SudokuSolver Forum

Prelims

a) R1C45 = {59/68}
b) R12C6 = {18/27/36/45}, no 9
c) R89C4 = {29/38/47/56}, no 1
d) R9C56 = {39/48/57}, no 1,2,6
e) 6(3) cage in N1 = {123}
f) 20(3) cage at R4C9 = {389/479/569/578}, no 1,2
g) 9(3) cage in N7 = {126/135/234}, no 7,8,9
h) 13(4) cage in N3 = {1237/1246/1345}, no 8,9

Steps resulting from Prelims
1a. Naked triple {123} in 6(3) cage, locked for N1
1b. 13(4) cage in N3 = {1237/1246/1345}, 1 locked for N3

2. 9(3) cage in N7 = {126/135/234}
2a. 4,5,6 must be in R89C3 (R89C3 cannot be {12/13/23} which clash with 6(3) cage in N1, ALS block), no 4,5,6 in R8C2
2b. Killer triple 1,2,3 in R12C3 and R89C3, locked for C3

3. 45 rule on N1 2 innies R3C23 = 15 = {69/78}, no 4,5
3a. 45 rule on N1 4 outies R3C4 + R4C234 = 13 = {1237/1246/1345}, no 8,9, no 1 in R4C5 (CPE plus anti-King for R3C4)

4. 45 rule on N3 2 innies R3C78 = 15 = {69/78}
4a. Naked quad 6,7,8,9 in R3C2378, locked for R3

5. 45 rule on N7 2 innies R7C23 = 13 = {49/58/67}, no 1,2,3

6. 45 rule on N9 2 innies R7C78 = 9 = {18/27/36/45}, no 9

7. 45 rule on N2 2 outies R3C46 = 4 = {13}, locked for R3 and N2, clean-up: no 6,8 in R12C6
7c. 18(3) cage at R2C4 = {279/459/468} (cannot be {567} which clashes with R1C45)
7d. 2 of {279} must be in R3C5 -> no 2 in R2C45

8. 45 rule on C1234 2 innies R12C4 = 17 = {89}, locked for C4 and N2, clean-up: no 2,3 in R89C4
8a. 6 in N2 only in R12C5, locked for C5

9. 45 rule on N8 2 innies R7C46 = 6 = {15/24}
9a . Killer pair 4,5 in R7C46 and R89C4, locked for N8, clean-up: no 7,8 in R9C56
9b. Naked pair {39} in R9C56, locked for R9 and N8
9c. R7C78 (step 6) = {18/27/36} (cannot be {45} which clashes with R7C46), no 4,5

10. 45 rule on C6789 2 innies R89C6 = 11 = [29/83], no 1,6,7 in R8C6

11. 6 in N8 only in R89C4 = {56}, locked for C4 and N8, no 5,6 in R89C3 (anti-King), clean-up: no 1 in R7C46 (step 9)

12. Naked pair {24} in R7C46, locked for R7 and N8 -> R8C6 = 8, R9C6 = 3 (step 10), R9C5 = 9, R3C46 = [31], no 3 in R2C3 + R4C5, no 8 in R79C7, no 3 in R8C7 (all anti-King), clean-up: no 1 in R7C8 (step 6)

13. Naked pair {17} in R78C5, locked for C5, clean-up: no 2 in 18(3) cage in N2 (step 7c)

14. Naked triple {456} in R123C5, locked for C5 and N2

15. Naked pair {27} in R12C6, locked for C6 -> R7C46 = [24], no 2,7 in R12C7, no 2 in R6C5 + R8C3, no 4 in R8C7 (all anti-King), clean-up: no 9 in R7C23 (step 5), no 7 in R7C78 (step 6)
15a. Killer pair 6,8 in R7C23 and R7C78, locked for R7
15b. Naked triple {147} in R456C4, no 4,7 in R5C3 (anti-King)
15c. Naked triple {569} in R456C6, no 5,6,9 in R5C7 (anti-King)
15d. 9 in N7 only in R78C1, locked for C1

16. 9(3) cage in N7 = {234} (only remaining combination), locked for N7, also 4 locked for C3, 3 locked for R8

17. 45 rule on R89 3 outies R7C159 = 17 = {179} (only remaining combination, cannot be {359} because no 3,5,9 in R7C5), locked for R7, clean-up: no 6 in R7C23 (step 5), no 8 in R7C8 (step 6)

18. Naked pair {58} in R7C23, locked for N7, no 5,8 in R6C23 (anti-King)

19. Naked pair {36} in R7C78, locked for N9 and 25(6) cage at R6C6, no 3,6 in R6C678

20. 25(6) cage at R6C6 = {123469} (only remaining combination) -> R6C6 = 9, R6C78 = {12}, locked for R6 and N6
20a. Naked pair {56} in R45C6, no 5,6 in R4C7 (anti-King)

21. R3C4 = 3 -> R3C4 + R4C234 (step 3a) = {1237/1345}, no 6, 1 locked for R4
21a. 2 of {1237} only in R4C2 -> no 7 in R4C2
[I missed an anti-King step here, see note at the end. I’ll re-work the later steps when I have time.]

22. R3C6 = 1 -> 23(4) cage at R3C6 = {1589/1679}, no 3,4, 9 locked for C7, no 9 in R34C8 (anti-King), clean-up: no 6 in R3C7 (step 4)

23. R3C9 = 2 (hidden single in R3)
[That’s been there for a while.]

24. 15(3) cage in N9 = {159/249}, no 7 -> R8C8 = 9
24a. R7C1 = 9 (hidden single in C1)

25. 23(4) cage at R6C3 = {2678} (only remaining combination) -> R6C4 = 7, R6C3 = 6, R7C3 = 8, R7C2 = 5, R5C3 = 9, R3C3 = 7, R3C2 = 8 (step 3), R3C78 = [96], R7C78 = [63], R5C6 = 5, R4C6 = 6, R4C7 = 7 (step 21), R4C3 = 5

26. R4C24 (step 21) = {14}, locked for R4 -> R45C8 = [84], R5C7 = 3, R5C4 = 1, R4C4 = 4, R4C2 = 1, no 4 in R3C5 (anti-King)

27. R3C5 = 5, R1C5 = 6, R1C4 = 8, R2C45 = [94]

28. 17(3) cage in N3 = {458} (only remaining combination) -> R1C7 = 4, R2C78 = [85]

29. 15(3) cage in N9 (step 24) = {159} (only remaining combination), 1,5 locked for C7 and N9 -> R6C78 = [21]

30. 26(5) cage at R5C2 = {14579} (only remaining combination) -> R56C2 = [74]

and the rest is naked singles.

5 9 1 8 6 2 4 7 3
6 3 2 9 4 7 8 5 1
4 8 7 3 5 1 9 6 2
3 1 5 4 2 6 7 8 9
2 7 9 1 8 5 3 4 6
8 4 6 7 3 9 2 1 5
9 5 8 2 1 4 6 3 7
1 2 3 6 7 8 5 9 4
7 6 4 5 9 3 1 2 8

Techniques used:
76 Naked Singles
4 Hidden Singles
8 Unique Pairs
7 Intersections
1 Complex Hidden Singles
2 Unique Triplets
1 Hidden Triplets
3 Odd Pairs
3 Odd Triplets
10 Double Innies & Outies
1 Mandatory Inclusions
3 Odd Quads
2 Complex Intersections
3 Triple Innies & Outies
2 Double Outies minus Innies
1 Complex Naked Pairs

The grid is solved!
Techniques used:
67 Naked Singles
14 Hidden Singles
2 Unique Pairs
3 Naked Pairs
18 Intersections
1 Unique Triplets
5 Odd Pairs
1 Odd Triplets
1 Double Innies & Outies
2 Mandatory Inclusions
4 Odd Quads
1 Complex Intersections
1 Triple Innies & Outies
3 Double Outies minus Innies
3 Complex Naked Pairs
2 Complex Hidden Pairs
1 Conflicting Pairs
4 Quadruple Innies & Outies
1 Triple Outies minus Innies
6 Pointing Pairs
1 Unique Combinations
6 Pointing Triplets
1 Locked Cages
3 Two String Kites
2 Finned X-Wing
3 Empty Rectangles
2 Finned Swordfish
1 Grouped X-Wing


Techniques used:
75 Naked Singles
6 Hidden Singles
4 Unique Pairs
4 Intersections
4 Unique Triplets
6 Odd Pairs
10 Odd Triplets
1 Mandatory Inclusions
7 Odd Quads
4 Complex Intersections
2 Triple Innies & Outies
2 Complex Naked Pairs
11 Complex Hidden Pairs
4 Conflicting Pairs
8 Quadruple Innies & Outies
1 Pointing Pairs
8 Odd Combinations
3 Pointing Triplets
2 Conflicting Triplets
5 Grouped XY-Chains up to 3 links
6 Conflicting Partial Pairs
26 Multiple Innies & Outies
40 Multiple Outies minus Innies
7 Complex XY-Chains up to 3 links
1 Complex XY-Chains
92 Conflicting Partial Triplets
3 Cages Grouping

Techniques used:
64 Naked Singles
15 Hidden Singles
2 Single Innies & Outies
7 Unique Pairs
1 Hidden Pairs
5 Intersections
6 Odd Pairs
2 Double Innies & Outies
1 Mandatory Inclusions
1 Naked Quads
1 Complex Intersections
2 Triple Innies & Outies

1.
INNIE in N2: R3C46 (red cells) = 4 = {13} => cage 9/2 in N2 = {27|45}

OUTIE in C56789: R12C4 (blue cells) = 17 = {89} => cage 11/2 in C4 = {47|56}


2.
INNIE in N8: R7C46 (red cells) = 6 = {15|24}

COMPLEX NAKED PAIR on {45} with cage 11/2 and R7C46 in N8 => 12/3 in N8 = {39}

OUTIE on C5: R89C6 (blue cells) = 11 => R8C6 = 2|8


3.
6 in N2 locked in C5 => 6 in N8 must be in cage 11/2 => cage 11/2 = {56}, R7C46 = {24}, R89C6 = [83]

NAKED SINGLES (blue)


4.
R78C5 (red cells) must be {17}: 7 cannot be elsewhere in C5 => 7 in N2 must be in cage 9/2.

cage 9/2 in C6={27} => R7C6 = 4

INNIE in N9: R7C78 (blue cells) = 9 = {18|36}

INNIE in N7: R7C23 (green cells) = 13 = {58|67}

R7C2378 (blue and green cells together) = {1678|3568}: {1678} removes all possibilities from R7C5 => R7C78 = {36}, R7C23 = {58}


5.
R123C5 = {456}, R456C5 = {238}

SPLIT CAGE: R6C678 (red cells) = 12 = {129} (only possible combination) => R6C6 = 9

cage 23/4 in C34 = {2579|2489|2678}, cannot contain a 9 so only {2678} is possible

R78C5 = [17] (anti-king with R6C4)


6.
cage 6/3 = {123}

INNIE in N1: R3C23 (red cells) = 15 = {69|78}

R4C3 = 5, R3C3 = 7 (HIDDEN SINGLES in C3)
R45C6 = [65] (NAKED SINGLES in N5)
SPLIT CAGE: R34C7 = 16 = [97]

OUTIE on C1: R19C2 (blue cells) = 15 = [96]

NAKED SINGLES in N2 (blue)


7.
Basic singles and cage combinations from here.