 -> no 7 in r9c1 since it would mean no 6 for r9 (Locking-out cages)
20a. no 8 in r8c1 (h15(2)r89c1)

Missing routine clean-up from here
21. Naked pair {69} in r89c1: both locked for c1 and 6 for n7
21a. r3c1 = 7, r7c1 = 2 (h9(2)r37c1), r19c2 = [68]

22. 16(4)r9c2, must have 3 or 4 for r9c3 = {1348} only, 3 & 4 locked for r9
22a. 13(2)r9c6 = {67} only combo: both locked for r9
22b. r89c1 = [69]
22c. Naked pair {89} in r8c78: both locked for n9
22d. Naked pair {89} in r18c8: both locked for c8

23. Naked pair {25} in r9c89: 5 locked for n9
23a. r8c29 = [51], r7c2 = 1

24. 3 and 4 in n9 only in r7, locked for r7
24a. r7c3 = 7
24b. Hidden single 7 in n9 -> r9c7 = 7, r9c6 = 6
24c. 10(2)r6c7 = {46} only valid combo: both locked for c7
24d. 7(2)r3c7 = {25} only valid combo: both locked for C7
24e. r12c7 = [13], r1c6 = 3, r2c8 = 6 (cage sum), r23c2 = [23]

25. Hidden single 1 in c8 -> r4c8 = 1

26. "45" on n3: 1 remaining outie r4c7 = 2, r3c7 = 5
26a. r3c89 = {24}: 4 locked for r3 and n3
26b. r2c9 = 7 (Alternatively, could have done this from 1 remaining innie r12)

27. "45" on n69: 2 remaining innies r5c78 = 14 = [95] only permutation
27a. r5c78 = 14 -> r5c56 = 12 (cage sum) = {48} only: both locked for r5 and n5

28. 40(7)r3c3 cannot have both 8 & 9 in c4 because r3c3 must have one of (89) -> only other place for 8 or 9 in c4 is r2c4 -> r2c4 = (89) (Hidden killer pair)
28a. Naked pair {89} in r2c34: both locked for r2

29. 15(4)r6c8 = {1347} only combination -> r6c8 = 7, r7c89 = {34}: 4 locked for r7
29a. r67c7 = [46]

30. 24(4)r6c5 = {1689/2589}(no 3) only valid combos
30a. must have 8 which is only in r7: 8 locked for r7
30b. 8 which must be in 40(7)r3c3 only in r3: 8 locked r3

31. 22(4)r3c5 = {1579} only valid combo: Naked pair {19} in r3c56: both locked for r3 and n2 and 22(4) cage
31a. no 9 in r4c56

32. 3 in n5 only in 40(7)r3c3: 3 locked for c4
32a. -> 40(7) must have 3 = {2356789} only (no 1)
32b. Hidden single 1 in c4 -> r9c4 = 1
33c. Hidden single 6 in c5 -> r6c5 = 6
33d. Hidden single 1 in c5 -> r3c5 = 1


Singles from there.

Thanks Ed for pointing out my omission is step 16.

Prelims

a) R1C67 = {13}
b) R34C7 = {16/25/34}, no 7,8,9
c) R67C7 = {19/28/37/46}, no 5
d) R9C67 = {49/58/67}, no 1,2,3
e) 10(4) cage at R1C1 = {1234}
f) 26(4) cage at R1C8 = {2789/3689/4589/4679/5678}, no 1
g) 26(4) cage at R2C3 = {2789/3689/4589/4679/5678}, no 1
h) 14(4) cage at R2C9 = {1238/1247/1256/1346/2345}, no 9
i) 26(4) cage at R5C5 = {2789/3689/4589/4679/5678}, no 1

1. 45 rule on R1 3 innies R1C189 = 21 = {489} (only possible combination) -> R1C1 = 4, R1C89 = {89}, locked for R1 and N3
1a. Naked pair {13} in R1C67, locked for R1
1b. Naked triple {123} in R2C12 + R3C2, locked for N1
1c. Naked quad {2567} in 20(4) cage at R1C2, 2 locked for N2

2. 45 rule on C1 3 remaining innies R289C1 = 16 = {169/178/259/268/358/367}
2a. R2C1 = {123} -> no 1,2,3 in R89C1

3. 45 rule on C1 4 outies R2378C2 = 11 = {1235}, locked for C2, 5 also locked for N7
3a. Min R456C2 = {468} = 18 (cannot be {467} which clashes with R1C2) -> max R456C3 = 11 but cannot be {128} which would clash with {468} -> no 8,9 in R456C3

4. 45 rule on N4 2 outies R37C1 = 9 = [63/72/81]
4a. 5,9 in N1 only in R123C3, locked for C3

5. 45 rule on R12 1 innie R2C9 = 1 outie R3C2 + 4, R3C2 = {123} -> R2C9 = {567}
5a. 1 in R2 only in R2C12, locked for N1, clean-up: no 5 in R2C9

6. 14(4) cage at R2C9 = {1247/1256/1346} (cannot be {1238/2345} because R2C9 only contains 6,7), no 8
6a. R2C9 = {67} -> no 6,7 in R3C89 + R4C8
6b. 7 in N3 only in R2C789, locked for R2

7. 45 rule on N3 2 outies R4C78 = 1 innie R1C7 + 2, IOU no 2 in R4C8
7a. R1C7 = {13} -> R4C78 = 3,5 = [21/14/23/41], R4C7 = {124}, R4C8 = {134}, R3C7 = {356}

8. 45 rule on N9 2 outies R6C78 = 1 innie R9C7 + 4, IOU no 4 in R6C8

9. 45 rule on R89 1 outie R7C2 = 1 innie R8C9 -> R8C9 = {1235}

10. Hidden killer triple 1,2,3 in R2C1, R456C1 and R7C1 for C1, R2C1 and R7C1 = {123} -> R456C1 must contain one of 1,2,3
10a. 29(6) cage at R4C2 = {124679/134678} (cannot be {123689} which clashes with R456C1), 1 locked for C3 and N4, 6,7 also locked for N4
10b. Hidden killer pair 8,9 in R456C2 and R9C2 for C2, R456C2 contain one of 8,9 -> R9C2 = {89}
10c. R9C2 = {89} -> R9C345 = 7,8 = {124/125/134}, no 6,7,8,9, 1 locked for R9 and N8
10d. 4 in C2 only in R456C2, locked for N4

11. 16(4) cage at R8C3 = {2347} (only possible combination, cannot be {2356} which clashes with R8C29, ALS block), locked for R8, clean-up: no 2,3 in R7C2 (step 9)
11a. Naked pair {15} in R8C29, locked for R8
11b. Naked pair {15} in R78C2, locked for C2 and N7, clean-up: no 8 in R3C1 (step 4)
11c. R78C2 = {15} = 6 -> R89C1 = 15 = R89C1 = [69/87/96], no 8 in R9C1
11d. Killer pair 8,9 in R89C1 and R9C2, locked for N7

12. Naked pair {67} in R1C2 + R3C1, locked for N1 -> R1C3 = 5

13. 8,9 in N3 only in 26(4) cage at R1C8 = {2789/3689/4589}, 14(4) cage at R2C9 (step 6) = {1247/1256/1346}
13a. Hidden killer pair 2,7 in 26(4) cage and 14(4) cage for N3, 26(4) cage contains both or neither of 2,7 -> 14(4) cage must contain both or neither of 2,7 -> 14(4) cage = {1247/1346} (cannot be {1256} which only contains one of 2,7), no 5
13b. 14(4) cage = {1247/1346}, CPE no 4 in R2C8
13c. Hidden killer triple 5,6,7 in 26(4) cage, R2C9 and R3C7 for N3, 26(4) cage contains one of 5,6,7, R2C69 = {67} -> R3C7 = {56}, clean-up: no 4 in R4C7

14. 24(4) cage at R8C7 = {2589/2679/3489/3678} (cannot be {3579/4578} because R8C78 only contain 6,8,9, cannot be {4569} which clashes with R9C345)
14a. R8C78 = {689} -> no 6,8,9 in R9C89
14b. 24(4) cage at R8C7 = {2589/3489/3678} (cannot be {2679} = {69}{27} which clashes with R89C1), 8 locked for R8 and N9, clean-up: no 2 in R6C7, no 7 in R9C1 (step 11c), no 5 in R9C6

15. Naked pair {69} in R89C1, locked for C1 and N7 -> R3C1 = 7, R7C1 = 2 (step 4), R19C2 = [68], clean-up: no 8 in R6C7, no 5 in R9C7
15a. Naked triple {347} in R789C3, locked for C3

16. 2 in N9 only in 24(4) cage at R8C7 (step 14b) = {2589} (only remaining combination) -> R8C78 = {89}, locked for R8 and N9, R9C89 = {25}, locked for R9 and N9 -> R89C1 = [69], R8C9 = 1, R78C2 = [15], clean-up: no 1,9 in R6C7, no 4 in R9C67
16a. R7C456 = {589} (hidden triple in R7)

17. Killer pair 6,7 in R67C7 and R9C7, locked for C7 -> R3C7 = 5, R4C7 = 2
17a. Killer pair 3,4 in R2C7 and R67C7, locked for C7 -> R1C67 = [31], R4C8 = 1 (hidden single in N6), R4C3 = 6

18. R1C89 = {89} = 17 -> R2C78 = 9 = [36], R2C9 = 7, clean-up: no 7 in R67C7
18a. Naked pair {24} in R3C89, locked for R3
18b. Naked pair {46} in R67C7, locked for C7 -> R9C67 = [67]

19. 45 rule on N69 2 remaining innies R5C78 = 14 = [95]
19a. R5C78 = 14 -> R5C56 = 12 = {48}, locked for R5 and N5

20. R6C78 = R9C7 + 4 (step 8)
20a. R9C7 = 7 -> R6C78 = 11 = [47]

21. R5C1 = 3, R5C9 = 6, R5C2 = 7 -> R5C4 = {12}
21a. 40(7) cage at R3C3 can only contain one of 1,2 -> no 1,2 in R36C4

22. 1 in R3 only in 22(4) cage at R3C5 = {1579} (only remaining combination, cannot be {1678} because 1,6,8 only in R3C56) -> R3C56 = {19}, locked for R3 and N2, R4C56 = {57}, locked for R4 and N5 -> R4C1 = 8

and the rest is naked singles.

I agree with the SS score of 1.25.