SudokuSolver Forum

3x3::k:6144:6144:11521:11521:11521:11521:11521:11521:11521:6144:2818:8195:3844:1285:1285:2310:2823:11521:6144:2818:8195:3844:5128:5128:2310:2823:11521:8195:8195:8195:9993:5128:5128:9993:2823:2823:2826:9993:9993:9993:9993:9993:9993:3851:3851:2826:2826:5644:5644:5644:6157:6157:6157:3851:11534:3599:5644:3344:2833:6157:5650:5650:4371:11534:3599:3599:3344:2833:5650:5650:5650:4371:11534:11534:11534:11534:11534:11534:11534:4371:4371:

1. O-I for N1 => r1c3={1,2}.
If r1c3=2 then r4c123={789}, r4c47={3,5}, r4c89={1,2}, r4c56={4,6}, ... and several more steps lead to a contradiction.
2. So r1c3=1, r4c123={689}, r4c47={35|17}.
If r4c47={35}, then r4c89={12}, r4c56={47}, ... and several more steps lead to a contradiction.
3. So r4c47={17}.

Prelims

a) R23C2 = {29/38/47/56}, no 1
b) R23C4 = {69/78}
c) R2C56 = {14/23}
d) R23C7 = {18/27/36/45}, no 9
e) R78C4 = {49/58/67}, no 1,2,3
f) R78C5 = {29/38/47/56}, no 1
g) 11(3) cage at R5C1 = {128/137/146/236/245}, no 9
h) 11(4) cage at R2C8 = {1235}
i) 32(5) cage at R2C3 = {26789/35789/45689}, no 1
j) 39(8) cage at R4C4 = {12345789}, no 6
k) and, of course, both 45(9) cages = {123456789}

Steps resulting from Prelims
1a. 11(4) cage at R2C8 = {1235}, CPE no 1,2,3,5 in R56C8
1b. 9 in N3 only in R1C789 + R23C9, locked for 45(9) cage at R1C3, no 9 in R1C3456

2. R78C4 = {49/58} (cannot be {67} which clashes with R23C4), no 6,7
2a. Killer pair 8,9 in R23C4 and R78C4, locked for C4

3. 45 rule on N1 3 outies R4C123 = 1 innie R1C3 + 22
3a. Max R4C123 = 24 -> max R1C3 = 2
3b. R1C3 = {12} -> R4C123 = {689/789}, 8,9 locked for R4, N4 and 32(5) cage at R2C3, no 8,9 in R23C3

4. 45 rule on N1 3 innies R123C3 = 10 = {127/145/235} (cannot be {136} because 32(5) cage at R2C3 contains 3 or 6), no 6 in R23C3

5. 45 rule on N9 1 innie R9C7 = 1 outie R8C6 + 6, R8C6 = {123}, R9C7 = {789}

6. 45 rule on R1234 2 innies R4C47 = 8 = {17/35}, no 2,4
6a. 39(8) cage at R4C4 = {12345789}, 2,4,8,9 locked for R5
6b. 4 in R4 only in R4C56, locked for N5 and 20(4) cage at R3C5, no 4 in R3C56

7. 45 rule on R789 2 innies R7C36 = 13 = {49/58/67}, no 1,2,3

8. 15(3) cage at R5C8 = {168/267/357/456} (cannot be {159/249/258/348} because R5C8 only contains 6,7), no 9
[Ed pointed out that I’d missed that 7 of {267/357} cannot be in R6C9; in that case I wouldn’t have needed step 10a but the interesting "clone pair" was still helpful for step 10f.]

9. 45 rule on N14 4(3+1) innies R156C3 + R5C2 = 12, min R156C3 = 6 -> no 7 in R5C2

10. R5C189 must contain 6 plus the same pair that are in R4C47 ("clone pair" plus the 6), because remainder of R5 occupied by 39(8) cage at R4C4
10a. Consider permutations for R4C47 (step 6) = {17/35}
R4C47 = [17] => R5C8 = 6
or R4C47 = {35} => R5C19 = {35}
or R4C7 = [71] => R5C1 = 1 (hidden single in R5)
-> no 6 in R5C1
10b. 6 in R5 only in R5C89, locked for N6
10c. 15(3) cage at R5C8 (step 8) = {168/267/456}, no 3
10d. R5C189 = {167/356}
10e. 3 of {356} must be in R5C1 -> no 5 in R5C1
10f. 5 of {356} must be in R5C9 -> no 5 in R4C7 (because R5C189 must contain 6 plus the same pair that are in R4C47), clean-up: no 3 in R4C4 (step 6)

11. 11(3) cage at R5C1 = {137/146/236} (cannot be {245} because R5C1 only contains 1,3,7), no 5
11a. Killer pair 6,7 in R4C123 and 11(3) cage, locked for N4

12. 5 in N4 only in R56C3 + R5C2
12a. R156C3 + R5C2 = 12 (step 9) = {1245} (cannot be 1{35}3, cannot be 2{35}2 which clashes with R123C3 = 2{35}, CCC), no 3, 4 locked for N4

13. 32(5) cage at R2C3 = {26789/35789} (cannot be {45689} which clashes with R156C3 + R5C2, ALS block), no 4

14. R123C3 (step 4) = {127/235}, 2 locked for C3 and N1, clean-up: no 9 in R23C2
14a. 9 in N1 only in 24(4) cage at R1C1 = {1689/3489/4569} (cannot be {3579} which clashes with R123C3), no 7

15. Killer pair 1,5 in R123C3 and R56C3, locked for C3, clean-up: no 8 in R7C6 (step 7)
15a. Hidden killer pair 1,5 in R123C3 and R56C3, R123C3 contains one of 1,5 -> R56C3 must contain one of 1,5 -> R56C3 = {14/45}, 4 locked for C3 and N4, clean-up: no 9 in R7C6 (step 7)
15b. R156C3 + R5C2 (step 12a) = {1245}, 1 in C3 only in R156C3 -> no 1 in R5C2

16. 45 rule on R6789 1 innie R6C9 = 1 outie R5C1 + 1 -> R6C9 = {248}

17. 11(3) cage at R5C1 (step 11) = {137/236}, R5C189 (step 10d) = {167/356}
17a. Consider placements for R5C2
R5C2 = 2 => 11(3) cage = {137}
or R5C2 = 5 => R5C189 = {167} => R5C1 = {17} => 11(3) cage = {137} (only remaining combination)
-> 11(3) cage = {137}, locked for N4

18. R1C3 = 1 (hidden single in C3)
18a. Naked pair {45} in R56C3, locked for C3 and N4 -> R5C2 = 2

19. 32(5) cage at R2C3 (step 13) = {26789} (only remaining combination) -> R23C3 = {27}, locked for C3 and N1, R4C123 = {689}, locked for R4, clean-up: no 4 in R23C2, no 6 in R7C6 (step 7)
19a. 2,7 in R1 only in R1C456789, locked for 45(9) cage at R1C3, no 2,7 in R23C9
19b. 2 in N7 only in R789C1, locked for 45(9) cage at R7C1, no 2 in R9C456
19c. 3 in C3 only in R89C3, locked for N7
19d. 3 in 45(9) cage at R7C1 only in R9C3456, locked for R9

20. 14(3) cage at R7C2 = {149/158/167/347/356}
20a. 8,9 of {149/158} must be in R8C3 -> no 8,9 in R78C2

21. 24(4) cage at R6C6 = {2589/2679/3489/3579/3678/4569/4578} (cannot be {1689} because R7C6 only contains 4,5,7), no 1

22. R78C5 = {29/38/56} (cannot be {47} which clashes with R78C4 + R7C6, killer ALS block), no 4,7

23. R4C47 = 8 (step 6), min R4C89 = 3, max R4C89 = 8 -> min R4C4789 = 11, max R4C4789 = 16, all 4-cell combinations between 11 and 16 without 4,6,8,9 must contain 1 -> 1 in R4C4789, locked for R4
[Alternatively can form combined cage from R4C47 = {17/35} and R4C89 which must be {12} when R4C47 = {35}.]

24. 20(4) cage at R3C5 contains 4 = {1478/2459/3458/3467} (cannot be {1469} because 1,6,9 only in R3C56, cannot be {2468} which clashes with R23C4)
24a. 9 in N2 only in R23C4 = {69} or in R3C56 -> 20(4) cage at {1478/2459/3458} (cannot be {3467}, locking-out cages), no 6
24b. 4,7 of {1478} must be in R4C56 -> no 7 in R3C56
24c. Killer pair 8,9 in R23C4 and 20(4) cage, locked for N2

25. 8,9 in 45(9) cage at R1C3 only in R1C789 + R23C9, locked for N3, clean-up: no 1 in R23C7

26. 1 in N3 only in R23C8, locked for C8 and 11(4) cage at R2C8, no 1 in R4C9

27. 1 in R4 only in R4C47 (step 6) = {17}, locked for R4 and 39(8) cage at R4C4, no 1,7 in R5C3456
27a. Naked quint {34589} in R5C34567, locked for R5
27b. 3 in N4 only in R6C12, locked for R6
27c. Naked triple {167} in R4C7 + R5C89, locked for N6
27d. 15(3) cage at R5C8 (step 8) = {168/267}, no 4

28. 20(4) cage at R3C5 (step 24a) = {2459/3458}, no 1

29. 1 in N2 only in R2C56 = {14}, locked for R2 and N2
29a. 4 in 45(9) cage at R1C3 only in R1C789 + R23C9, locked for N3, clean-up: no 5 in R23C7
29b. 4 in C4 only in R789C4, locked for N8, clean-up: no 9 in R7C3 (step 7)

30. 22(4) cage at R6C3 = {1489/2569/2578/4567} (cannot be {1579/2479} because R7C3 only contains 6,8, cannot be {1678} because R6C3 only contains 4,5)
30a. R7C3 = {68} -> no 6,8 in R6C45
30b. 9 of {1489} must be in R6C5 -> no 1 in R6C5
30c. 1 in N5 only in R46C4, locked for C4

31. R6C6 = 6 (hidden single in R6)
31a. 24(4) cage at R6C6 (step 21) = {2679/4569}, no 8, 9 locked for R6 and N6
31b. R7C6 = {57} -> no 5 in R6C7

32. R6C9 = 8 (hidden single in R6)
32a. 15(3) cage at R5C8 (step 27d) = {168} (only remaining combination) -> R5C89 = [61], R5C1 = 7, R4C47 = [17], clean-up: no 2 in R23C7, no 1 in R8C6 (step 5)
32b. Naked pair {36} in R23C7, locked for C7 and N3
32c. 7 in N3 only in R1C89, locked for R1
32d. 1 in N8 only in R9C56, locked for R9 and 45(9) cage at R7C1, no 1 in R78C1

33. 7 in N2 only in R23C4 = {78}, locked for C4 and N2, clean-up: no 5 in R78C4
33a. Naked pair {49} in R78C4, locked for N8, clean-up: no 2 in R78C5

34. 1 in N7 only in R78C2, locked for C2 -> R6C12 = [13], clean-up: no 8 in R23C2
34a. Naked pair {56} in R23C2, locked for C2 and N1
34b. 14(3) cage at R7C2 (step 20) = {149/167}, no 3,8
34c. R9C3 = 3 (hidden single in N7)

35. R78C5 = {38} (only remaining combination, cannot be {56} which clashes with R9C4), locked for C5 and N8 -> R8C6 = 2, R9C7 = 8 (step 5)
35a. 6 in N8 only in R9C45, locked for R9C45, locked for R9 and 45(9) cage at R7C1, no 6 in R78C1

36. 6 in N9 only in 17(4) cage at R7C9 = {2456} (only remaining combination), locked for N9

37. Naked pair {19} in R78C7, locked for C7 and N9
37a. Naked pair {37} in R78C8, locked for C8
37b. Naked pair {25} in R24C8, locked for C8 and 11(4) cage at R2C8, no 2,5 in R4C9 -> R9C8 = 4, R6C8 = 9, R1C8 = 8

38. R5C56 = [98] (hidden pair in R5)
38a. R3C6 = 9 (hidden single in N2)
38b. 20(4) cage at R3C5 (step 28) = {2459} (only remaining combination), no 3, 2 locked for C5
38c. R5C4 = 3 (hidden single in N5), R1C6 = 3 (hidden single in N2)
38d. Naked pair {49} in R1C12, locked for R1 and N1
38e. Naked pair {25} in R1C7 + R2C8, locked for N3

39. 9 in R9 only in R9C12, locked for N7 -> R8C3 = 6

40. R7C3 = 8 -> 22(4) cage at R6C3 (step 30) = {2578} (only remaining combination) -> R6C345 = [527]

and the rest is naked singles.

I'll rate my walkthrough for A240 at Hard 1.5; I used a short forcing chain combined with a "clone pair" and later another short forcing chain. Step 12 was also a tricky, but slightly lower rated step.

Many thanks to Andrew for some corrections and replacing some fuzzy logic.

1. "45" on n1: 3 outies r4c123 - 22 = 1 innie r1c3
1a. max. 3 outies = 24 -> max. r1c3 = 2
1b. r4c123 = 23/24 = {689/789}(no 1..5)
1c. 8 & 9 locked for r4, n4 and 32(5)r2c3
1d. no 8,9 in r23c3

2. 32(5)r2c3: no 1
2a. = {26789/35789/45689}= [2/5..]

3. 39(8)r4c4 (no 6)
3a. "45" on r1234: 2 innies r4c47 = 8 = {17/35}(no 2,4)

4. 39(8)r4c4 must have 2,4,8 & 9: all locked for r5

5. 11(4)r2c8 = {1235} only
5a. -> no 1,2,3,5 in r56c8

6. 15(3)r5c8 must have 6/7 for r5c8 = {168/267/357/456}(no 9)
6a. {267} must have 2 in r6c9; {357} must have 7 in r5c8 -> no 7 in r6c9

7. "45" on r6789: 1 outie r5c1 + 1 = 1 innie r6c9
7a. no 6 in r5c1
7b. no 1,3,5 in r6c9

8. 6 in r5 only in 15(3)r5c8: 6 locked for n6
8a. 15(3) must have 6 = {168/267/456}(no 3)
8b. no 6 in r6c9 -> no 5 in r5c1 (IODr6789=-1)

9. 11(3)n4 must have 1/3/7 for r5c1 = {137/146/236}(no 5)

10. "45" on n14: 4 innies r1c3+r5c2+r56c3 = 12 and must have 5 for n4.
10a. -> other three innies = 7
10b. deleted
10c. [22]{15} clashes with [2/5] in 32(5)r2c3 (step 2a)
10d. = {1245} only
10e. 2 or 5 must be in r5c2 to avoid clashing with [2/5] in 32(5)
10f. one of 2 or 5 must be in r156c3 -> Killer pair 2,5 with r23c3: both locked for c3
10g. 4 must be in r56c3: locked for n4 and c3
10h. 1 must be in r156c3: locked for c3

11. "45" on n1: 3 innies r123c3 = 10 = {127/235}: 2 locked for n1 & c3
11a. no 1,9 in 11(2)n1

This is a huge blocking move. It reminds me of Killer ALSxz except that it is blocking rather than locking.
12. 32(5)r2c3 = {26789/35789} = [6->2..]
12a. h8(2)r4c47 = {17/35} = [5/7..]
12b. r4c123 & h8(2) cannot both be 7 -> at least one must be [6->2] or {35}(could be both) -> innies n14 = {1245} cannot be [25]{14} since r5c2 sees 5 in r4c47 and 2 in r23c3
12c. no 5 in r5c2, no 2 in r1c3
12d. r1c3 = 1, r5c2 = 2
12e. r23c3 = {27} (step 11): 7 locked for n1, c3 and no 7 in r4c12
12f. no 4 in 11(2)n1

13. Naked triple {689} in r4c123: 6 locked for r4 & n4

This step might not be absolutely necessary but it's too much fun to leave out!
14. 9(2)n3 = {18/27/36/45}(no 9)=[1/2/3/5..]
14a. ->r23c78 have exactly three of 1,2,3,5 -> the 45(9)r1c3 must have exactly one of 2,3,5 in n3
14b. -> r1c456 must have two of 2,3,5 for 45(9)
14c. -> {23} blocked from 5(2)n2
14d. 5(2)n2 = {14} only: both locked for r2 and n2
14e. no 5,8 in r3c7

15. 4 in r4 only in n5: locked for n5

16. 4 in c4 only in n8: 4 locked for n8
16a. no 1,7 in 11(2)n8

17. "45" on r789: 2 innies r7c36 = 13 = [67/85]

18. 15(2)n2 = {69/78}(no 2,3,5) = [6/7..]
18a. -> {67} blocked from 13(2)n8
18b. 13(2) = {49/58}(no 1,2,3,6,7)
18c. Killer pair 8,9 in those two cages: both locked for c4

These next steps are ones that Andrew is really good at. I'm not fond of them really since there are too many combinations for my taste. (Note: Andrew worked differently with these cages so his way is better)
19. 22(4)r6c3:{3478} blocked by r6c12 = (137) (ALS block)
19a. must have 4/5 & 6/8 = {1489/2569/2578/3469/3568/4567} = [5/9 in r6]
19b. note: cannot have both 6&9 in r6c45 since min. r67c3 = 10

20. Hidden killer pair 6,9 -> 24(4)r6c6 must have at least one of 6/9 -> {4578} blocked
20a. 24(4): must have 5/7 but {359/379} in r6 clashes with 22(4)r6c3 (step 19a) or r6c12 (ALS block) -> {3579} blocked from 24(4)
20b. 24(4) = {2589/2679/3678/4569} (no 1)
20c. has one of 5/7 but not both -> no 5,7 in r6c678

21. 5 in r6 only in 22(4)r6c3 = {2569/2578/3568/4567}(no 1)

22. 1 in r6 only in n4: no 1 in r5c1
22a. no 2 in r6c9 (IOD r6789 = -1)

23. 15(3)n6 must have 4/8 = {168/456}(no 7)
23a. r5c8 = 6

24. 7 in n6 only in c7: 7 locked for c7 & 39(8)r4c4
24a. no 7 in r4c4, r5c456
24b. no 2 in 9(2)n3
24c. no 1 in r4c7 (h8(2)r4c47)

25. 7&9 in n3 only in 45(9)r1c3: locked for cage
25a. no 7,9 in r1c456

26. "45" on n23: 2 remaining outies r4c56 (& must have 4) - 3 = 2 innies r23c8
26a. min. r23c8 = 3 -> min. r4c56 = 6 (no 1)

27. Hidden killer pair 7,9 in n2: 15(2) has one of 7,9 -> 20(4) has one of 7,9
27a. ->20(4)r3c5 = {2459/3467}
27b. but {67} in r3c56 clashes with 15(2)n2 = [6/7..]
27c. 20(4) = {2459} only
27d. 9 locked for r3 and n2
27e. -> 15(2)n2 = {78} only: locked for n2 and c3
27f. -> 13(2)n8 = {49} only: locked for c3 and n8
27g. no 2 in 11(2)n8

28. "45" on n9: 1 outie r8c6 + 6 = 1 outie r9c7
28a. = [28/39]

29. 2 in n7 only in 45(9)r7c1: no 2 in r9c456
29a. r8c6 = 2 (hsingle n8)
29b. r9c7 = 8 (step 28a)


On from there.