SudokuSolver Forum

3x3::k:4097:4097:4097:1794:1794:1794:5635:5635:5635:1284:1284:6149:6149:11270:3591:3591:3336:3336:2569:6149:6149:11270:11270:11270:3591:3591:2314:2569:11270:11270:7179:7692:5901:11270:11270:2314:7179:7179:7179:7179:7692:5901:5901:5901:5901:3598:3598:3087:3087:7692:4112:4112:1553:1553:3598:3090:5139:7692:5140:7692:5397:2070:1553:3095:3090:5139:5139:5140:5397:5397:2070:4888:3095:3095:5139:5140:5140:5140:5397:4888:4888:

there are so many CPEs. Several times I re-worked when I spotted ones which I’d missed earlier, including the ones resulting from the Prelims. This eventually led to my solving path. The CPEs in steps 1i and 1j seem to be the most important ones.

I'll rate my walkthrough for A212 at 1.5. Some of the steps and analysis seemed to be too hard for the 1.25 range.

I've recently downloaded SudokuSolver, but only so that I can use it to calculate SS scores for use in the Archive.

The SS score for A212 is 1.22.

Prelims

a) R2C12 = {14/23}
b) R2C89 = {49/58/67}, no 1,2,3
c) R34C1 = {19/28/37/46}, no 5
d) R34C9 = {18/27/36/45}, no 9
e) R6C34 = {39/48/57}, no 1,2,6
f) R6C67 = {79}
g) R78C2 = {39/48/57}, no 1,2,6
h) R78C8 = {17/26/35}, no 4,8,9
i) 7(3) cage at R1C4 = {124}
j) 22(3) cage at R1C7 = {589/679}
k) 6(3) cage at R6C8 = {123}
l) 19(3) cage at R8C9 = {289/379/469/478/568}, no 1
m) 14(4) cage at R2C6 = {1238/1247/1256/1346/2345}, no 9
n) 44(8) cage at R1C5 = {23456789}, no 1

Steps resulting from Prelims
1a. Naked triple {124} in 7(3) cage at R1C4, locked for R1 and N2
1b. 22(3) cage at R1C7 = {589/679}, 9 locked for R1 and N3, clean-up: no 4 in R2C89
1c. Naked pair {79} in R6C67, locked for R6, clean-up: no 3,5 in R6C34
1d. Naked pair {48} in R6C34, locked for R6
1e. Naked triple {123} in 6(3) cage at R6C8, CPE no 1,2,3 in R45C9, clean-up: no 6,7,8 in R23C9
1f. Naked quint {56789} in R1C789 + R2C89, locked for N3, clean-up: no 4 in R4C9
1g. 3 in R1 only in R1C123, locked for N1, clean-up: no 2 in R2C12, no 7 in R4C1
1h. Naked pair {14} in R2C12, locked for R2 and N1, clean-up: no 6,9 in R4C1
1i. 7,9 in N4 only in R4C23 + R5C123, CPE no 7,9 in R4C4
1j. 4,5,6,8 in N6 only in R45C789, CPE no 4,5,6,8 in R4C6

2. 44(8) cage at R1C5 = {23456789}, 2,4 only in R4C2378, locked for R4, clean-up: no 6,8 in R3C1
2a. 2 in N5 only in R5C456, locked for R5

3. 45 rule on R6789 3 innies R6C5 + R7C46 = 20 = {389/569/578} (cannot be {479} because no 4,7,9 in R6C5), no 1,2,4
3a. 3 of {389} must be in R6C5 -> no 3 in R7C46

4. 45 rule on R6789 2 outies R45C5 = 10 = {19/37}/[64/82], no 5, no 6,8 in R5C5

5. 45 rule on N1 1 outie R2C4 = 1 innie R3C1 -> R2C4 = R3C1 = {79}, clean-up: no 8 in R4C1

6. 45 rule on N3 1 outie R2C6 = 1 innie R3C9 + 4, no 3 in R2C6
6a. 3 in N2 only in R2C5 + R3C456, locked for 44(8) cage at R1C5, no 3 in R4C2378

7. 45 rule on N7 1 innie R7C1 = 1 outie R8C4 + 1, no 1 in R7C1, no 9 in R8C4

8. 45 rule on N9 1 outie R8C6 = 1 innie R7C9 + 3 -> R8C6 = {456}

9. 14(3) cage at R6C1 = {158/167/257/356} (cannot be {149/248/347} because 4,7,8,9 only in R7C1, cannot be {239} = {23}9 which clashes with R6C89, ALS block), no 4,9, clean-up: no 3,8 in R8C4 (step 7)
9a. 7 of {257} must be in R7C1 -> no 2 in R7C1, clean-up: no 1 in R8C4 (step 7)

10. 45 rule on R1234 3 innies R4C456 = 16
10a. R4C56 cannot be 10 which would clash with R45C5, CCC -> no 6 in R4C4
[Alternatively 45 rule on R5 2 outies R4C46 = 1 innie R5C5 + 6, IOU no 6 in R4C4 is simpler, but I spotted the CCC first.]
10b. R4C456 = {169/178/358/367}
10c. 6 of {169} must be in R4C5 -> no 9 in R4C5, clean-up; no 1 in R5C5 (step 4)
[There’s a bit more but I didn’t spot it until step 18.]

11. 21(4) cage at R7C7 must contain at least one of 1,2,3
11a. Killer triple 1,2,3 in 21(4) cage at R7C7, R78C8 and R7C9, locked for N9

12. Hidden killer triple 1,2,3 in R3C9 and R67C9 for C9, R67C9 contains two of 1,2,3 -> R3C9 = {123}, clean-up: no 8 in R2C6 (step 6), no 5 in R4C9

13. 8 in N2 only in R2C5 + R3C456, locked for 44(8) cage at R2C5, no 8 in R4C2378

14. 45 rule on R6 2 outies R7C19 = 1 innie R6C5 + 3
14a. Min R6C5 = 3 -> min R7C19 = 6 -> no 3 in R7C1 (R7C19 cannot be [33], clean-up no 2 in R8C4 (step 7)

15. 45 rule on N8 4 innies R78C46 = 25 = {4579/4678}, 4,7 locked for N8, 4 also locked for R8, CPE no 7 in R7C3, clean-up: no 8 in R7C2

16. R78C46 (step 15) = {4579/4678} contains one of 8,9 in R7C46 -> R6C5 + R7C46 (step 3) = {569/578} (cannot be {389} which contains both of 8,9 in R7C46), no 3
16a. R6C5 + R7C46 = {569/578}, CPE no 5 in R789C5
[Can make candidate eliminations from R78C46 using interactions with R6C5 + R7C46 but I’ll leave them for now and hope to find an easier way to do them, even though they appear to be immediately helpful.]

[Then I noticed
R2C6 = {567}, R2C89 = {58/67} = 13 -> variable combined cage R2C678 = 18,19,29 = {567/568/578}, 5 locked for R2
Alternatively there’s the “human” “chainy” way
R2C6 = 5 or R2C6 = {67}, locked for R2 => R2C89 = {58}, locked for R2
I felt that this interesting step probably wasn’t essential to the solving path so I’ve just made it a comment; later it simplified to a killer pair (step 27e).]

17. 14(3) cage at R6C1 = {158/257/356} (cannot be {167} = {16}7 which clashes with R34C1), CPE no 5 in R5C1
17a. Caged X-Wing for 5 in 14(3) cage at R6C1 and R6C5 + R7C46, no other 5 in R67, clean-up: no 7 in R8C2, no 3 in R8C8
[I’d spotted step 17 earlier but hadn’t used it since it didn’t give any candidate eliminations in the 14(3) cage; now I’ve spotted that it leads to the CPE in step 17 and to step 17a.]

18. R4C456 (step 10b) = {169/178/358/367}
18a. 6 of {169} must be in R4C5, 7 or 8 of {178} must be R4C5 (R45C5 = 10 (step 4) cannot be [19] because R4C456 + R5C5 = [7189/8179] clash with R6C6) -> no 1 in R4C5, clean-up: no 9 in R5C5 (step 4)

19. R6C5 + R7C46 (step 16a) = {569/578} contains 5 -> 30(5) cage at R4C5 = {25689/35679/45678}, CPE no 6 in R789C5

20. 45 rule on N8 (using R6C5 + R7C46 = 20, step 3), 2 innies R8C46 = 1 outie R6C5 + 5
20a. R6C5 = {56} -> R8C46 = 10,11 and must contain 4 (step 15) = {46}/[74], no 5, clean-up: no 6 in R7C1 (step 7), no 2 in R7C9 (step 8)
20b. R6C5 + R8C46 = 5{46}/6[74], CPE no 6 in R7C46
[These are the eliminations I mentioned after step 16. Steps 20a and 20b are a neater way to make them but not any simpler.]
20c. 30(5) cage at R4C5 (step 20) = {25689/35679/45678}, 6 locked for C5 and N5

21. 6(3) cage at R6C8 = {123}, 2 locked for R6 and N6

22. 14(3) cage at R6C1 (step 17) = {158/356}, no 7, clean-up: no 6 in R8C4 (step 7)
22a. Killer pair 1,3 in R4C1 and 14(3) cage at R6C1, locked for N4
22b. 6 in N8 only in R8C6 + R9C46, CPE no 6 in R9C7

23. R78C46 (step 15) = {4579/4678} = {59}[74]/{78}[46]
23a. Killer pair 5,8 in R7C1 and R7C46, locked for R7

24. 1 in C3 only in R789C3, locked for N7
24a. 2 in C1 only in R89C1, locked for N7
24b. 12(3) cage in N7 = {237/246}, no 5,8,9

25. 20(4) cage at R7C3 must contain 1 = {1379/1469/1478} (cannot be {1568} because R8C4 only contains 4,7), no 5
25a. 4 of {1469/1478} must be in R8C4 (R789C3 cannot be {148} which clashes with R6C3) -> no 4 in R79C3
25b. 4 in C3 only in R456C3, locked for N4

26. 20(5) cage in N8 = {12359/12368}
26a. One of 5,6 must be in R9C46 -> there cannot be more than one of 1,2,3 in R9C46 -> at least two of 1,2,3 must be in R789C5
26b. Killer quad 1,2,3,4 in R1C5, R45C5 and R789C5, locked for C5
26c. Hidden killer quad 1,2,3,4 in R1C5, R45C5 and R789C5 for C5, R1C5 and R45C5 each contain one of 1,2,3,4 -> R789C5 must contain two of 1,2,3 -> R789C5 must also contain one of 8,9 -> no 8,9 in R9C46

27. 3 in N2 only in R3C46, locked for R3, clean-up: no 7 in R2C6 (step 6), no 6 in R4C9
27a. Naked triple {124} in R3C789, locked for R3 and N3 -> R2C7 = 3
27b. R2C3 = 2 (hidden single in R2), R4C2 = 2 (hidden single in R4)
27c. 9 in R2 only in R2C45, locked for N2
27d. 3 in C9 only in R67C9, locked for 6(3) cage at R6C8, no 3 in R6C8
27e. Killer pair 5,6 in R2C6 and R2C89, locked for R2

28. R4C456 (step 10b) = {169/358/367} (cannot be {178} which clashes with R4C9)
28a. 9 of {169} must be in R4C6 -> no 1 in R4C6
28b. 3 of {358} must be in R4C6, 6 of {367} must be in R4C5 -> no 3 in R4C5, clean-up: no 7 in R5C5 (step 4)
28c. 6 of {367} must be in R4C5 -> no 7 in R4C5, clean-up: no 3 in R5C5 (step 4)
28d. 5 of {358} must be in R4C4 -> no 8 in R4C4
28e. Killer pair 5,6 in R4C456 and R6C5, locked for N5
28f. R45C5 (step 4) = [64/82]
28g. Killer pair 4,8 in R45C5 and R6C4, locked for N5

29. 3 in C5 only in R789C5, locked for N8

30. 3 in N9 only in R7C89, locked for R7, clean-up: no 9 in R8C2

31. 7 in C5 only in R23C5, locked for N2 -> R2C4 = 9, R3C1 = 9 (step 5), R4C1 = 1, R2C12 = [41]
31a. R4C456 (step 28) = {358/367}, no 9

32. 9 in C5 only in R789C5, locked for N8
32a. 20(5) cage at R7C5 (step 26) must contain 5 = {12359} (only remaining combination), no 6,8, 5 locked for R9 and N8

33. Naked pair {78} in R7C46, locked for R7 -> R7C1 = 5, R8C4 = 4 (step 7), R8C6 = 6, R6C34 = [48], R7C46 = [78], R4C5 = 6, R5C5 = 4 (step 4), R6C5 = 5, R4C4 = 3, R4C6 = 7, R6C67 = [97], R2C6 = 5, R3C46 = [63], R4C9 = 8, R3C9 = 1, R7C9 = 3, R6C89 = [12], clean-up: no 8 in R2C8, no 2 in R7C8, no 5,7 in R8C8

34. R78C8 = [62] -> R2C89 = [76], R23C5 = [87], R3C78 = [24]

35. 19(3) cage at R8C9 = {478} (only remaining combination) -> R9C8 = 8, R9C9 = 4, R8C9 = 7

36. R8C1 = 3, R9C1 = 2 (hidden single in C1), R9C2 = 7 (step 24b)

and the rest is naked singles.

Note: this is an optimised solution so many obvious eliminations have been left out. However, I try and do clean-up as I go. Please tell me about any corrections or clarifications.

prelims
i.7(3)n2 = {124}
ii. 22(3)n3 = {589/679}(no 1,2,3,4)
iii. 5(2)n1 = {14/23}
iv. 44(8)r2c5: no 1
v. 14(4)r2c6: no 9
vi. 13(2)n3: no 1,2,3
vii. 10(2)r3c1: no 5
viii. 9(2)r3c9: no 9
ix. 12(2)r6c3: no 1,2,6
x. 16(2)r6c6 = {79}
xi. 6(3)r6c8 = {123}
xii. 12(2)r7c2: no 1,2,6
xiii. 8(2)r7c8: no 4,8,9
xiv. 19(3)n9: no 1

1. 7(3)n2 = {124}: all locked for r1 and n2

2. 3 in r1 only in n1: locked for n1
2a. -> 5(2)n1 = {14}: both locked for r2 and n1
2b. no 6,7,9 in r4c1
2c. no 9 in 13(2)n3

3. 44(8)r2c5 must have 2 & 4 which are only in r4: both locked for r4
3a. no 6,8 in r3c1
3b. no 5,7 in r3c9

4. "45" on n1: 1 outie r2c4 = 1 innie r3c1 = (79) only
4a. r4c1 = (13)

5. "45" on n3: 1 outie r2c6 - 4 = 1 innie r3c9
5a. -> min. r2c6 = 5; max r3c9 = 4
5b. no 1,3 in r4c9

6. 16(2)r6c6 = {79}: both locked for r6
6a. -> 12(2)r6c3 = {48}: both locked for r6

[Andrew's way of doing the next step is better. See his step 14]
7. "45" on r789: 4 innies r7c1469 = 23
7a. max. r7c9 = 3 -> min. other 3 cells, r7c146 = 20 (no 1,2,3: can't have 3 since max. 3 is in r7c9)

8. Hidden killer pair 5,6 in r6 in r6c125 since 14(3)r6c1 can't have both 5 & 6 (no 3 in r7c1)
8a. -> r6c5 = (56)
8b. 14(3)r6c1 must have 5/6 = {158/167/257/356}(no 4,9)

9. "45" on n7: 1 outie r8c4 + 1 = 1 innie r7c1
9a. r8c4 = (4..7)

10. "45" on n9: 1 outie r8c6 - 3 = 1 innie r7c9
10a. r7c9 = (456)

11. 1,2,3 in n8 only in 20(5) = 123{59/68}(no 4,7)

12. "45" on r6789: 2 outies r45c5 = 10 (no 5)
12a. no 6,8 in r5c5

[Andrew's alt. step 10a is a simpler IOU than the following step 13a]
13. "45" on r1234 including the h10(2)r45c5; 1 outie r5c5 + 6 = 2 innies r4c46
13a. -> no 6 in r4c46 (IOU)
13b. and no 9 in r5c5 since the innies r4c46 = 15 = {78}; but {78}[9]clashes with r6c6 (and of course can't be {69}[9])
13b. -> no 1 in r4c5 (h10(2)r45c5)

14. 5 & 8 in n6 only in r45c789 -> no 5, 8 in r4c6 (CPE)

15. "45" on r1234: 3 innies r4c456 = 16 = {169/178/358/367}
15a. 6 in {169} must be in r4c5 -> no 9 in r4c5
15b. no 1 in r5c5 (h10(2)r45c5)
15c. 3 in {358} must be in r4c6; 6 in {367} must be in r4c5 -> no 3 in r4c5
15d. -> no 7 in r5c6 (h10(2)r45c5)

16. "45" on r6789: 3 innies r6c5 + r7c46 = 20 and must have 5/6 for r6c5 = {569/578}(no 4)
16a. must have 5 -> no 5 in r789c5 (CPE)
16b. and 30(5)r4c5 must have 5 = 5{2689/3679/4678}
16c. must have 6 -> no 6 in r789c5 (CPE)

17. r15c5 must have 4 for c5 -> = 4{1/2/3} ie, has one of 1,2,3 (no eliminations yet)
17a. -> r789c5 cannot = {123}(ALS block)
17b. -> r789c5 must have 8/9
17c. -> no 8/9 in r9c46 since the 20(5)n8 can only have one of 8/9 (step 11)

18. 8 in n8 only in r7c46 + r789c5
18b. -> no 8 in r4c5 (CPE)(Love that rocket shape cage!)
18c. no 2 in r5c5 (h10(2)r45c5)

19. from step 17c, the 20(5)n8 can only have one of 8/9
19a. -> r789c5 must have two of 1, 2 or 3
19b. -> Killer triple 1,2,3 with r15c5 (step 17): 3 locked for c5

20. r2c7 = 3 (hsingle r2)
20a. r2c3 = 2 (hsingle r2)
20b. no 6 in r4c9
20c. no 7 in r2c6 (IOD n3 = +4)

21. 44(8)r2c5 must have both 6 & 7 -> at least one must be in r3c46 to not clash with r4c5
21a. -> 24(4)r2c3 cannot be [2]{679}, can only be [2]{589}
21b. -> r2c4 = 9, r3c23 = {58}: both locked for n1, r3

22. "45" on n1: 1 innie r3c1 = 9

Cracked.

1. r1c456 = {124}, r1c789 = {9(58|67)}, r1c123 = {3(67|58)}
-> r2c12 = {14} and r2c89 = {67|58}
-> r3c9 from (1234) -> r4c9 from (8765) -> r2c6 from (5678)

2. Innies - Outies n1 -> r3c1 = r2c4. Whatever it is must be in r1c789. i.e. from (56789).
r3c1 cannot be 5 must be from (6789) -> r4c1 from (4321).
2 and 4 in 44/1 must be in r4c2378 -> not in r4c1 -> r4c1 from (13) -> r3c1 from (97) -> r2c4 from (97).

3. r6c67 = {79} -> r6c34 = {48}.
Innies - outies n9 -> r8c6 = r7c9+3 -> r8c6 from (456).
Innies - outies n7 -> r7c1 = r8c4+1.
Innies r6789 -> r6c5+r7c46 = +20 -> r6c5 from (356).
If r6c5 = 3 -> r7c1 and r7c9 both = 3 -> r6c5 from (56).

a) If r6c5 = 5 -> r7c46 = +15 -> r8c46 = +10 must be {46} -> r7c46 = {78}
-> r7c1 not = 7 -> r8c4 not = 6 -> r8c46 = [46] -> r6c34 = [48] -> r7c46 = [78].

b) If r6c5 = 6 -> r7c46 = {59} -> r8c46 = +11 must be [74].

-> r8c6 from (46) -> r7c9 from (13) -> 2 in r6c89.

Also, -> 7 in r78c4 -> r2c4 = 9 -> r34c1 = [91] -> 1 in r6c89 -> r7c9 = 3 -> r8c6 = 6.
-> a) it is. -> r6 = [{36}[48]5{79}{12}], r7c146 = [578], r8c46 = [46].

4. Innies r1234 -> r4c456 = +16. Outies r6789 -> r45c5 = +10.
Whatever is in r4c4 must go in n4 in r6. i.e., from (346).
Cannot be 4 (already in r8c4). Cannot be 6 since that would make r4c6 = r5c5 -> r4c4 = 3.
-> r4c56 = +13. 9 in 44/8 must be in r4c2378 and 5 already in n5 -> r4c56 = [67].
-> r5c5 = 4, r6c67 = [97], r5c46 = {12}.
-> r5c123 = +23 or +24. Cannot be +23 -> r5c123 = {789} and r5c46 = [12], r5c789 = {356}.
-> r4c23 = {25} -> r2c6 = 5 -> r4c9 = 8 and r2c89 = {67}, r1c789 = {589}, r1c123 = {367}.

Rest is cleanup.

The first key move was proving 7 is in r78c4 using the alternate chains a) and b) above (which I know "proper" solvers don't like). This shortened the solving path considerably, but maybe that can or should be done in a better way.

Another key move was proving 3 is in r4c4.