SudokuSolver Forum

1. Innies n3 -> r3c7 = 9
Combined cage 14(2)c5 and 15(2)c5 = +29(4) -> 14(2)c5 = [95] and 15(2)c5 = {78}
Since (59) in n1 cannot both go in 12(3) -> 25(4)n1 not {4678} - must have a 9
-> 9 in r1c12

2. Innies n7 -> r7c3 = 6
Innies n8 = r7c46 + r8c6 = +18(3) can only be [{39}6] or {459}
But the former leaves no solution for (IOD n9) r7c789 = +9(3)
-> Innies n8 = {459}
-> 5(2)n8 = {23} and 7(2)n8 = {16}

3. Innies n124 = r3c6 + r6c3 = +12(2)
-> Min r3c6 = 3
11(2)n2 from [83] or [74]
-> (HS 2 in n2) 8(2)n2 = {26}
-> 7(2)n8 = [61]

4. Remaining outies r12 = r3c89 = +11(2) which must be from {38} or {47}
-> (26) in r3 in n1 in r3c123 with 6 in r3c12.
11(2)n2 also from [83] or [74], and since at least one of those values must be in r2 in n1 - that value must also be in r3c89 in n3.
Since r3c89 = +11(2) -> Whichever of [83] or [74] is in 11(2)n2 is also in r3c89.
-> Those two values are also in r2c123 in n1.

5. Innies n2 = r23c4 + r3c6 = +12(3) = {138} or {147} with 1 in r23c4
IOD n14 -> r6c3 = r2c4 + r3c4
5 in n1 either in 25(4) or in 12(3)
Given previous placements, the only solution for 12(3)n1 with a 5 is [{35}4]
But that puts r3c4 = 1 and (IOD n14) another 5 in c3 in r6c3
-> 5 in n1 in 25(4)
-> the other two values in 25(4) = +11(2) which must be the same two values as in 11(2)n2
-> r1c12 = {59} and r2c12 = {38} or {47}
-> The two values in r3c46 are in n1 in r12c3

6. 11(2)n2 from [74] or [83]
Since 3 already in c4 -> Innies n2 = r23c4 + r3c6 = [813] or {147} with 1 in r23c4
I.e., 8 in n2 in r12c8 and r3c6 from (347)
-> (Innies n124) r6c3 from (985)
r456c5 = {123} or {124}
-> Innies n5 = r4c6,r6c4 = +9(2) = {45} or [36]
-> 8 in n5/c6 in r56c6

7. Innies n689 = r4c7 + r7c4 = +10(2)
Since r7c4 from (459) -> r4c7 from (651)
-> 8 in r4 only in r4c2 or r4c89

8! Can 16(3)r3c1 be [{26}8]?
This puts remaining Innies n4 = r46c3 = +10(2) = [19] which puts r3c6 = 3
But this leaves no solution for 10(3)r3c3
-> (HS 8 in r4) 9(2)n6 = {18}

9. -> 2 in r3/n1 not in r3c12 -> must be r3c3 = 2
-> since r3c4 cannot be from (35) nor r4c3 = 1 -> 10(3)r3c3 = [217]
-> Remaining Innies n4 = r4c2,r6c3 = [29] or [38]
-> (Innies n124) r3c6 from (34)
-> r12c4 = {78}
-> r56c6 = {78}
-> 9 in n5 in r45c4
-> 9 in n8 in r78c6
-> r7c4 from (45)
-> (Innies n689) r4c7 from (65)
Also 9 in n4 only in r6
-> 9 in n6 in r5c89
-> r4c4 = 9

10. r7c4 cannot be 5 since that puts (Innies n689) r4c7 = 5 which leaves no place for 5 in n5
-> r7c4 = 4 and r4c7 = 6
-> 13(2)n6 = [49] and 12(2)n6 = {57}
-> r6c78 = {23}
-> (HS 4 in n4) 10(2)n4 = {46}
-> r56c4 = [65] and r4c6 = 4
Also 8(2)n4 = {35}
-> 9(2)n4 = {18}
-> [r4c2,r6c3] = [29]
etc.

Prelims

a) R1C45 = {29/38/47/56}, no 1
b) R12C6 = {17/26/35}, no 4,8,9
c) R23C5 = {59/68}
d) R45C1 = {17/26/35}, no 4,8,9
e) R4C89 = {18/27/36/45}, no 9
f) R5C23 = {18/27/36/45}, no 9
g) R5C78 = {49/58/67}, no 1,2,3
h) R56C9 = {39/48/57}, no 1,2,6
i) R6C12 = {19/28/37/46}, no 5
j) R78C5 = {69/78}
k) R89C4 = {14/23}
l) R9C56 = {16/25/34}, no 7,8,9
m) 10(3) cage at R3C3 = {127/136/145/235}, no 8,9
n) 10(3) cage at R6C8 = {127/136/145/235}, no 8,9
o) 27(4) cage at R8C8 = {3789/4689/5679}, no 1,2

1a. 45 rule on N3 1 innie R3C3 = 9, clean-up: no 5 in R2C5, no 4 in R5C8
1b. 45 rule on N7 1 innies R7C7 = 6, clean-up: no 3 in R5C2, no 9 in R8C5
1c. R78C5 = {78} (cannot be [96] which clashes with R23C5), locked for C5 and N8, clean-up: no 3,4 in R1C4, no 6 in R23C5
1d. R23C5 = [95], clean-up: no 2,6 in R1C45, no 3 in R12C6, no 2 in R9C6
1e. 45 rule on N5 2 innies R4C6 + R6C4 = 9 = {18/27/36/45}, no 9
1f. 45 rule on R12 2 remaining outies R3C89 = 11 = {38/47}
1g. R3C89 = 11 -> R23C8 cannot total 11 (CCC, combo crossover clash) -> no 3 in R2C7
1h. Similarly R2C7 + R3C8 cannot total 11 -> no 3 in R2C8
1i. 9 in N6 only in R5C78 = [49] or R56C9 = {39} -> no 4 in R56C9 (locking-out cages), clean-up: no 8 in R56C9
1j. R9C56 = {16}/[25] (cannot be {34} which clashes with R89C4), no 3,4
1k. Killer pair 1,2 in R89C4 and R9C56, locked for N8
1l. 45 rule on N8 3 innies R7C46 + R8C6 = 18 = {369/459}
1m. 6 of {369} only in R8C6 -> no 3 in R8C6

2a. 45 rule on N124 2(1+1) innies R3C6 + R6C3 = 12 = [39/48/75/84] (cannot be [66] because no 6 in R6C3), R3C6 = {3478}, R6C3 = {4589}
2b. 45 rule on N2 3 innies R2C4 + R3C46 = 12 = {138/147/246} (cannot be {237} which clashes with R1C45 and with R12C6)
2c. 8 of {138} must be in R2C4 (cannot be {13}8 which clashes with R89C4), no 8 in R3C6, clean-up: no 4 in R6C3
2d. 3 of {138} must be in R3C6 -> no 3 in R23C4
2e. Killer pair 1,2 in R23C4 and R89C4, locked for C4, clean-up: no 7,8 in R4C6 (step 1e)
2f. 8 in N2 only in R12C4, locked for C4, clean-up: no 1 in R4C6 (step 1e)
2g. 8 in N2 only in R12C4, CPE no 8 in R1C3
2h. 45 rule on N689 2(1+1) innies R4C7 + R7C4 = 10 = [19/55/64/73], R4C7 = {1567}
2i. 45 rule on N6 3 innies R4C7 + R6C78 = 11 = {128/146/236/245} (cannot be {137} which clashes with R56C9), no 7, clean-up: no 3 in R7C4
2j. 5 of {245} must be in R4C7 -> no 5 in R6C78

3a. Consider placement for 9 in R4
R4C2 = 9 => 8 in R4 only in R4C89 = {18}, 1 locked for R4 => no 9 in R7C4 (step 2h)
or R4C4 = 9
-> no 9 in R7C4, clean-up: no 1 in R4C7 (step 2h)
3b. 9 in N8 only in R89C6, locked for C6
3c. R4C7 + R6C78 (step 2i) = {146/236/245} (cannot be {128} because R4C7 only contains 5,6), no 8
3d. 6 of {146/236} must be in R4C7 -> no 6 in R6C78
3e. R7C46 + R8C6 (step 1l) = {459} (cannot be {369} because R7C4 only contains 4,5), 4,5 locked for N8, clean-up: no 1 in R89C4, no 2 in R9C5
3f. Naked pair {23} in R89C4, locked for C4, clean-up: no 6 in R4C6 (step 1e)
3g. 2 in N2 only in R12C6 = {26}, locked for C6, 6 locked for N2 -> R9C56 = [61], clean-up: no 7 in R6C4 (step 1e)
3h. 2 in R3 only in R3C123, locked for N1
3i. 12(3) cage at R1C3 = {138/147/345}, no 9
3j. 6 in R3 only in R3C12, locked for N1 and 16(3) cage at R3C1, no 6 in R4C2
3k. 16(3) cage = {169/268/367}, no 4,5
3l. 9 of {169} must be in R4C2 -> no 1 in R4C2
3m. 45 rule on N4 3 innies R4C23 + R6C3 = 18 = {189/279/378/459}
3n. 4 of {459} must be in R4C3 -> no 5 in R4C3
3o. 10(3) cage at R3C3 = {127}, 2 locked for C3, clean-up: no 7 in R5C2
3p. R4C23 + R6C3 = {189/279/378}, no 5, clean-up: no 7 in R3C6 (step 2a)
3q. Naked pair {34} in R1C5 + R3C6, locked for N2
3r. 7 in N2 only in R123C4, locked for C4
3s. 12(3) cage at R1C3 = {138/147} (cannot be {345} because 3,4,5 only in R12C3), no 5
3t. 12(3) cage = {138/147}, CPE no 1 in R2C12
3u. Caged X-Wing for 1 in 12(3) cage and 10(3) cage, no other 1 in C34, clean-up: no 8 in R5C2
3v. Killer pair 3,4 in R3C6 and R3C89, locked for R3

4a. R4C6 + R6C4 (step 1e) = [36/45] (cannot be [54] which clashes with R4C7 + R7C4 (step 2h) = [55/64])
4b. Naked pair {34} in R34C6, locked for C6
4c. Naked pair {59} in R89C6, 5 locked for C6 and N8
4d. R7C4 = 4 -> R4C7 = 6 (step 2h), clean-up: no 3 in R4C89, no 2 in R5C1, no 7 in R5C78
4e. R4C89 = {18/27} (cannot be {45} which clashes with R5C78), no 4,5
4f. R5C23 = [18/27/63] (cannot be {45} which clashes with R5C78), no 4,5
4g. 4 in N4 only in R6C12 = {46}, locked for R6, 6 locked for N4, R6C4 = 5 -> R4C6 = 4 (step 1e), R3C6 = 3, R6C3 = 9 (step 2a), R1C5 = 4 -> R1C4 = 7, R3C4 = 1, R2C4 = 8 -> R12C3 = {13}, locked for N1, 3 locked for C3, clean-up: no 8 in R3C89 (step 1f), no 2 in R4C1, no 3,7 in R5C9
4h. Naked pair {47} in R3C89, locked for N3, 7 locked for R3 -> R34C4 = [27], R5C3 = 8 -> R5C2 = 1, clean-up: no 2 in R4C89, no 5 in R5C78
4i. R5C78 = [49], R5C9 = 5 -> R6C9 = 7
4j. Naked pair {18} in R4C89, 1 locked for R4 and N6
4k. R45C1 = [53] -> R4C2 = 2
4l. Naked pair {68} in R3C12, 8 locked for N1 -> R1C12 = [95]
4m. 5,7 in N3 only in 14(3) cage at R2C7 = {257} = {25}7, 2 locked for R2 and N3 -> R12C6 = [26]
4n. 10(3) cage at R6C8 = {235} (only remaining combination) -> R7C8 = 5, R7C6 = 9 -> R67C7 = [21]
4o. R8C6 = 5 -> R89C7 = {37}, 3 locked for C7 and N9
4p. R8C3 = 4 -> R78C2 = {37}, locked for N7, 7 locked for C2

and the rest is naked singles.

.-------------------------------.-------------------------------.-------------------------------.
| 123456789 123456789 1234579   | 78        34        1267      | 12345678  12345678  12345678  |
| 12345678  12345678  1234578   | 124678    9         1267      | 1245678   1245678   12345678  |
| 1234678   1234678   12347     | 12467     5         347       | 9         3478      3478      |
:-------------------------------+-------------------------------+-------------------------------:
| 123567    123456789 123457    | 345679    12346     23456     | 1567      12345678  12345678  |
| 123567    12345678  1234578   | 345679    12346     123456789 | 45678     456789    3579      |
| 12346789  12346789  589       | 34567     12346     123456789 | 12345678  1234567   3579      |
:-------------------------------+-------------------------------+-------------------------------:
| 12345789  12345789  6         | 3459      78        3459      | 1234578   123457    123457    |
| 12345789  12345789  12345789  | 1234      78        4569      | 12345678  3456789   3456789   |
| 12345789  12345789  12345789  | 1234      126       156       | 12345678  3456789   3456789   |
'-------------------------------.-------------------------------.-------------------------------'

End of Andrew's step 2h above
3. 10(3)r3c3 = {127/136/145/235}
3a. 5 in {145} must be in r4c3 -> no 4 in r4c3

4. 6 in r3 in r3c4 -> r34c3 = 4 = {13}
4a. or 6 in r3 is in 16(3)r3c1 -> no 6 in r4c2
4b. and "45" on n4: 3 innies r4c23 + r6c3 = 18
4c. but [675] leaves no 6 for r3
4d. the only way for r4c23 to sum to 13 is [85]
4e. -> no 5 in r6c3
4f. -> no 7 in r3c6

5. 8 in r3 is in h11(2) = {38} -> h12(2)r3c6+r6c3 = [48]
5a. or 8 in r3 is in 16(3)r3c1
5b. both ways, no 8 in r4c2

6. now have hidden single 8 in r4 -> 9(2)n6 = {18}

On from there