SudokuSolver Forum

1. D\/
a) Innie R5 = R5C5 = 9
b) 1,2 locked in 20(4) @ D\ = {1289} -> R3C3+R4C4 = (12), R3C4+R4C3 = (89)
c) 1,2 locked in 20(4) @ D/ = {1289} locked for N3 -> R1C9+R2C8 = (12), R1C8+R2C9 = (89)
d) 3 locked in 27(4) @ D\ = {3789} locked for N1; R1C2+R2C1 <> 3
e) 8,9 locked in 27(4) + 20(4) for R12
f) 22(4) = {4567} -> 7 locked for C4+N2
g) 8,9 locked in 20(4) + 28(4) for C89+N9

2. R6789+C46 !
a) 8(3) @ N2 = 1{25/34} -> 1 locked for C6+N2
b) 8(3) @ N9 = 1{25/34} -> 1 locked for C7+N9
c) 9(3) @ R9: R9C4 <> 6 since {12}6 blocked by R3C3 = (12)
d) Innies R89 = 17(4)
e) ! Using Innies R89: Innies N8 = 28(5): R7C5 <> 1 since R9C46 <= 9
f) 1 locked in 9(3) @ R7 = 1{26/35} for N7+9(3)

3. C1234 !
a) Innies+Outies C12: 8 = R5C34 - R6C2 -> R5C4 <> 8 (IOU @ N4)
b) Innies+Outies C12: 8 = R5C34 - R6C2: R5C3 <> 1,2 since R5C4 <= 6
c) Hidden Single: R3C3 = 1 @ C3 -> R4C4 = 2
d) Innies C1234 = 4(1+1) = {13} -> CPE: R6C4 <> 3
e) ! Killer pair (13) locked in 16(4) + R6C2 for N4
f) 2,3 locked in R789C3 @ C3 for N7
g) 9(3) = {126} -> R6C1 = 2; 6 locked for R7+N7
h) 27(4) @ C3: R7C3 <> 3 since [7839] blocked by R3C4 = (89)
i) 2,3 locked in 9(3) @ C3 = {234} for 9(3) -> R9C4 = 4

4. N379
a) 28(4) @ N7 = {4789} locked for N7; 4 also locked for R8
b) 27(4) @ N7 = {5679} since R7C3 = 5 -> R7C3 = 5, R6C4 = 6, R7C4 = 9, R6C3 = 7
c) Innies+Outies N9: 3 = R6C9+R9C6 - R7C7: R6C9 <> 1 since R9C6 <= 5 and R7C7 >= 4
d) 7 locked in R3C789 @ R3 for N3
e) Innies R1234 = 10(2) = [37]/{46}
f) R6C9 <> 3 since it sees all 3 of C8
g) 12(3) @ N9 = {345} since R6C9 <> 2,3,7 -> R6C9 = 5; 3,4 locked for R7+N9
h) 12(3) @ N3 = {147} -> 1 locked for R4+N6
i) Killer pair (47) locked in Innies R1234 + R3C9 for N3
j) R7C7 = 7

5. Rest is singles without considering diagonals.

Hard 1.0. I used some hidden cages, IOU and CPE.

1. Innies r5 -> r5c5 = 9

2. HP (12) in D\ -> r3c3,r4c4 = {12} -> 20(4)D\ = {1289}
-> HS 3 in D\ -> 27(4)n1 = {3789}

3. HP (12) in D/ -> r1c9,r2c8 = {12} -> 20(4)n3 = {1289}
-> 22(4)n12 = {4567} with 7 in r12c4.
Also 1 in r12c6
-> 1 in r89c7

4! All the 28(4)s are {4789} or {5689}
-> 8s locked in r1289c1289
-> 8 in r5 in c34 or c67
If the former (since another 8 already in c34) -> 27(4)r6c3 = {5679}
and if the latter (since another 8 already in c67) -> 27(4)r3c6 = {5679}

i.e., one of the 27(4)s on D/ is {5679}

3 in D/ either in r67 or r34
-> the other of the 27(4)s on D/ is {3789}

-> Both of the 27(4)s on D/ contain a 7 and neither contains a 4.
-> HS 4 in D/ in r89 -> 28(4)n7 = {4789}!

Easy from here...

5 -> HS 7 in 27(4)r6c3 -> r6c3 = 7
-> r7c4 = 9
-> r4c3 = 9 and r3c4 = 8 -> r3c6 = 9 -> r6c7 = 9
-> HS 8 in c3 -> r5c3 = 8
-> r6c4,r7c3 = {56}
-> 27(4)r3c6 = {3789}

6. Remaining innies c34 = r5c4 + r8c4 = +4 = {13}
-> r3c3 = 1, r4c4 = 2
-> 9(3)r8c3 = [{23}4]
-> r8c2 = 4
-> 4 in r7c789
-> 28(4)n9 = {5689}
Also 8(3)r8c7 = {125} with 5 in r9c6
Also 28(4)r6c6 = {4789}

7. 7 in n9 only in r7c789
-> HS 7 in 28(4)r6c6 -> r7c7 = 7
-> r7c89 = {34}
-> r6c9 = 5
Also r6c6 = 4

Also HS 7 in 27(4)r3c6 -> r4c6 = 7
-> r3c7 = 3
-> Innies r1234 = r2c7 + r3c8 = +10 = {46}
-> r3c9 = 7 and r1c7 = 5

etc.

Prelims

a) 27(4) cage at R1C1 = {3789/4689/5679}, no 1,2
b) 8(3) cage at R1C6 = {125/134}
c) 27(4) cage at R3C6 = {3789/4689/5679}, no 1,2
d) 9(3) cage at R6C1 = {126/135/234}, no 7,8,9
e) 27(4) cage at R6C3 = {3789/4689/5679}, no 1,2
f) 28(4) cage at R6C6 = {4789/5689}, no 1,2,3
g) 28(4) cage at R8C1 = {4789/5689}, no 1,2,3
h) 9(3) cage at R8C3 = {126/135/234}, no 7,8,9
i) 8(3) cage at R8C7 = {125/134}
j) 28(4) cage at R8C8 = {4789/5689}, no 1,2,3

Steps resulting from Prelims
1a. 27(4) cage at R1C1 = {3789/4689/5679}, 9 locked for N1
1b. 8(3) cage at R1C6 = {125/134}, CPE no 1 in R1C45
1c. 28(4) cage at R8C1 = {4789/5689}, 8,9 locked for N7
1d. 28(4) cage at R8C8 = {4789/5689}, 8,9 locked for N9
1e. 8,9 in R7 only in R7C456, locked for N8

2. 45 rule on R5/C5 1 innie R5C5 = 9, placed for both diagonals

3. R3C3 + R4C4 = {12} (hidden pair on D\) -> 20(4) cage at R3C3 = {1289}
3a. X-Wing for 9 in 20(4) cage and 27(4) cage at R6C3, no other 9 in C4
3b. 9 in N2 only in R3C46, locked for R3

4. 3 on D\ only in R1C1 + R2C2, locked for N1 -> 27(4) cage at R1C1 = {3789}, locked for N1
4a. 8,9 in C3 only in R456C3, locked for N4

5. R1C9 + R2C8 = {12} (hidden pair on D\), locked for N3 -> 20(4) cage at R1C8 = {1289}, locked for N3
5a. 8(3) cage at R1C6 = {125/134}, 1 locked for C6 and N2
5b. 5 of {125} must be in R1C7 -> no 5 in R12C6
5c. 8(3) cage at R8C7 = {125/134}, 1 locked for C7 and N9

6. 8 in R3 only in R3C456, locked for N2
6a. 8,9 in C7 only in R456C7, locked for N6

7. 22(4) cage at R1C3 = {4567} (only possible combination), 7 locked for C4 and N2
7a. 2 in N1 only in R3C123, locked for R3
7b. 7 in R3 only in R3C789, locked for N3

8. 45 rule on R1234 2 innies R2C7 + R3C8 = 10 = [37/46/64]

9. 45 rule on C1234 2(1+1) innies R6C2 + R8C4 = 4 = [13/22/31]
9a. 45 rule on R6789 using R6C2 + R8C4 = 4, 2 remaining innies R6C8 + R8C6 = 9 = {27/36/45}, no 1

10. Hidden killer triple 3,4,5 in 27(4) cage at R3C6, 27(4) cage at R6C3 and 28(4) cage at R8C1 for D/, 27(4) and 28(4) cages can only contain one of 3,4,5 -> 3,4,5 in these cages must be on D/ -> no 3,4,5 in R3C6, R4C7, R6C3, R7C4, R8C1 and R9C2

11. Hidden killer pair 4,5 in 28(4) cage at R6C6 and 28(4) cage at R8C8 for D\, 28(4) cages can only contain one of 4,5 -> one of the 28(4) cages must be {4789} with 4 on D\ and the other 28(4) cage must be {5689} with 5,6 on D\ -> no 4,5,6 in R6C7, R7C6, R8C9 and R9C8

12. 9(3) cage at R8C3 = {126/135/234}
12a. 6 of {126} must be in R89C3 (R89C3 cannot be {12} which clashes with R3C3) -> no 6 in R9C4

13. 27(4) cage at R6C3 = {3789/4689/5679}
13a. 3 of {3789} must be in R6C4 (R67C4 cannot be [89] which clashes with R3C4) -> no 3 in R7C3

14. One of 27(4) cage at R3C6 and 27(4) cage at R6C3 must be {5679} (otherwise no place for 8 in R5 and C5) -> 5 in of these 27(4) cages, locked for D/
14a. 28(4) cage at R8C1 = {4789} (only remaining combination), locked for N7, 4 locked for D/
[Cracked. The rest is fairly straightforward.]

15. 27(4) cage at R6C3 = {5679} (only remaining combination because R7C3 only contains 5,6), no 3,8
15a. Naked pair {56} in R6C4 + R7C3, locked for D/ and 27(4) cage -> R7C4 = 9, R6C3 = 7, R3C4 = 8, R4C3 = 9, clean-up: no 2 in R8C6 (step 9a)

16. 3 on D/ only in 27(4) cage at R3C6 = {3789} -> R3C6 = 9, 8 locked for R4

17. R5C3 = 8 (hidden single in C3)
17a. 45 rule on C12 1 remaining outie R5C4 = 1 innie R6C2, R6C2 = {123} -> R5C4 = {123}

18. Naked triple {456} in R127C3, locked for C3, 4 also locked for 22(4) cage at R1C3
18a. R9C4 = 4 (hidden single in C4) -> R89C3 = 5 = {23}, locked for C3 and N7 -> R3C3 = 1, R4C4 = 2, clean-up: no 2 in R6C2 (step 17a)

19. R8C2 = 4 (hidden single in N7)
19a. 28(4) cage at R8C8 = {5689} (only remaining combination) -> R8C8 + R9C9 = {56}, locked for N9 and D\
19b. 8(3) cage at R8C7 = {125} (only remaining combination) -> R9C6 = 5, R89C7 = {12}, locked for C7 and N9

20. R7C56 = [28] (hidden pair in R7), R6C67 = [49], R7C7 = 7, placed for D\, R3C7 = 3, R4C6 = 7, placed for D/, clean-up: no 7 in R3C8 (step 8), no 2,5 in R6C8 (step 9a)
20a. Naked pair {46} in R2C7 + R3C8, locked for N3 -> R1C7 = 5, R12C6 = 3 = {12}
20b. R3C9 = 7 -> R4C89 = 5 = {14}, locked for R4 and N6
20c. R5C67 = [36], R5C8 = 7 (hidden single in R5) -> R5C9 = 2 (cage sum)

and the rest is naked singles, without using the diagonals.

I'll rate my walkthrough for A327X at Easy 1.5. Step 14 feels like an implied forcing chain. I also used hidden killers.