SudokuSolver Forum

3x3:d:k:4352:4609:4609:4609:4609:5122:5122:5122:3075:6148:4352:4609:2565:3078:4871:4871:3075:6920:1289:6148:4352:2565:3078:4871:3075:6920:6920:1289:6148:7178:2059:2059:5644:6920:6920:3853:6148:7178:7178:7178:5644:5644:5644:6920:3853:11534:11534:7178:5903:5903:5644:2064:2064:2064:2321:11534:11534:11534:5903:11534:3602:3859:3859:2321:3604:5397:11534:11534:11534:4886:3602:3859:3604:5397:5397:5397:4886:4886:4886:4886:3602:

I enjoyed finding the clashes with R9C5 and R9C6.

Prelims

a) R23C4 = {19/28/37/46}, no 5
b) R23C5 = {39/48/57}, no 1,2,6
c) R34C1 = {14/23}
d) R4C45 = {17/26/35}, no 4,8,9
e) R45C9 = {69/78}
f) R78C1 = {18/27/36/45}, no 9
g) 14(2) cage at R8C2 = {59/68}
h) 20(3) cage at R1C6 = {389/479/569/578}, no 1,2
i) 19(3) cage at R2C6 = {289/379/469/478/568}, no 1
j) 23(3) cage at R6C4 = {689}
k) 8(3) cage at R6C7 = {125/134}
l) 18(5) cage at R1C2 = {12348/12357/12456}, no 9
m) and, of course, 45(9) cage at R6C1 = {123456789}

1a. 8(3) cage at R6C7 = {125/134}, 1 locked for R6 and N6
1b. 23(3) cage at R6C4 = {689}, CPE no 6,8,9 in R45C5, clean-up: no 2 in R4C4
1c. 45 rule on N9 2 outies R9C56 = 3 = {12}, locked for R9, N8 and 19(5) cage at R8C7
1d. 45(9) cage at R6C1 = {123456789}, 1 locked for R7 and N7, clean-up: no 8 in R78C1
1e. 45 rule on R6789 2 innies R6C36 = 12 = {39/48/57}, no 1,2,6
1f. 18(5) cage at R1C2 = {12348/12357/12456}, CPE no 1,2 in R1C1
1g. R7C5 ‘sees’ R7C2346 + R8C456 of 45(9) cage at R6C1 -> whichever of 6,8,9 is in R7C5 must also be in R6C12
1h. 45 rule on R789 2 outies R6C12 = 1 innie R7C5 -> R6C12 = {26/28/29}, 2 locked for R6, N4 and 45(9) cage, clean-up: no 3 in R3C1, no 5 in 8(3) cage at R6C7
1i. R6C36 = {57} (hidden pair in R6)
1j. Naked pair {57} in R6C36, CPE no 5,7 in R3C3 using D\
1k. Naked triple {134} in 8(3) cage at R6C7, 3,4 locked for N6
1l. Combined cage R34C1 + R78C1 = {14}{27}/{14}{36}/[23]{45}, 4 locked for C1
1m. Combined cage R6C4 + 14(2) cage at R8C2 = 6{59}/8{59}/9{68}, 9 locked for D/
1n. 45 rule on D\ 3 innies R4C4 + R5C5 + R6C6 = 14 = {257/347/356} (cannot be {167} = [617] because R45C5 = [12] clashes with R9C5), no 1, clean-up: no 7 in R4C5
1o. 7 of {257/347} must be in R6C6 (cannot be [725] because R45C5 = [12] clashes with R9C5), no 7 in R4C4, clean-up: no 1 in R4C5
1p. 2,4 of {257/347} must be in R5C5 -> no 7 in R5C5
1q. Hidden killer pair 8,9 in 17(3) cage at R1C1 and 14(3) cage at R7C7 for D\, neither cage can contain both of 8,9, no 8,9 in R4C4 + R5C5 + R6C6 -> each cage must contain one of 8,9
1r. 14(3) cage at R7C7 = {149/158/248} (cannot be {239} which clashes with R4C4 + R5C5 + R6C6), no 3,6,7
1s. Killer pair 4,5 in R4C4 + R5C5 + R6C6 and 14(3) cage, locked for D\
1t. Hidden killer pair 1,2 in 14(3) cage and 15(3) cage at R5C8 for N9, 14(3) cage contains one of 1,2 -> 15(3) cage must contain one of 1,2 = {159/249/258/267} (cannot be {168} which clashes with 14(3) cage), no 3
1u. Consider combinations for R4C45 = {35}/[62]
R4C45 = {35}, locked for N5
or R4C45 = [62] => R5C5 + R6C6 = [35]
-> 3,5 in R4C45 + R5C5 + R6C6, locked for N5, no 5 in R5C5
1v. 7 in N5 only in R456C6 + R5C4, CPE no 7 in R5C7
1w. 7 in N6 only in R4C789 + R5C89, CPE no 7 in R23C9

2a. 45 rule on N36 3+1 innies R1C78 + R25C7 = 28
2b. Max R1C78 + R2C7 = 24 -> no 2 in R5C7
2c. Max R125C7 = 24 -> min R1C8 = 4
2d. 2 in N6 only in R4C78 + R5C8, locked for 27(6) cage at R2C9, no 2 in R2C9 + R3C89
2e. Min R5C7 + R6C6 = 11 -> max R4C6 + R5C56 = 11, no 9 in R5C6
2f. R4C4 + R5C5 + R6C6 (step 1n) = {257/356} (cannot be {347} = [347] because R45C6 + R5C7 = 11 = {12}8 clashes with R9C6), no 4, 5 locked for N5 and D\, clean-up: no 3 in R4C4
2g. Naked pair {23} in R45C5, locked for C5, 2 locked for N5 -> R9C56 = [12], clean-up: no 9 in R23C5
2h. 4 on D\ only in 14(3) cage at R7C7, locked for N9
2i. 3 in N9 only in 19(5) cage at R8C7 = {12358/12367}, no 9
2j. 1 in C6 only in R45C6, locked for N5
2k. Hidden killer pair 1,2 in R1C4 and R23C4 for N2, neither can contain both of 1,2 -> R1C4 = {12}, R23C4 = {19/28}
2l. 19(3) cage at R2C6 = {379/469/478/568} (cannot be {289} which clashes with R23C4), no 2
2m. 2 in N3 only in 12(3) cage at R1C9, locked for D/ -> R5C5 = 3, placed for both diagonals, R4C5 = 2, R4C4 = 6, placed for D\, R6C6 = 5, R6C3 = 7, clean-up: no 9 in R5C9
2n. Naked pair {89} in R6C45, locked for R6, N5 and 23(3) cage at R6C4 -> R5C4 = 4, R7C5 = 6, clean-up: no 3 in R8C1
2o. Naked pair {17} in R45C6, 7 locked for C6, R5C5 = 3, R6C6 = 5 -> R5C7 = 6 (cage sum), clean-up: no 9 in R4C9
2p. Naked pair {78} in R45C9, locked for C9 and N6
2q. Naked pair {59} in R4C78, locked for R4, C6 and 27(6) cage at R2C9
2r. 27(6) cage at R2C9 = {123579/124569} (only combinations containing 2,5,9), no 8, 1 locked for N3
2s. 2 in N3 only in 12(3) cage at R1C9 = {246} (only remaining combination), 4,6 locked for N3 and D/, clean-up: no 8 in 14(2) cage at R8C2
2t. Naked pair {13} in R23C9, locked for C3 and N3 -> R3C8 = 7, R6C9 = 4, R9C9 = 9, placed for D\, R7C7 + R8C8 = [41], 1 placed for D\, 12(3) cage = [642], 17(3) cage at R1C1 = [728], R6C78 = [13], 14(2) cage = [95], R7C3 = 1, 1,9 placed for D/ -> R45C6 = [71], R6C45 = [89], naked pair {19} in R23C4, locked for N2, 9 locked for C4 -> R1C4 = 2, clean-up: no 5 in R2C5, no 2 in R7C1, no 2,4 in R8C1
2u. R1C2 = 1 (hidden single in R1) -> R34C1 = [41], R3C5 = 5 -> R2C5 = 7
2v. Naked pair {25} in R78C9, locked for N9

and the rest is naked singles.

Preliminaries by SudokuSolver
Cage 5(2) n14 - cells only uses 1234
Cage 14(2) n7 - cells only uses 5689
Cage 15(2) n6 - cells only uses 6789
Cage 8(2) n5 - cells do not use 489
Cage 12(2) n2 - cells do not use 126
Cage 9(2) n7 - cells do not use 9
Cage 10(2) n2 - cells do not use 5
Cage 23(3) n58 - cells ={689}
Cage 8(3) n6 - cells do not use 6789
Cage 20(3) n23 - cells do not use 12
Cage 19(3) n23 - cells do not use 1
Cage 18(5) n12 - cells do not use 9

Note: no clean-up done unless stated.
1. "45" on n9: 2 outies r9c56 = 3 = {12}: both locked for r9, n8 and 19(5)

2. 8(3)r6c7 = {125/134}: 1 locked for r6 and n6

3. "45" on n5: 1 innie r5c4 + 8 = 2 outies r5c7 + r7c5.
3a. 1 innie sees one outie so must be unequal -> no 8 in r7c5 (Innie Outie Unequal IOU)

4. 45(9)r6c1 must have both 6,9 and r7c5 = (69) -> r7c5 repeats in r6c12
4a. -> an implied 23(3) in r6c1245
4b. -> 6,8,9 locked for r6, 8 locked for n5

5. "45" on r789: 4 outies r6c1245 = 25 and must have 6,8,9 = 23 for r6
5a. -> must be a 2 in r6c12: 2 locked for r6, n4 and 45(9)

6. 8(3)n6 = {134}: 3,4 locked for n6 and r6

7. naked pair {57} in r6c36: r5c4 sees both those -> no 5,7 in r5c4 (Common Peer Elimination CPE)
7a. r3c3 sees both {57} -> no 5,7 in r3c3 (CPE)

8. from step 3, r5c4 + 8 = r5c7 + r7c5
8a. = [356/329/466/686/989]
8b. r5c4 = (3469), r5c7 = (2568)

9. "45" on n36: 1 outie r1c6 + 8 = 2 innies r25c7
9a. min. r1c6 = 3 -> min. r25c7 = 11
9b. -> no 2 in r2c7

10. if 2 in n3 in 12(3) -> no 2 in r5c5 (through d/)
10i. or 2 in n3 in 27(6) -> 2 in n6 in r5c7
10a. result: no 2 in r5c5

11. "45" on d\: 1 outie r4c5 + 6 = 2 innies r5c5 + r6c6
11a. -> no 6 in r5c5 (IOU)
[Alternatively, from the 23(3)r6c4 -> no 6 (or 9) in r45c5]

12. "45" on d\: 3 innies r4c4 + r5c5 + r6c6 = 14
12a. but {167} as [617] only, blocked by [22] in r49c5 (combined cages h14(3)+8(2))
12b. must have 5 or 7 for r6c6
12c. = {257/347/356}(no 1,9)
12d. = [257/275/347/635]
12e. ie r4c4 = (236)
12f. -> 8(2)n5 = [35]/{26}

13. 1,7 in n5 only in 22(5)
13a. {12379/12478/13567}
13b. 1 locked for c6
13c. r9c56 = [12]
13d. {12478} blocked since 2,8 are only in r5c7
13e. 22(5) = {12379/13567}(no 4)

14. r5c4 = 4 (hsingle n5)
14a. -> r5c7 + r7c5 = 12 (iodn5=-8)
14b. = [66] only
14c. -> 22(4)n5 in n5 = {1357}[6](no 9)
14d. 5 locked for n5
14e. r4c45 = [62] (6 placed for d\)

15. 15(2)n6 = {78} only: both locked for n6, c9

16. r4c78 = {59}: both locked for 27(6) and r4
16a. r5c8 = 2

17. h14(3)n5 = [635] only, 3 placed for both d/\ and 5 for d\
17a. r6c3 = 7

18. "45" on d/: 3 remaining innies r4c6 + r6c4 + r7c3 = 16 = [781] only: all placed for d/

On from there

1. Outies n9 = r9c56 = +3(2) = {12}
8(3)n6 contains a 1 -> 1 in 45(9) in r7c23
Remaining IOD n7 -> whatever else is in r7c23 is also in r9c4 (Not 2)
-> 2 in 45(9) in r6c12
-> 8(3)n6 = {134}

2. -> Innies r6789 = r6c36 = +12(2) = {57}

3. 23(3)r6c4 = {689}
IOD n5 -> r5c4 + 8 = r5c7 + r7c5
-> r7c5 not 8
-> 8 in r6c45

4. Innies n36 = r1c78 + r2c7 + r5c7 = +28(4)
-> Min r5c7 = 5 (4 already in n6)
-> 2 in n6 in 27(6)

5! IOD n5:
If r6c45 = {68} -> r5c4 = r5c7 + 1
If r6c45 = {89} -> r5c4 = r5c7 - 2
Since r5c4 'sees' both r6c36 -> r5c4 not (57) (or 8)
Whatever goes in r5c7 goes in n5 in r4c45 or r6c45
-> Possibilities for r5c47:
a) r6c45 = {68} -> r5c47 = [98]
b) r6c45 = {89} -> r5c47 = [46] or [68] (Cannot be [35] since that leave no place for 5 in n5)
-> r5c7 from (68)
-> 5 in n6 in 27(6)
Also since remaining Innies n36 in n3 are +22(3) or +20(3) -> 2 in n3 in 12(3)n3
-> 5 in n3 in Innies n3 (r1c78 or r2c7)

6! 1 in n2 in r1c45 or in 10(2)n2 = {19}
Innies n236 -> r1c45 + r5c7 = +12(3)
r5c7 from (68)

a) r5c7 = 8 puts r1c45 = {13} which puts whichever of (57) is in r6c6 into n2 in 12(2)n2 = {57}
b) r5c7 = 6 and 5 in r2c7 puts r123c6 = [3{68}] (since r123c6 = +17(3) in this case) which puts 12(2)n2 = {57}
c) r5c7 = 6 and 5 in r1c78 puts r1c45 = {24} which puts 10(2)n2 = {19} (HS 1 in n2) and 12(2)n2 = {57}
-> In all cases 12(2)n2 = {57}

7! Innies D\ = [r4c4,r5c5,r6c6] = +14(3)
Since 2 already in D/ -> [r4c4,r5c5,r6c6] cannot be <527>
-> No (57) in 8(2)n5
-> 8(2)n5 = [62] and 23(3)n5 = [{89}6]
-> r5c47 = [46]
-> 15(2)n6 = {78}
-> r1c45 = [24], 10(2)n2 = {19}
etc.