SudokuSolver Forum

3x3::k:4864:4864:4864:3841:2306:2306:2306:5635:5635:2052:2309:2309:3841:5635:5635:2310:5635:6151:2052:5640:5640:2825:2825:2825:2310:6151:6151:3082:3082:5640:1803:6412:6412:5389:5389:5389:3082:5134:5640:1803:6412:3343:5136:5389:3601:5134:5134:5134:6412:6412:3343:5136:3601:3601:4626:4626:787:3604:3604:3604:5136:5136:2581:4626:7446:787:7446:7446:3351:3096:3096:2581:7446:7446:3353:3353:3353:3351:3354:3354:3354:

Prelims

a) R12C4 = {69/78}
b) R23C1 = {17/26/35}, no 4,8,9
c) R2C23 = {18/27/36/45}, no 9
d) R23C7 = {18/27/36/45}, no 9
e) R45C4 = {16/25/34}, no 7,8,9
f) R56C6 = {49/58/67}, no 1,2,3
g) R78C3 = {12}
h) R78C9 = {19/28/37/46}, no 5
i) R89C6 = {49/58/67}, no 1,2,3
j) R8C78 = {39/48/57}, no 1,2,6
k) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
l) 9(3) cage at R1C5 = {126/135/234}, no 7,8,9
m) 24(3) cage at R2C9 = {789}
n) 11(3) cage at R3C4 = {128/137/146/236/245}, no 9

1a. Naked pair {12} in R78C3, locked for C3 and N7, clean-up: no 7,8 in R2C2
1b. Naked triple {789} in 24(3) cage at R2C9, locked for N3, clean-up: no 1,2 in R23C7
1c. 45 rule on N1 2 outies R45C3 = 13 = {49/58/67}, no 3
1d. 45 rule on N1 2 innies R3C23 = 9 = [18/27]/{36/45}, no 9, no 7,8 in R3C2
1e. 9 in N1 only in 19(3) cage at R1C1 = {289/379/469}, no 5, 9 locked for R1, clean-up: no 6 in R2C4
1f. 9(3) cage at R1C5 = {126/135} (cannot be {234} which clashes with 19(3) cage), no 4, 1 locked for R1
1g. 9 in R3 only in R3C89, locked for N3
1h. 45 rule on N6 2 innies R56C7 = 10 = {19/28/37/46}, no 5
1i. 45 rule on N6 2 outies R7C78 = 10 = {19/28/37/46}, no 5
1j. 45 rule on N3 2 outies R2C56 = 1 innie R1C7 + 10, min R2C56 = 11, no 1 in R2C56
1k. 45 rule on R89 3 innies R8C139 = 10 = {127/136/145/235}, no 8,9, clean-up: no 1,2 in R7C9
1l. 5 of {145} must be in R8C1 -> no 4 in R8C1

2a. 19(3) cage at R1C1 (step 1e) = {289/379/469}
2b. Consider combinations for R12C4 = [69]/{78}
R12C4 = [69]
or R12C4 = {78}, naked pair {78} in R2C49, locked for R2 => R2C23 = {36/45} => 19(3) cage = {289/379} (cannot be {469} which clashes with R2C23)
-> 19(3) cage = {289/379}, no 4,6
2c. 4 in R1 only in R1C89, locked for N3 and 29(5) disjoint cage at R1C8, clean-up: no 5 in R23C7 (step 1d)
2d. Naked pair {36} in R23C7, locked for C7 and N3, clean-up: no 4,7 in R56C7 (step 1h), no 4,7 in R7C8 (step 1i), no 9 in R8C8
2e. 4 in N2 only in 11(3) cage at R3C4 = {146/245}, 4 locked for R3, no 3,7,8
2f. 4 in N1 only in R2C23 = {45}, 5 locked for R2 and N1, clean-up: no 3 in R23C1
2g. R3C23 = 9 (step 1d) = [18] (cannot be [27] which clashes with R23C1, cannot be {36} which clashes with R3C7), clean-up: no 7 in R23C1, no 6 in 11(3) cage, no 5 in R45C3 (step 1c)
2h. Naked triple {245} in 11(3) cage, 2,5 locked for N2, 2 locked for R3 -> R23C1 = [26], R23C7 = [63], R2C8 = 1, clean-up: no 9 in R7C7 (step 1i)
2i. Naked triple {379} in 19(3) cage, 3,7 locked for R1, clean-up: no 8 in R2C4
2j. Naked pair {16} in R1C56 -> R1C7 = 2 (cage sum), clean-up: no 8 in R56C7 (step 1h), no 8 in R7C8 (step 1i)
2k. R1C4 = 8 (hidden single in R1) -> R2C49 = [78], clean-up: no 2 in R8C9
2l. Naked pair {19} in R56C7, locked for C7, N6 and 20(4) cage at R5C7, clean-up: no 3 in R8C8
2m. Naked quad (4578) in R7C7 + R8C78 + R9C7, locked for N9, clean-up: no 3,6 in R78C9
2n. R78C9 = [91] -> R3C89 = [97], R78C3 = [12]
2o. R8C39 = [21] -> R8C1 = 7 (step 1k), clean-up: no 5 in R8C78, no 6 in R9C6
2p. Naked pair {48} in R8C78, locked for R8 and N9 -> R79C7 = [75], R7C8 = 3 (step 1i), clean-up: no 9 in R9C6
2q. Naked pair {26} in R9C89, locked for R9
2r. R8C1 = 7 -> R7C12 = 11 = [56]
2s. Naked triple {248} in 14(3) cage at R7C4, 4,8 locked for N8, R9C6 = 7 -> R8C6 = 6, R1C56 = [61]
2t. Naked triple {359} in R8C245, 3,9 locked for 22(5) disjoint cage at R8C2 -> R9C12 = {48}, 4 locked for N7
2u. Killer triple 3,7,9 in R1C3, R45C3 and R9C3, locked for C3

[It’s flowed up to here; now it’s time so start nibbling at the combinations in the middle three rows.]
3a. 12(3) cage at R4C1 = {129/138} (cannot be {147} which clashes with R45C3, cannot be {237} because 2,7 only in R4C2, cannot be {345} which clashes with R45C3 + R6C3, killer ALS block), no 4,5,7, 1 locked for N4
3b. 2 of {129} must be in R4C2 -> no 9 in R4C2
3c. R45C4 = {16/34} (cannot be {25} which clashes with R37C4 (ALS block), no 2,5
3d. 14(3) cage at R5C9 = {248/257/347/356}
3e. 7,8 of {248/257/347} must be in R6C8 -> no 2,4 in R6C8
3f. Consider permutations for R8C78 = {48}
R8C78 = [48] => 14(3) cage = {257/347/356}
or R8C78 = [84] => R1C89 = [54] => 14(3) cage = {257/356}
-> 14(3) cage = {257/347/356}, no 8

4a. 2,7 in N5 only in 25(5) cage at R4C5 = {12679/23479/23578}
4b. 6 of {12679} only in R6C4 -> no 1 in R6C4
4c. 20(4) cage at R5C2 = {2459/3458} (cannot be {2369/2468/3467} which clash with R45C3, cannot be {2378} which clashes with 12(3) cage at R4C1 (step 3a), cannot be {2567} because no 2,5,6,7 in R6C1), 4 locked for N4, clean-up: no 9 in R45C3 (step 1c)
4d. 3 of {3458} must be in R56C2 (R6C1 cannot be 3 which clashes with R1C1 = {39} + R45C1 {19}), no 3 in R6C1
4e. Naked pair {67} in R45C3, locked for C3, 7 locked for N4
4f. 8 in N6 only in 21(4) cage at R4C7 = {2478/2568/3468}
4g. 5 of {2568} must be in R45C8 (R45C8 cannot be {26} which clashes with R9C8), no 5 in R4C9
4h. 14(3) cage at R5C9 (step 3f) = {257/347/356}
4i. Consider combinations for 21(4) cage = {2478/2568/3468}
21(4) cage = {2478} => 5 in R4 only in R4C56, locked for N5 => R56C6 = {49}, 4 locked for N5
or 21(4) cage = {2568/3468}, 6 locked for N6 => R6C4 = 6 (hidden single in R6) => R45C5 = {34}, 4 locked for N5
-> 25(5) cage at R4C5 = {12679/23578}, no 4
4j. 6 of {12679} must be in R6C4 -> no 9 in R6C4
4k. 6 of {12679} in R6C4 -> R45C4 = {34}, 4 locked for N5, R56C6 = {58}, locked for C6, R37C6 = {24}, locked for C6 -> 9 of {12679} only in R4C6 = 9 -> no 9 in R456C5
4l. 9 in C4 only in R89C4, locked for N8
4m. R2C5 = 9 (hidden single in C5) -> R2C6 = 3
4n. R1C2 = 7 (hidden single in R1)

[Step 5b took me a while to find as a clean forcing chain]
5a. 12(3) cage at R4C1 (step 3a) = {129/138}, 14(3) cage at R5C9 (step 3f) = {257/347/356}
5b. Consider combinations for R56C6 = {49/58}
R56C6 = {49}, 9 locked for N5 => R4C1 = 9 (hidden single in R4), R4C2 + R5C1 = [21]
or R56C6 = {58}, 5 locked for N5 => R4C8 = 5 (hidden single in R4), 14(3) cage = {347}, 4 locked for N6 => R4C7 = 8
-> R4C2 = {23}, R45C1 = {18/19}
[Cracked, at last.]
5c. R1C1 = 3 (hidden single in C1) -> R1C3 = 9, R9C345 = [391]
5d. 1 in N5 only in R45C4 = {16}, 6 locked for N5
5e. 4 in N5 only in R56C6 = {49}, 4 locked for C6, 9 locked for N5
5f. R4C1 = 9 (hidden single in R4) -> R4C2 + R5C1 = [21], R45C4 = [16], R45C3 = [67], R56C7 = [91], R56C7 = [49]
5g. 4 in R4 only in R4C789, locked for N6
5h. 21(4) cage at R4C7 (step 4f) = {2478} (only remaining combination) -> R4C789 = [874], R5C8 = 2

and the rest is naked singles.

.-------------------------------.-------------------------------.-------------------------------.
| 2346789   2346789   346789    | 678       12356     12356     | 12356     23456     23456     |
| 123567    123456    345678    | 789       123456789 123456789 | 3456      123456    78        |
| 123567    123456    345678    | 12345678  12345678  12345678  | 3456      789       789       |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 123456789 456789    | 123456    123456789 123456789 | 123456789 123456789 123456789 |
| 123456789 123456789 456789    | 123456    123456789 456789    | 123456789 123456789 123456789 |
| 123456789 123456789 3456789   | 123456789 123456789 456789    | 123456789 123456789 123456789 |
:-------------------------------+-------------------------------+-------------------------------:
| 3456789   3456789   12        | 123456789 123456789 123456789 | 123456789 123456789 12346789  |
| 3456789   3456789   12        | 123456789 123456789 456789    | 345789    345789    12346789  |
| 3456789   3456789   3456789   | 123456789 123456789 456789    | 123456789 123456789 123456789 |
'-------------------------------.-------------------------------.-------------------------------'

.

Then
2. "45" on r12: 2 innies r2c19 - 7 = 1 outie r3c7
2a. since 1 innie and 1 outie are in n3 -> difference can't be 0
2b. -> r2c1 can't be 7 (Innie Outie Difference Unequal IOU)
2c. -> no 1 in r3c1

3. "45" on r12: 3 innies r2c179 = 16
3a. must have 7,8 for r2c9 but can't have both since there are no others
3b. -> no 1 in r2c1
3c. -> no 7 in r3c1

4. 1 in n1 is in either the 9(2) or h9(2)r3c23 -> 8 locked in r23c3 for n1 and c3 (Locking cages)
4a. 1 also locked for c2
4b. no 5 in r45c3 (h13(2))

5. r1c4 = 8 (hsingle r1)
5a. r2c49 = [78]

6. 9(2)r2c2: {36} blocked by the 8(2)n1 which needs one of those
6a. 9(2) = {45} only: both locked for r2, n1

7. two remaining innies r12 = 8 = [26] only

On from there with the easy middle.

.-------------------------------.-------------------------------.-------------------------------.
| 39        379       39        | 8         6         1         | 2         45        45        |
| 2         45        45        | 7         39        39        | 6         1         8         |
| 6         1         8         | 245       245       245       | 3         9         7         |
:-------------------------------+-------------------------------+-------------------------------:
| 1389      238       67        | 1346      12345789  234589    | 48        245678    23456     |
| 1389      234589    67        | 1346      12345789  4589      | 19        245678    23456     |
| 489       234589    45        | 234569    12345789  4589      | 19        567       23456     |
:-------------------------------+-------------------------------+-------------------------------:
| 5         6         1         | 24        248       248       | 7         3         9         |
| 7         39        2         | 359       359       6         | 48        48        1         |
| 48        48        39        | 139       139       7         | 5         26        26        |
'-------------------------------.-------------------------------.-------------------------------'

Then
5. hsingle 7 n1 -> r1c2 = 7

6. 20(4)n4 has both 4 & 5 (Andrew's step 4c)
6a. 13(2)n5 has one of 4 or 5
6b. 14(3)n6 has one of 4 or 5 (step 3f)
6c. -> 4 and 5 locked in those cages for r56
6e. -> no 3 r4c4

Key step. This step could be written as a straight forcing chain but I didn't see it like that. I like to "see them in my head" so have to break the logic up into clumps to remember where things are up to.
7. if 5 in 21(4)r4c7 -> = {2568} only (Andrew's step 4f): ie must have 8 in r4c7
7a. if {58} in 13(2)n5 -> 5 in r4 only in 21(4) -> must have 8 r4c7
7b. -> r4c2 = (23)
7c. if 20(4)n4 has 9 must be {2459} (Andrew's step 4c)
i. also 9 locked for r56 because r56c7 = {19} -> {58} in 13(2)
ii. -> 8 in r4c7 (step 7a)
iii. and 3 in r4c2 (steps 7b and 7c)
iv. and 9 in r8c2
v. -> {2459} has 9 in r6c1 only
7d. result -> no 9 in r56c2

8. hsingle 9 c2 -> r8c2 = 9, r9c3 = 3, r1c13 = [39]

9. 14(3)n6 = {257/347/356}: has at most one of 6 or 7
9a. -> hidden killer pair 6,7 in r6 -> r6c45 must have at least one of 6 or 7
9b. "45" on r6789: 3 outies r5c269 - 2 = 3 innies r6c457
9c. innies must have 1 for r6 and at least one of 6,7
9d. outies must have both 4 & 5 for r5
9e. -> max. 3 outies = {459} = 18 -> max. 3 innies = 16
9f. -> [971] = 17 blocked from r6c457
9g. -> no 9 in r6c4 (step 9c)

10. hsingle 9 c4 -> r9c4 = 9, r9c5 = 1

11. 1 in n5 only in 7(2) = {16}: 6 locked for n5
11a. -> r6c5 = 7 (step 9a)

12. hsingle 1 in r6 -> r56c7 = [91]

13. "45" on r6789: 3 outies r5c269 - 10 = 1 remaining innie r6c4
13a. remembering 3 outies must have both 4 & 5 for r5
13b. -> can't total 13
13c. -> no 3 in r6c4
13d. -> r6c4 = 2
13e. -> 3 outies = 12 = {345} only: 3 locked for r5
13f. -> r5c5 = 8

singles now.

1. Start is straightforward to get to the position shown at end of Andrew's Step 4f except that r45c3 is not resolved.

2. 12(3)n4 -> r456c1 cannot = +12(3) -> r19c9 cannot be +13(2)
-> if r1c1 = 9 this puts r9c12 = [84]

r45c3 = +13(2) = {49} or {67}
Trying r45c3 = {49} puts r8c2 = 9 and r1c1 = 9
It also puts r2c23 = [45]
This leaves no place for 4 in n9

-> r45c3 = {67}
-> r1c2 = 7
Also -> r26c3 = {45]
-> 12(3)n4 only from <129> or {138} with 1 in r45c1

3! 4 and 5!
(45) in n4 both in 20(4)n4. I.e., both in r56
13(2)n5 from {49} or {58} (I.e., also in r56)
-> One of (45) in r4 in n6 (In r4c789)
But (since 9 elsewhere in n6) not both (45) in 21(4)n6
-> The other of (45) is in 14(3)n6 (I.e., in r56)
-> That other of (45) in r4 in n5 (in r4c456)

Putting the above another way, one of the following is correct:
13(2)n5 = {58) and 4 in 14(3)n6
13(2)n5 = {49} and 5 in 14(3)n6

4. (23) not both in 14(3)n6
Since 21(4)n6 already contains one of (45) it cannot also contain both (23)
-> One each of (23) in 21(4)n6 and 14(3)n6
Note that if 2 in 14(3)n6 that can only be {257} (Since r4c7 from (48) and 9 already in n6)

5. r36c4 are two of (245)
-> 7(2)n5 from {16} or {34}
(A) Trying 7(2)n5 = {34} puts 13(2)n5 = {58} puts 5 in n6 in r4c89
(B) Trying 7(2)n5 = {16} puts 6 in n6 in r6c89
In neither case can 14(3)n6 be {257}
-> 2 in n6 in 21(4) and 3 in n6 in 14(3) in r56c9

6. Extending those two cases...
(A) Trying 7(2)n5 = {34} puts 13(2)n5 = {58}
(B) Trying 7(2)n5 = {16} puts 6 in n6 in r6c89 puts 14(3)n6 = {356}
-> (Last lines in Step 3) 13(2)n5 = {49}
-> One of the following is correct:
(A) 7(2)n5 = {34} and 13(2)n5 = {58}
(B) 7(2)n5 = {16} and 13(2)n5 = {49}

7! Extending those cases still further (using Step 3) ...
(A) 13(2)n5 = {58} puts 4 in 14(3)n6 puts r4c7 = 8
(B) 13(2)n5 = {49} puts 14(3)n6 = {356} and also puts 9 in r4 in 12(3)n4 = [921]
This puts 21(4)n6 = [8742]
In both cases r4c7 = 8
-> 12(2)n9 = [48]

8! Finally! Use Step 2 again.
-> 12(3)n4 only from <129> or [138]
-> r19c9 cannot be a 9 since that would put r9c1 = 8 (Step 2) leaving no solution for 12(3)n4
-> 13(3)n1 = [379]
-> 29(5)r8c2 = [9{35}{48}]
-> 13(3)r9c3 = [3{19}]
-> r378c4 is three of {2345}
-> 7(2)n5 = {16}
etc.

Phew!