SudokuSolver Forum

I'm glad you appreciated this Assassin! Truth is that the rating of 0.9 also seemed to be a little off for me, but hey, I'm no rating expert.

Now I'll have to look at all your nice walkthroughs. I'm almost sure that one of you found an easier way than I did...

Cheers,
Nasenbaer

Thanks Nasenbaer for an enjoyable and challenging Assassin.

I'm slowly catching up with them after our holiday.

I found a different multiple nonet in my solution. It gives the same reductions to two-candidate cells as those used by Afmob and Ed but without the CPEs for 5s. However after going through both posted walkthroughs I don't think this made much difference. The key early (or fairly early) breakthrough is to fix R6C9. However after that there's still a lot of work to do but in my walkthrough I'd done a lot of that before I found that breakthrough so the remaining steps were easier.

Ed's step 4 was a clever move. Maybe Mike can tell us what sort of step this is?

I'll also rate A108 as Hard 1.25.

Here is my walkthrough for A108.

Prelims

a) R2C67 = {14/23}
b) R23C4 = {49/58/67}, no 1,2,3
c) R34C1 = {17/26/35}, no 4,8,9
d) R34C5 = {39/48/57}, no 1,2,6
e) R34C8 = {19/28/37/46}, no 5
f) R5C12 = {39/48/57}, no 1,2,6
g) R5C67 = {49/58/67}, no 1,2,3
h) R6C78 = {19/28/37/46}, no 5
i) R89C5 = {59/68}
j) R9C78 = {39/48/57}, no 1,2,6
k) 20(3) cage in N1 = {389/479/569/578}, no 1,2
l) 9(3) cage in N2 = {126/135/234}, no 7,8,9
m) 11(3) cage at R4C9 = {128/137/146/236/245}, no 9

1. 45 rule on N1 1 outie R1C4 = 1 innies R3C1 + 5, R3C1 = {123}, R1C4 = {678}, clean-up: no 1,2,3 in R4C1

2. 45 rule on N9 1 outie R6C9 = 1 innie R9C7, no 1,2,6 in R6C9

3. 45 rule on N2356 2 innies R1C4 + R6C9 = 12 = [75/84], no 6 in R1C4, no 3,7,8,9 in R6C9, clean-up: no 1 in R3C1 (step 1), no 7 in R4C1, R9C7 = {45} (step 2), R9C6 = {78}

4. 11(3) cage at R4C9 = {128/137/146/236} (cannot be {245} which clashes with R6C9), no 5

5. 45 rule on N1 3 innies R12C3 + R3C1 = 8 = {125/134}, 1 locked in R12C3, locked for C3 and N1
5a. R3C1 = {23} -> no 2,3 in R12C3

6. 14(3) cage at R6C9 = {149/158/248/257/347/356} (cannot be {167/239} because R6C9 only contains 4,5)
6a. R6C9 = {45} -> no 4,5 in R7C89

7. 45 rule on C123 4 outies R1789C4 = 16 = {1258/1267/1348/1357/2347} (cannot be {1249/1456/2356} because R1C4 only contains 7,8), no 9
7a. R1C4 = {78} -> no 7,8 in R789C4

8. 9(3) cage in N2 = {126/135/234}
8a. If {234} => R2C67 = [14] -> 1 locked in R12C56, locked for N2
8b. If {234} => R2C67 = [14] -> no 4 in R2C5
[I hope this doesn’t look like T&E, it seems to be the clearest way to look at the combination of the 9(3) and 5(2) cages. I see that Ed also used this in his WT.
If you don’t like this approach, my original move here was 45 rule on N2, 4 innies R1C4 + R2C6 + R3C56 = 23 (as in step 17) when combinations including 1 must have the 1 in R2C6 -> no 1 in R3C6. This doesn’t eliminate 4 from R2C5 but it doesn’t matter because step 26 eliminates 4 from the 9(3) cage.]

9. 45 rule on R6789 3 outies R4C4 + R5C45 = 10 = {127/136/145/235}, no 8,9
9a. 45 rule on C1234 3 innies R456C4 = 16
9b. From steps 9 and 9a, R6C4 = R5C5 + 6, R6C4 = {789}, R5C5 = {123}

10. Killer triple 7,8,9 in R1C4, R23C4 and R6C4, locked for C4

11. Grouped X-Wing on 1 in R12C3 and R12C56, locked for R12, clean-up: no 4 in R2C6
[Alternatively 1 in R3 locked in R3C789, locked for N3.]

12. Killer triple 1,2,3 in 9(3) cage and R2C6, locked for N2, clean-up: no 9 in R4C5

13. Hidden killer triple 4,5,6 in R1789C4, R23C4 and R45C4 -> R45C4 must contain one of 4,5,6
13a. R4C4 + R5C45 (step 9) = {136/235} (cannot be {145} which would contain both of 4,5 in R45C4), no 4, 3 locked for N5, clean-up: no 9 in R3C5

14. Killer pair 5,8 in R34C5 and R89C5, locked for C5

15. 45 rule on N5 3 innies R4C56 + R5C6 = 17 = {179/278/458/467} (cannot be {269} because R4C5 only contains 4,5,7,8)
15a. 1 of {179} must be in R4C6 -> no 9 in R4C6

16. 45 rule on R6789 3 innies R6C456 = 18 = {189/279/468} (cannot be {459} which clashes with R6C9, cannot be {567} which clashes with R4C4 + R5C45), no 5
16a. R4C56 + R5C6 (step 15) = {179/458/467} (cannot be {278} which clashes with R6C456), no 2

17. 45 rule on N2 4 innies R1C4 + R2C6 + R3C56 = 23 = {1589/2489/2678/3479/3578} (cannot be {1679} because 6,9 only in R3C6, cannot be {2579} which clashes with 9(3) cage, cannot be {3569} because R1C4 only contains 7,8, cannot be {4568} because R2C6 only contains 1,2,3)
17a. 9 of {2489/3479} must be in R3C6 -> no 4 in R3C6

18. 45 rule on N4 4 innies R4C1 + R6C123 = 18 = {1269/1359/1368/1458/1467/2358/2367/2457/3456} (cannot be {1278/2349} because R4C1 only contains 5,6)
18a. Only combination containing both of 5,6 is {3456} which must be 5{346} (cannot be 6{345} when R6C123 clashes with R6C9) -> no 5 in R6C123

19. R6C9 = 5 (hidden single in R6), R1C4 = 7 (step 3), R3C1 = 2 (step 1), R4C1 = 6, R9C7 = 5 (step 2), R9C6 = 7, clean-up: no 6 in R23C4, no 4 in R3C8, no 5 in R4C5, no 8 in R4C8, no 1 in R5C5 (step 9b), no 8 in R5C6, no 6 in R5C7, no 9 in R8C5
19a. R3C1 = 2 -> R12C3 = {15} (step 5), locked for C3 and N1

20. 20(3) cage in N1 = {389/479}, no 6, 9 locked for N1
20a. 7 of {479} must be in R2C1 -> no 4 in R2C1

21. R4C4 + R5C45 (step 13a) = {136/235}
21a. 6 of {136} must be in R5C4 -> no 1 in R5C4

22. 1 in R5 locked in R5C89, locked for N6, clean-up: no 9 in R3C8, no 9 in R6C78

23. 11(3) cage at R4C9 (step 4) = {128/137/146}
23a. 4 of {146} must be in R4C9 -> no 4 in R5C89

24. R1C4 + R2C6 + R3C56 (step 17) = {2678/3479/3578} (cannot be {1589/2489} which don’t contain 7), no 1, clean-up: no 4 in R2C7

25. Naked pair {23} in R2C67, locked for R2

26. 9(3) cage in N2 = {126/135} (cannot be {234} which clashes with R2C6), no 4
[Alternatively 1 in N2 locked in 9(3) cage = {126/135}]
26a. 5 of {135} must be in R1C6 -> 3 in R1C6

27. 9 in N6 locked in R4C78 + R5C7
27a. 45 rule on N6 3 remaining innies R4C78 + R5C7 = 19 = {289/379}, no 4, clean-up: no 6 in R3C8, no 9 in R5C6

28. 9 in N5 locked in R6C456, locked for R6
28a. R6C456 (step 16) = {189/279}, no 4,6

29. 4 in N5 locked in R4C56 + R5C6 (step 15) = {458/467}, no 1

30. 15(3) cage in N4 = {249/258/348/357} (cannot be {159} because 1,5 only in R4C2), no 1
30a. 5 of {357} must be in R4C2 -> no 7 in R4C2

31. 1 in N4 locked in R6C12, locked for R6, clean-up: no 8 in R6C46 (step 28a)
31a. R6C456 = [972], R2C67 = [32], R5C5 = 3, clean-up: no 4 in R23C4, no 5 in R45C4 (step 21), no 9 in R5C12, no 3,8 in R6C78
31b. R45C4 = [16], clean-up: no 4 in R4C9 (step 23)

32. Naked pair {58} in R23C4, locked for C4 and N2 -> R3C5 = 4, R4C5 = 8, clean-up: no 6 in R89C5
32a. R89C5 = [59]

33. 9 in N4 locked in 15(3) cage = {249} (only remaining combination, step 30), locked for N4, clean-up: no 8 in R5C12
33a. Naked pair {57} in R5C12, locked for R5 -> R5C6 = 4, R4C6 = 5, R5C7 = 9, R5C3 = 2, clean-up: no 1 in R3C8

34. 19(5) cage at R6C2 must contain 1 -> R6C2 = 1
34a. 19(5) cage = {12349/12367}, no 8 -> R6C3 = 3, R6C1 = 8
34b. R78C4 = {24} -> R7C3 = 9, R4C23 = [94]

35. R1C5 = 2, R3C6 = 9 (hidden singles in N2)
35a. R9C4 = 3 (hidden single in C4)
35b. R89C3 = 13 = [76], R3C3 = 8, R23C4 = [85], clean-up: no 3 in N1 (step 20), no 2 in R4C8
35c. R1C12 = [94], R2C1 = 7, R23C2 = [63], R5C12 = [57], R2C5 = 1, R1C6 = 6, R12C3 = [15], R2C89 = [49], R34C8 = [73], R4C79 = [72], R6C78 = [46], R7C5 = 6

36. 18(3) cage in N9 = {468} (only remaining combination) -> R9C89 = [84], R8C9 = 6

and the rest is naked singles

I'm not sure whether I'll try the V2, certainly not until I've caught up on the V1s and other forum puzzles. I can see that combining those two cages takes away the clean-ups in R6C78 which reduced that cage to a naked pair.

Another variant from my backlog of unfinished puzzles. Thanks Nasenbaer for a challenging variant.

A108 V2 was a very hard puzzle. However Afmob, udosuk and I managed to solve it with different solving paths; udosuk used heavy combination analysis in conjunction with a "clone", Afmob and I both used short contradiction moves but in different parts of the grid.


Here is my walkthrough for A108 V2.

Prelims

a) R2C67 = {14/23}
b) R23C4 = {49/58/67}, no 1,2,3
c) R34C1 = {17/26/35}, no 4,8,9
d) R34C5 = {39/48/57}, no 1,2,6
e) R34C8 = {19/28/37/46}, no 5
f) R5C12 = {39/48/57}, no 1,2,6
g) R5C67 = {49/58/67}, no 1,2,3
h) R89C5 = {59/68}
i) R9C78 = {39/48/57}, no 1,2,6
j) 20(3) cage in N1 = {389/479/569/578}, no 1,2
k) 9(3) cage in N2 = {126/135/234}, no 7,8,9
l) 11(3) cage in N6 = {128/137/146/236/245}, no 9

1. 45 rule on N1 3 innies R12C3 + R3C1 = 8 = {125/134}, 1 locked for N1, clean-up: no 1,2 in R4C1
1a. 17(3) cage in N1 = {269/278/368/467} (cannot be {359/458} which clash with R12C3 + R3C1), no 5

2. 45 rule on N1 2 outies R1C4 + R4C1 = 13 = [67/76/85], clean-up: no 5 in R3C1

3. 45 rule on R6789 3 outies R45C4 + R5C5 = 10 = {127/136/145/235}, no 8,9
3a. 45 rule on C1234 3 innies R456C4 = 16
3b. -> R6C4 = R5C5 + 6, R6C4 = {789}, R5C5 = {123}

4. 45 rule on N478 1 innie R4C1 = 1 outie R9C7 + 1, no 7 in R4C1, R9C7 = {45}, clean-up: no 1 in R3C1, no 6 in R1C4 (step 2), R9C6 = {78}
4a. R4C1 + R9C7 = [54/65], CPE no 5 in R4C7 + R9C1
4b. Killer triple 7,8,9 in R1C4, R23C4 and R6C4, locked for C4

5. R12C3 + R3C1 (step 1) = {125/134}, 1 locked for C3
5a. R3C1 = {23} -> no 2,3 in R12C3
5b. 1 in N4 only in R4C2 + R6C12, CPE no 1 in R7C2

6. Max 9(3) cage in N2 + R2C6 = 13(4) must contain 1, locked for N2
6a. 1 in R3 only in R3C789, locked for N3, clean-up: no 4 in R2C6
6b. Killer triple 1,2,3 in 9(3) cage and R2C6, locked for N2, clean-up: no 9 in R4C5

7. 45 rule on C123 4 outies R1789C4 = 16 = {1258/1267/1348/1357/2347} (cannot be {1456/2356} because R1C4 must contain one of 7,8)
7a. Hidden killer triple 1,2,3 in R45C4 and R1789C4 for C4, R1789C4 contains two of 1,2,3 -> R45C4 must contain one of 1,2,3
7b. R45C4 + R5C5 (step 3) = {136/235} (cannot be {145} which doesn’t contain one of 1,2,3 in R45C4), no 4, 3 locked for N5, clean-up: no 9 in R3C5

8. Killer pair 5,8 in R34C5 and R89C5, locked for C5

9. 45 rule on N5 3 innies R4C56 + R5C6 = 17 = {179/278/458/467} (cannot be {269} because no 2,6,9 in R4C5)
9a. 1 of {179} must be in R4C6 -> no 9 in R4C6

10. 45 rule on N4 4 innies R4C1 + R6C123 = 18 = {1269/1359/1368/1458/1467/2367/2457} (cannot be {1278/2349} because R4C1 only contains 5,6, cannot be {2358/3456} which clash with R5C12)
10a. R4C1 = {56} -> no 5,6 in R6C123

11. 45 rule on N2 4 innies R1C4 + R2C6 + R3C56 = 23 = {1589/2489/2678/3479/3578} (cannot be {1679} because 6,9 only in R3C6, cannot be {2579/4568} which clash with R23C4, cannot be {3569} because R1C4 only contains 7,8)
11a. 9 of {2489/3479} must be in R3C6 -> no 4 in R3C6

12. 45 rule on R12 3 innies R2C248 = 1 outie R3C9 + 17
12a. Max R2C248 = 24 -> max R3C9 = 7

13. 13(3) cage at R1C3 = {148/157}
13a. 20(3) cage in N1 = {389/479/569/578}
13b. 7 of {479} must be in R2C1 (cannot be {47}9/{79}4 which clash with 13(3) cage) -> no 4 in R2C1

14. 45 rule on N69 4(3+1) innies R4C78 + R5C7 + R9C7 = 24
14a. Max R9C7 = 5 -> min R4C78 + R5C7 = 19, no 1 in R4C78, clean-up: no 9 in R3C8

15. R2C67 cannot be [14], here’s how
R2C67 = [14] => R2C3 = 5, R1C34 = [17] (step 13) is impossible because no combinations for R1C4 + R2C6 + R3C56 (step 11) contain both 1 and 7
15a. -> R2C67 = {23}, locked for R2
15b. 1 in N2 only in 9(3) cage at R1C5 = {126/135}, no 4
15c. 5 of {135} must be in R1C6 -> no 3 in R1C6
15d. 2 in N1 only in R3C123, locked for R3, clean-up: no 8 in R4C8

16. R12C3 + R3C1 (step 1) = {125/134} -> R12C3 + R34C1 = {14}[35]/{15}[26], CPE no 5 in R12C1 + R45C3
16a. 20(3) cage in N1 = {389/479/569/578}
16b. 5 of {569} must be in R1C2 -> no 6 in R1C2
16c. 7 of {479} must be in R2C1 (step 13b), 5 of {578} must be in R1C2 -> no 7 in R1C2

[At this stage I found two forcing chains but have omitted them after finding step 17a, which took me a long time to see. One chain has been replaced by steps 17a and 18; the other one by step 21.]

17. 45 rule on R1234 4 innies R4C2349 = 16 = {1249/1258/1267/1348/1357/2347} (cannot be {1456/2356} which clash with R4C1)
17a. R4C56 + R5C6 (step 9) = {278/458/467} (cannot be {179} = [719] which clashes with R4C2349), no 1,9, clean-up: no 4 in R5C7

18. 9 in N5 only in R6C456, locked for R6
18a. 45 rule on R6789 3 innies R6C456 = 18 = {189/279/459}, no 6
18b. 5 of {459} must be in R6C6 -> no 4 in R6C6
18c. 6 in R6 only in R6C789, locked for N6, clean-up: no 4 in R3C8, no 7 in R5C6

19. 15(3) cage in N4 = {168/249/258/267/348/357} (cannot be {159} because 1,5 only in R2C4, cannot be {456} which clashes with R4C1)
19a. R4C1 + R6C123 (step 10) = {1368/1458/2457} (cannot be {1467} which clashes with R6C456, cannot be {2367} which clashes with 15(4) cage)
19b. 15(3) cage = {168/249/267} (cannot be {258/348/357} which clash with R4C1 + R6C123), no 3,5
19c. 1 of {168} must be in R4C2 -> no 8 in R4C2

20. R5C12 = {39/57} (cannot be {48} which clashes with R4C1 + R6C123), no 4,8

21. R4C1 + R6C123 (step 19a) = {1368/1458/2457}, R6C456 (step18a) = {189/279/459} -> combined cage R4C1 + R6C123456 = 6{138}{279}/6{138}{459}/5{148}{279}/5{247}{189}, 1,8 locked for R6
[I wondered whether I could use a killer pair or hidden killer pair but decided that I couldn’t because 1,8 must be in R6C123 or in R6C456.]
21a. 1 in N6 only in 11(3) cage in N6 = {128/137}, no 4,5
21b. R6C789 (from combinations for combined cage R4C1 + R6C123456) = {356/456} (cannot be {267} which clashes with 11(3) cage) -> no 2,7 in R6C789, 5 locked for R6 and N6, clean-up: no 8 in R5C6, no 4 in R6C5 (step 18a)

22. 4 in N5 only in R4C56 + R5C6 (step 17a) = {458/467}, no 2

23. 4 in R5 only in R5C36 -> 15(3) cage in N4 = {249} or R5C67 = [49], CPE no 9 in R4C7

24. 9 in N6 only in R4C8 + R5C7
-> R4C78 + R5C7 + R9C7 = 24 (step 14) = {289/379}5/{389/479}4
24a. 4 of {479}4 must be in R4C8 + R9C7 -> no 4 in R4C7

25. 1 in R4 only in R4C249
25a. R4C2349 (step 17) = {1249/1258/1267} (cannot be {1348} because R4C23 in 15(3) cage cannot be {14/48}, cannot be {1357} because R4C23 in 15(3) cage cannot contain both of 1,7), no 3
25b. 8 of {1258} must be in R4C3 (R4C23 cannot contain both of 1,2) -> no 8 in R4C9

[The puzzle is now cracked.]

26. 3 in N5 only in R5C45, locked for R5, clean-up: no 7 in 11(3) cage in N6 (step 21a), no 9 in R5C12
26a. Naked pair {57} in R5C12, locked for R5 and N4 -> R4C1 = 6, R3C1 = 2, R9C7 = 5 (step 5), R9C6 = 7, R1C4 = 7 (step 2), clean-up: no 4 in R12C3 (step 13), no 6 in R23C4, no 5 in R4C5, no 6 in R5C6, no 8 in R5C7

27. R5C67 = [49], clean-up: no 8 in R3C5, no 1 in R3C8

28. Naked triple {128} in R5C389, locked for R5, 1 also locked for N6 -> R4C9 = 2, R5C45 = [63], R4C4 = 1 (step 3), clean-up: no 8 in R3C8
28a. Naked pair {18} in R5C89, locked for R5 and N6 -> R5C3 = 2

29. R6C456 (step 18a) = {279} (only remaining combination) -> R6C4 = 9, R6C6 = 2, R6C5 = 7, R4C5 = 8, R3C5 = 4, R4C6 = 5, R2C67 = [32], clean-up: no 6 in R89C5
29a. R89C5 = [59]

30. Naked pair {49} in R4C23, locked for R4 and N4, clean-up: no 6 in R3C8
30a. Naked pair {37} in R4C78, locked for N6
30b. Naked pair {37} in R34C8, locked for C8

31. R3C6 = 9 (hidden single in C6)
31a. R34C6 = [95] -> 31(5) cage at R2C8 = {35689/45679}, no 1, 6 locked for N3
31b. R4C7 = {37} -> no 3,7 in R3C7

32. R3C9 = 1 (hidden single in R3), R5C89 = [18]

33. Naked pair {15} in R12C3, locked for C3 and N1

34. 19(5) cage at R6C2 = {12349} (only remaining combination, cannot be {12367} because 6,7 only in R7C3) -> R6C2 = 1, R6C3 = 3, R7C3 = 9, R78C4 = {24}, locked for N8 -> R9C4 = 3, R4C23 = [94], R6C1 = 8

34. R9C4 = 3 -> R89C3 = 13 = [76], R9C9 = 4, R9C1 = 1

35. R6C1 = 8 -> R7C12 = 8 = {35}, locked for R7 and N7 -> R8C1 = 4, R78C4 = [42], R89C2 = [82]

36. 20(3) cage in N1 = {479} (only remaining combination) -> R1C1 = 9, R1C2 = 4

and the rest is naked singles.