SudokuSolver Forum

Techniques used:
73 Naked Single
6 Hidden Single
1 Single Innies & Outies
3 Unique Pair
1 Naked Pair
2 Hidden Pair
1 Complex Hidden Single
3 Unique Triplet
12 Intersection
14 Odd Pairs
8 Odd Triplets
3 Double Innies & Outies
3 Mandatory Inclusion
5 Odd Quads
6 Complex Intersection
8 Triple Innies & Outies
3 Double Outies minus Innies
6 Complex Naked Pair
10 Complex Hidden Pair
3 Conflicting Pair
12 Quadruple Innies & Outies
2 Triple Outies minus Innies
1 Pointing Pair
1 Odd Combinations
1 Pointing Triplet
6 Locked Cages
1 Generalized X-Wing
3 Conflicting Partial Pair
1 Turbot Fish
1 Empty Rectangle
12 Multiple Innies & Outies
3 Multiple Outies minus Innies

3x3:d:k:7680:7680:4865:4865:4865:4865:3842:3842:3842:9219:7680:7680:7680:3076:1797:4102:3591:3591:9219:9219:9219:9219:3076:1797:4102:3591:6408:4617:9219:9219:1546:3076:2571:3084:4102:6408:4617:4617:6157:1546:3076:2571:3084:6408:6408:1806:1806:6157:7695:7695:7695:7695:7695:6928:4369:4369:6157:2834:2834:2834:6928:6928:6928:4369:5651:6157:6676:6676:6676:6676:6676:6928:4369:5651:5651:5651:3861:3861:3861:1558:1558:

1. Outies r6789 -> r5c3 = 5
Innies n36 = r6c789 = +8(3) (Must contain a 1)
-> 7(2)r6 = {34}
-> r6c789 = {125}
-> Max r6c78 = +7(2)
-> Min r6c456 = +23(3) = {689} or {789}
-> r6 from:
(A) [{34}6{789}{15}2] or
(B) [{34}7{689}{25}1]

Also, remaining outies n7 = r6c3,r9c4 = +13(2) = {67}

2! Either Step 1(A) is correct, in which case this puts ...
r6c789 = {789}, 10(2)n5 = {46}, and 7(2)n2 = {25}
-> 11(3)n8 cannot be {245}

Or Step 1(B) is correct, in which case this puts ...
r6c789 = {689}
and (since remaining Innies n5 = r45c5 = +6) 12(4)c5 = [{15}{24}] or [{24}{15}]
-> 11(3)n8 cannot be {245} (again!)

-> 5 in n8 in r89

3! 24(4)c3 cannot contain all of (567)
-> Whichever of (67) is in r9c4 must go in n7 in 17(4)n7 in r78
Outies r9 = r78c12 = +15(4)
IOD r9 -> r9c1 = r8c2 + 2

Either Step 1(A) is correct, in which case this puts ...
r6c3,r9c4 = [67], 24(4)c3 = [56{49}], r78c12 = {1257}, and r9c123 = {368}
which puts 17(4)n7 = [{257}3] and 22(4)n7 = [1687]

or Step 1(B) is correct in which case this puts...
r6c3,r9c4 = [76]
and Outies r9 = r78c12 = +15(4) contains a 6 -> must also contain a 3
which puts 24(4)c3 = [57{48}], r78c12 = {1356}, and r9c123 = {279}
which puts 17(4)n7 = [{136}7] and 22(4)n7 = [5{29}6]

BUT! this latter case puts 6(2)n9 = {15} which leaves no place for 5 in n8 (Step 2)
-> Step 1(A) is correct

4. -> r6 = [{34}6{789}{15}2]
24(4)c3 = [56{49}]
-> 17(4)n7 = [{257}3] and 22(4)n7 = [1687]
-> 7(2)n4 = [43]
Also 10(2)n5 = {46}
-> 6(2)n5 = [51]
-> 12(4)c5 = [{16}32]
Also 7(2)n2 = {25}

5. Innies r1 = r1c12 = +11(2) (No 1)
Remaining Innies n4 = r4c23 = +9(2) = [81] or {27}
But that cannot be [81] since that leaves no place for 1 in c1
-> r4c23 = {27}
-> 18(3)n4 = [1{89}]

6. 7 already in c4 -> r6c4 from (89)
Remaining IOD n2 -> r23c4 = r1c3 + 12. I.e., r23c3 is Min +13(2) (No 3)
-> 3 in n2 in r1c456
-> 19(4)r1 can only be [1{378}]
-> r23c4 = {49}

7. {27} in n1 only in 30(5)
-> 30(5)n12 = {24789}
-> (Since r1c12 = +11(2)) 30(5) = [{29}874]
-> 18(3)n4 = [189]
-> r1c12 = [92]
-> 36(7)r2c1 = [{56}43972]

8. 15(3)n3 = {456}
r2c156 = [562] or [615]
But the latter puts r2c789 = +14(3) contradicting existing 14(3) cage in n3
-> r23c1 = [56]
-> r23c5 = [61]
-> r23c6 = [25]
-> r2c789 = {139}
-> 14(3)n3 = [{39}2], 16(3)r2c7 = [178], r3c9 = 8
-> 12(3)n6 = [93]
etc.

Prelims

a) R23C6 = {16/25/34}, no 7,8,9
c) R45C4 = {15/24}
d) R45C6 = {19/28/37/46}, no 5
e) R45C7 = {39/48/57}, no 1,2,6
f) R6C12 = {16/25/34}, no 7,8,9
g) R9C89 = {15/24}
h) 11(3) cage at R7C4 = {128/137/146/236/245}, no 9
i) 12(4) cage at R2C5 = {1236/1245}, no 7,8,9

1a. 12(4) cage at R2C5 = {1236/1245}, 1,2 locked for C5
1b. 45 rule on R6789 1 outie R5C3 = 5, clean-up: no 1 in R4C4, no 7 in R4C7, no 2 in R6C12
1c. 45 rule on N36 3 innies R6C789 = 8 = {125} (cannot be {134} which clashes with R6C12), locked for R6 and N6, clean-up: no 7 in R5C7, no 6 in R6C12
1d. Naked pair {34} in R6C12, locked for R6 and N4
1e. 45 rule on R6 2 innies R6C39 = 8 = [62/71]
1f. R56C3 = 11,12 -> R78C3 = 12,13 = {39/48/49} (cannot be {67} which clashes with R6C3), no 1,2,6,7
1g. 8,9 in R6 only in R6C456, locked for N5, clean-up: no 1,2 in R45C6
1h. R23C6 = {16/25} (cannot be {34} which clashes with R45C6), no 3,4
1i. 18(3) cage at R4C1 = {189/279}, no 6, 9 locked for N4
1j. 45 rule on R1 2 innies R1C12 = 11 = {29/38/47/56}, no 1
1k. 45 rule on R1 3 outies R2C234 = 19 = {289/379/469/478/568}, no 1
1l. 45 rule on N3 1 innie R3C9 = 1 outie R4C8, no 1,2,5 in R3C9

2a. 45 rule on N7 2 remaining outies R6C3 + R9C4 = 13 = {67}, CPE no 6,7 in R6C4 + R9C3
2b. 45 rule on R9 1 innie R9C1 = 1 outie R8C2 + 2, no 8,9 in R8C2, no 1,2 in R9C1
2c. Hidden killer pair 8,9 in R9C123 and 15(3) cage at R9C5 for R9 -> R9C123 and 15(3) cage must each contain one of 8,9 (both of 8,9 in R9C123 would clash with R78C3)
2d. Killer pair 8,9 in R89C3 and R9C123, locked for N7
2e. 15(3) cage at R9C5 = {159/168/249/348} (cannot be {258} which clashes with R9C89, cannot be {267/357/456} which don't contain one of 8,9), no 7
2f. Killer pair 1,4 in 15(3) cage and R9C89, locked for R9, clean-up: no 2 in R8C2
2g. 7 in R9 only in R9C1234, CPE no 7 in R8C2, clean-up: no 9 in R9C1
2h. 22(4) cage at R8C2 = {1579/1678/2479/2569/2578/3478/3568} (cannot be {1489/2389} because R9C4 only contains 6,7, cannot be {4567} because R9C3 only contains 2,3,8,9, cannot be {3469} = 4{369} which clashes with R8C2 + R9C1 = [46])
2i. Killer pair 8,9 in R9C23 and 15(3) cage, locked for R9, clean-up: no 6 in R8C2
2j. 17(4) cage at R7C1 = {1367/1457/2357/2456}
2k. Killer pair 3,4 in 17(4) cage and R78C3, locked for N9, clean-up: no 5,6 in R9C1
2l. 17(4) cage = {1367/1457/2357} (cannot be {2456} because R9C1 only contains 3,7), 7 locked for N7
2m. 22(4) cage = {1678/2569} (cannot be {1579} = {159}7 which clashes with 17(4) cage, cannot be {2578} = 5{28}7 which clashes with R8C2 + R9C1 = [57]), 6 locked for R9
2n. {1678/2569} = [1687]/5{29}6, no 5,8 in R9C2
2o. 8 in N7 only in R789C3, locked for C3

3a. 45 rule on N124 1 remaining innie R6C3 = 2 outies R45C5 + 1
3b. R6C3 = {67} -> R45C5 = 5,6 = {23/24}/[51] (cannot be {14} which clashes with R45C4), no 1,6 in R4C5, no 6 in R5C5
3c. 12(4) cage at R2C5 = {1236/1245} -> R45C5 = {23}/[51] (cannot be {24} because 12(4) cage = {15}{24} clashes with R23C6), no 4 in R45C5, R23C5 = {16/24}, no 3,5
3d. Combined cage R23C5 + R23C6 = {16}{25}/{24}{16}, 1,2,6 locked for N2
3e. Variable hidden killer pair 8,9 in R1C456 and R23C4 for N2, R23C4 cannot both of 8,9 (which would clash with R6C4), 19(4) cage at R1C3 cannot contain both of 8,9 -> R1C456 and R23C4 must each contain one of 8,9
3f. Killer pair 8,9 in R23C4 and R6C4, locked for C4
3g. 19(4) cage at R1C3 = {1378/2359} (cannot be {1567/2467/3457} which don’t contain one of 8,9 for R1C456, cannot be {1279/1369/1468/2368} because 1,2,6 only in R1C3, cannot be {1459/2458} which clash with R23C5 + R23C6) -> R1C3 = {12}, R1C456 = {359/378}, no 4, 3 locked for R1 and N1, clean-up: no 8 in R1C12 (step 1j)
3h. Combined cage R1C12 + 19(4) cage = {29}1{378}/{47}2{359}/{56}1{378}, 7 locked for R1
3i. Combined cage R1C12 + 19(4) cage = {29}1{378}/{47}2{359}/{56}1{378}, CPE, no 7 in R2C4
3j. Consider combinations for 12(4) cage = {1236/1245}
12(4) cage = {1236} => 4 in N2 only in R23C4, R45C5 = {23} => R45C6 = {46}
or 12(4) cage = {1245} = {24}[51], R23C6 = {16}, R45C4 = {24}
-> 4 in R2345C4, locked for C4, 6 in R2345C6, locked for C6
3k. 1 in R4 only in R4C123, locked for N4
3l. 1 in R4 only in R4C123, CPE no 1 in R23C1
3m. 1 in N1 only in R1C3 + R3C23, CPE no 1 in R4C3
3n. 1 in C3 only in R13C3, locked for N1
3o. 45 rule on N4 3 remaining innies R4C23 + R6C3 = 15 = {267} (only remaining combination, cannot be {168} because 1,8 only in R4C2), 2 locked for R4 and 36(7) cage at R2C1, clean-up: no 4 in R5C4, no 3 in R5C5 (step 3c)
3p. R4C23 + R6C3 = {267}, CPE no 6,7 in R3C3
3q. R4C1 = 1 (hidden single in N4) -> R5C12 = 17 = {89}, locked for R5, clean-up: no 3,4 in R4C7
3r. 6 in C4 only in R789C4, locked for N8
3s. 45 rule on N14 2 outies R23C4 = 2 remaining innies R16C3 + 6
3t. R16C3 = [16/17/26/27] = 7,8,9 -> R23C4 = 13,14,15 = {49/59}/[87], no 8 in R3C4
3u. R16C3 = [16/17/27] (cannot be [26] because 19(4) cage = 2{359} which clashes with R23C4 = 14 = {59})

4a. 15(3) cage at R1C7 = {168/249/456} (cannot be {159/258} which clash with 19(4) cage at R1C3
4b. 14(3) cage at R2C8 = {158/167/239/257/347/356} (cannot be {149/248} which clash with 15(3) cage)
4c. 16(3) cage at R2C7 = {169/178/259/268/367/457} (cannot be {349/358} which clash with R45C7)
4d. R3C9 = R4C8 (step 1l) -> 16(3) cage = R23C7 + R3C9 = {178/259/367/457} (cannot be {169/268} which clash with 15(3) cage)
4e. R16C3 (step 3u) = [16/17/27], R6C39 (step 1e) = [62/71]
4f. Consider placements for R1C3 = {12}
R1C3 = 1 => 15(3) cage = {249/456} => R23C7 + R3C9 = {178/367} (cannot be {259/457} which clash with 15(3) cage)
or R1C3 = 2 => R6C3 = 7, R6C9 = 1, R6C7 = {25} => 16(3) cage = {178/367/457} (cannot be {259} which clashes with R6C7)
-> R23C7 + R3C9 = 16(3) cage = {178/367/457}, no 2,9, 7 locked for N3

5a. R16C3 (step 3u) = [16/17/27], R6C3 + R9C4 (step 2a) = {67}, 22(4) cage at R8C2 (step 2m) = {1678/2569}
5b. Consider permutations for R4C23 + R6C3 (step 3o) = {267}
R4C23 + R6C3 = {267} = {26}7 => R9C4 = 6 => 22(4) cage = {2569}, caged X-Wing for 2 in R4C23 and R9C23, no other 2 in C23
or R4C23 + R6C3 = {267} = {27}6 => R1C3 = 1
-> R1C3 = 1
[That step could have been done instead of step 4f, but was harder to spot. Both depended on the important step 3u.
Almost cracked; fairly straightforward from here.]
5c. R1C3 = 1 -> 19(4) cage at R1C3 (step 3g) = 1{378}, 7,8 locked for R1 and N2, clean-up: no 4 in R1C12 (step 1j)
5d. 9 in N2 only in R23C4, locked for C4 -> R6C4 = 8, placed for D/
5e. 9 in N2 only in R23C4, CPE no 9 in R2C1
5f. 4 in R1 only in R1C789, locked for N3, clean-up: no 4 in R4C8 (step 1l)
5g. 36(7) cage at R2C1 = {2345679}, no 8, 3 locked for N1
5h. R2C2 = 8 (hidden single in N1), placed for D\, R5C12 = [89], clean-up: no 2 in R1C1 (step 1j)
5i. R2C234 = 19 (step 1k) = {289/478/568} -> R2C34 = [29/65/74], no 4,9 in R2C3
5j. Naked triple {267} in R246C3, 2 locked for C3
5k. 9 in N7 only in R789C3, locked for C3

6a. 2 in C1 only in R78C1, locked for N7 -> R9C24 = [67] = 13 -> R8C2 + R9C3 = 9 = [18], 1 placed for D/, R9C1 = 3, placed for D/, R6C12 = [43], R1C4 = 3, naked pair {49} in R78C3, 4 locked for C3 and N7, R6C3 = 6 (cage sum), R6C9 = 2 (step 1e), R3C3 = 3, placed for D\
[No routine clean-ups from here]
6b. R5C5 = 2, placed for both diagonals, R5C4 = 1, R4C4 = 5, placed for D\ -> R4C5 = 3, R23C5 = 7 = {16}, locked for N2
6c. R3C2 = 4 (hidden single in N1), R3C4 = 9, R1C1 = 9 (hidden single in N1), placed for D\, R1C2 = 2 (step 1j), R2C3 = 7, R6C6 = 7, placed for D\, R6C5 = 9, R4C2 = 7, R7C2 = 5
6d. Naked triple {146} in R7C7 + R8C8 + R9C9, locked for N9
6e. Naked triple {456} in R1C789, 5,6 locked for N3, R2C8 = 9, placed for D/, R7C3 = 4, placed for D/, R4C6 = 6, placed for D/
6f. R3C7 = 7, R4C8 = 8 -> R2C7 = 1 (cage sum)
6g. R7C5 = 8 -> R7C46 = [21]

and the rest is naked singles, not using the diagonals.

After my initial careless omission I thought that the important thing was to reduce R6C39 to [62]. However my solving path was almost there when I reduced 19(4) cage to one combination with R1C3 placed.

Preliminaries from SudokuSolver
Cage 6(2) n9 - cells only uses 1245
Cage 6(2) n5 - cells only uses 1245
Cage 7(2) n4 - cells do not use 789
Cage 7(2) n2 - cells do not use 789
Cage 12(2) n6 - cells do not use 126
Cage 10(2) n5 - cells do not use 5
Cage 11(3) n8 - cells do not use 9
Cage 12(4) n25 - cells do not use 789

Clean-up NOT done unless stated
1. "45" on r6789: 1 outie r5c3 = 5
1a. no 1 in r4c4

2. "45" on n36: 3 innies r6c789 = 8 = {125/134}
2a. 1 locked for n4 and r6

3. 7(2)r6c1 = {34} only: both locked for r6 and n4

4. "45" on r6: 2 innies r6c39 = 8 = [71/62]

5. r6c789 = {125}: 2 & 5 locked for n6, r6

6. 5 in n6 must be in 30(5) along with 1 or 2 = {789}{15}/{689}{25}
6a. -> r6c456 = {6789}
6b. 8 & 9 locked for n5
6c. no 1,2 in 10(2)n5

7. "45" on n7: 2 outies r6c3 + r9c4 = 13 = {67} only
7a. -> no 6,7 in r6c4 nor r9c3

8. split 19(3)r6c3 must have 6 or 7 for r6c3 = [6]{49}/[7]{39/48} = 3 or 4 in n7
8a. r78c3 from {3489}

9. "45" on r9: 4 outies r78c12 = 15
9a. and 6 or 7 in r9c4 must repeat in n7 there (in the 17(4) cage)
9b. but {1347/2346} both blocked by r78c3 = 3 or 4
9c. = {1257/1356}(no 4,8,9) = 3 or 7
9d. can't have both 6,7 -> no 6,7 in r8c2 (step 9a)
9e. 1 and 5 locked for n7

10. hidden killer pair 6,7 in n7 from 9b.
10a. -> r9c123 must have exactly one of 6 or 7
10b. -> killer pair 6,7 with r9c4: both locked for r9

11. "45" on r9: 4 innies r9c1234 = 24 and must have both 6 & 7 for r9
11a. = {2679/3678}(no 4)

12. "45" on r9: 1 outie r8c2 + 2 = 1 innie r9c1
12a. = [13/57]

13. killer pair 3,7 between h15(4)n7(step 9c) and r9c1, both locked for n7

14. 22(4)r8c2:
14a. must have 6 for r9 = {1678/2569}
14b. 8 in {1678} must be in r9c3 -> no 8 in r9c2
(took me a long time to see 14b)

15. 4 & 8 in n7 only in c3: locked for c3

Finally moving away from n7!
16. "45" on r1: 2 innies r1c12 = 11 (no 1)
16a. -> r2c234 = 19 (no 1)

17. 1 in n1 only in r1c3 or 36(7) -> no 1 in r4c3 (Common Peer Elimination CPE)

18. "45" on n4: 3 remaining innies r4c23 + r6c3 = 15
18a. must have 6 or 7 for r6c3
18b. but {168} blocked by no 1,8 for r4c3
18c. = {267} only: all locked for n4 and none in r3c3 (CPE)
18d. 2 locked for 36(7)r2c1 and r4
18e. no 4 in r5c4

19. 1 and 5 in n5 are both in 6(2) or both in r45c5 (hidden killer pair, or Locking-out cages)
19a. -> no 1 in r4c5

20. r4c1 = 1 (hsingle r4)
20a. r5c12 = {89}: both locked for r5
20b. hidden killer pair 8,9 in c1 -> r123c1 must have one of 8 or 9 (no eliminations yet)

cracker step
21. h19(3)r2c234 must have at least one of 8 or 9 (since {567}=18)
21a. 36(7)r2c1 must have 9
i. if 9 in r3c4 -> r36c4 = [98] -> h19(3)r2c234 must have 8 or 9 in n1 -> killer pair 8,9 with r123c1 (step 20b) -> no 8,9 in r1c23
ii. or 9 in n1 -> no 9 in r1c23
21b. -> no 9 in r1c23
21c. -> no 2 in r1c1 (h11(2))

22. 2 in c1 only in h15(4)n7 = {1257} only (step 9c): 2 & 7 locked for n9
22a. r9c1 = 3, placed for d/
22b. -> r8c2 = 1 (iodr9=+2), placed for d/
22c. -> r9c234 = [687](7 hsingle r9, then 6 hsingle r9/n7)
22d. -> r6c3 = 6 (outies n7=13)
22e. -> r6c9 = 2 (inniesr6=8)

23. r4c23 = {27}: 7 locked for r4 and 36(7)

24. 10(2)n5 = {46} only: both locked for c6 and n5
24a. -> r4c4 = 5 (placed for d\), r5c5 = 2 (placed for both d/\)
24b. r4c5 = 3

25. r23c5 = 7 = {16} only: both locked for n2 and c5

26. "45" on n2: 1 remaining outie r1c3 + 12 = 2 innies r23c4
26a. = [1]{49} only, 4 & 9 locked for c4 and n2
26b. and no 4,9 in r2c1 (CPE)

27. r78c3 = {49}: 9 locked for c3
27a. r3c3 = 3 (locked for d\)

28. 36(7)r2c1 must have 2,3,7 = {1236789/2345679}
28a. must have 6 which is only in r23c1: 6 locked for n1
28b. -> no 5 r1c2 (h11(2))

Pretty smooth from here