(Unofficial) Assassin 97

[Andrew]

uA97 V1.5 was a nice variant. Thanks J-C.

There is a lot of similarity between Afmob's walkthrough and mine but we took significantly different paths to break this puzzle open after an easy opening. It was surprising that the early stages were, in fact, easier than for Afmob's original puzzle. This variant didn't need clashes between cages and hidden cages in N8 which were used in the quickest start for the original puzzle. I must admit I was disappointed that I was unable to use the overlap of R7C3467 and R7C456; maybe it will be needed for J-C's V2 if I try it.

It looks as if a very short forcing chain, Afmob's step 7a, my step 30, is one of the key moves in this puzzle.

I must admit I couldn't follow Gary's "By using x-wings on the 1s in r123 can now show that in N2 9<>12(3) cage.So 11(2) cage N5={29}". I couldn't see any x-wings on those 1s. There is a forcing chain for R1 where either R1C5 must be 1 or R1C78 must be {19). That still leaves 9 to be eliminated from R3C5; in my walkthrough 9 was eliminated from R2C5 by CPE.

Afmob's step 7d is the sort of step that's probably more easily seen by those using software solvers than by me using an Excel worksheet and only looking up combinations in Ruud’s Combination Calculator or when previously included in my partial walkthrough.

After reading Afmob's walkthrough, I checked whether that step could have been used in my solving path but it couldn't. I still had {459} in 18(3) cage at R5C7. I made the same eliminations in my more complicated step 34.

I'll go slightly higher than Afmob and rate uA97 V1.5 as an Easier 1.5. Most of my steps 29 to 34 were fairly difficult to find. Maybe it's a Hard 1.25 for those using software solvers and an Easier 1.5 for those of us who don't. Still I've got no immediate plans to change my solving method.

Here is my walkthrough for uA97 V1.5.

Prelims

a) R45C5 = {29/38/47/56}, no 1
b) R67C5 = {69/78}
c) R89C1 = {39/48/57}, no 1,2,6
d) R89C9 = {18/27/36/45}, no 9
e) 9(3) cage at R7C3 = {126/135/234}, no 7,8,9
f) 9(3) cage at R8C2 = {126/135/234}, no 7,8,9
g) R8C456 = {389/479/569/578}, no 1,2
h) R9C456 = {128/137/146/236/245}, no 9
i) 21(3) cage in N9 = {489/579/678}, no 1,2,3
j) 27(4) cage at R1C4 = {3789/4689/5679}, no 1,2, CPE no 9 in R2C56

1. 45 rule on C1234 2 innies R89C4 = 10 = {37/46}/[82/91], no 5, no 8 in R9C4

2. 45 rule on C6789 2 innies R89C6 = 14 = {68}/[95], no 1,2,3,4,7, no 5 in R8C6

3. 45 rule on C5 2 innies R89C5 = 7 = {34}/[52/61], no 7,8,9, no 5,6 in R9C5

4. R9C456 = {128/146/236/245} (cannot be {137} because R9C6 only contains 5,6,8), no 7, clean-up: no 3 in R8C4 (step 1)
4a. R9C6 = {568} -> no 6 in R9C4, clean-up: no 4 in R8C4 (step 1)

5. R8C456 = {389/479/569/578}
5a. 3,4,5 only in R8C5 -> R8C5 = {345}, clean-up: no 1 in R9C5 (step 3)

6. 1 in C5 locked in R123C5, locked for C5
6a. R123C5 = 1{29/38/47/56}

7. 45 rule on N7 2 innies R7C12 = 1 outie R7C4 + 15, R7C12 = 16,17, R7C4 = {12}
7a. R7C12 = {79/89}, 9 locked for R7, N7 and 22(4) cage, clean-up: no 6 in R6C5, no 3 in R89C1

8. 9 in N8 locked in R8C46, locked for R8
8a. 9 in N9 locked in 21(3) cage = {489/579}, no 6

9. R9C456 (step 4) = {146/236/245} (cannot be {128} which clashes with R7C4), no 8, clean-up: no 6 in R8C4 (step 2)
9a. Killer pair 1,2 in R7C4 and R9C45, locked for N8

10. R8C456 must contain 9 (step 8) = {389/479} (cannot be {569} which clashes with R9C6), no 5,6, clean-up: no 4 in R9C4 (step 1), no 2 in R9C5 (step 3)

11. Naked pair {34} in R89C5, locked for C5 and N8, clean-up: no 7,8 in R123C5 (step 6a), no 7,8 in R45C5, no 7 in R8C4 (step 1)
11a. Naked pair {12} in R79C4, locked for C4

12. Naked pair {89} in R8C46, locked for R8 and N8, R8C5 = 3 (step 10), R9C5 = 4, clean-up: no 7 in R6C5, no 5 in R8C9, no 1,6 in R9C9

13. R67C5 = [87] (hidden pair in C5)
13a. Naked pair {56} in R79C6, locked for C6

14. Naked pair {89} in R7C12, locked for R7, N7 and 22(4) cage, clean-up: no 4 in R8C1

15. R7C12 = 17 -> R7C4 = 2 (step 7), R9C4 = 1, R9C6 = 6 (step 9), R7C6 = 5, R8C4 = 9 (step 1), R8C6 = 8

16. 27(4) cage at R1C4 = {3789/4689/5679}, 9 locked in R2C23, locked for R2 and N1

17. Naked pair {57} in R89C1, locked for C1 and N7

18. Naked pair {23} in R9C23, locked for R9 and N7, R8C2 = 4 (cage sum), clean-up: no 6,7 in R8C9, no 5 in R9C9

19. 21(3) cage in N9 = {579} (only remaining combination), locked for N9 -> R89C9 = [18], R78C3 = [16], R8C7 = 2, R7C7 = 6 (cage sum)

20. R7C89 = {34} = 7 -> R56C9 = 15 = {69}, locked for C9 and N6

21. R234C9 = {457} (only remaining combination), locked for C9 -> R7C89 = [43], R1C9 = 2
21a. R1C78 = 10 = {19/37}/[46], no 5,8

22. 17(4) cage at R3C3 = {2348/2357/2456}, no 9
22a. 2 locked in R34C3, locked for C3 -> R9C23 = [23]

23. 45 rule on C1 1 outie R7C2 = 1 innie R1C1 + 5, R1C1 = {34}
23a. R1C123 = {358/367/457} (cannot be {178} because R1C1 only contains 3,4), no 1
23b. R1C1 = {34} -> no 3 in R1C2, no 4 in R1C3
23c. R1C78 (step 21a) = {19}/[46] (cannot be {37} which clashes with R1C123), no 3,7

24. R7C12 = {89} = 17 -> R56C1 = 5 = {14/23}
24a. Killer pair 3,4 in R1C1 and R56C1, locked for C1

25. R234C1 = {169/268}
25a. 9 of {169) must be in R4C1 -> no 1 in R4C1

26. 45 rule on R1 3 innies R1C456 = 17 = {179/368/458/467} (cannot be {359} which clashes with R1C123)
26a. 1 of {179} must be in R1C5 -> no 9 in R1C5
26b. 8 of {368} must be in R1C4 -> no 3 in R1C4
26c. 5,6 of {368/458/467} must be in R1C5 -> no 5,6 in R1C4

27. R123C5 (step 6a) = 1{29/56}
27a. 9 of {129} must be in R3C5 -> no 2 in R3C5

28. 18(3) cage at R5C3 = {369/378/459/468/567}
28a. 8,9 of {459/468} must be in R5C3 -> no 4 in R5C3

29. 2 in C8 locked in R456C8 -> 21(4) cage at R3C8 must contain 2
29a. Hidden killer pair 3,8 in R2C8 and R3456C8 for C8 -> R3456C8 must contain at least one of 3,8
29b. 21(5) cage at R3C8 = {12378/12468} (cannot be {12369} because 6,9 only in R3C8, cannot be {12459/12567} because they don’t contain 3 or 8, cannot be {23457} which clashes with R8C8 because 4 must be in R6C7), no 5,9
29c. 8 locked in R345C8, locked for C8

30. 45 rule on C12 2 remaining outies R16C3 = 1 innie R2C2 + 6
30a. 9 in R2 must be in R2C23 -> R6C3 can only be 9 when R2C2 is 9
30b. If R2C2 = 9 => R16C3 = 15 = [87]
30c. Combining steps 30a and 30b -> no 9 in R6C3

31. Hidden killer pair 1,3 in R456C2 and R56C1 for N4 -> R456C2 must have one, and only one, of 1,3
31a. 24(5) cage at R3C2 = {13479/13569/13578/14568}
31b. Killer pair 8,9 in R3456C2 and R7C2, locked for C2
31c. R3C2 of {13479/13578} must contain 1 or 3 (step 31) -> no 7 in R3C2

32. R2C3 = 9 (hidden single in R2)

33. 27(4) cage at R1C4 = {3789/4689/5679}
33a. 3 of {3789} must be in R2C2 (cannot be in R2C4 when 7 or 8 of {3789} clashes with R1C123), no 3 in R2C4
33b. 3 of {3789} must be in R2C2, 7 of {5679} must be in R1C4 -> no 7 in R2C2

34. Hidden killer pair 5,6 in R123C5 and R23C4 -> either R123C5 or R23C4 must contain both of 5,6 because R123C5 cannot contain only one of 5,6
34a. 27(4) cage at R1C4 = {3789/4689/5679}
34b. If {5679} => R1C4 = 7, R2C24 = {56} => R3C4 = {56} (step 34) => no place for 8 in C4 -> cannot be {5679}
34c. -> 27(4) cage at R1C4 = {3789/4689}, no 5
34d. 6 of {4689} must be in R2C2 -> no 6 in R2C4, clean-up: no 5,6 in R3C4 (step 34)
34e. 8 locked in R12C4, locked for C4

35. 5,6 in N2 locked in R123C5 = {156}, locked for C5
35a. Naked pair {29} in R45C5, locked for N5

36. 14(3) cage at R5C6 = {347} (only remaining combination, cannot be {158} because 5,8 only in R5C7), CPE no 3,4,7 in R5C4

37. R4C6 = 1 (hidden single in C6)
37a. 1 in N6 locked in R5C8 + R6C78, locked for 21(5) cage -> no 1 in R3C8
37b. 5 in N6 locked in R4C79, locked for R4

38. 5 in C4 locked in R56C4, locked for 18(3) cage at R5C3 -> no 5 in R5C3
38a. 18(3) cage at R5C3 = {567} (only remaining combination) -> R5C3 = 7
38b. Naked pair {56} in R56C4, locked for C4

39. Naked pair {34} in R5C67, locked for R5 and 14(3) cage -> R6C6 = 7, clean-up: no 1,2 in R6C1 (step 24)

40. 17(4) cage at R3C3 (step 22) = {2348/2357}
40a. 3,4 of {2348} must be in R34C4 -> no 4 in R34C3

41. R6C3 = 4 (hidden single in C3), R6C1 = 3, R5C1 = 2 (step 24), R1C1 = 4, R45C5 = [29], R56C9 = [69], R56C4 = [56], R4C3 = 8, R1C3 = 5, R1C2 = 7 (step 23a), R3C3 = 2, R56C2 = [15], R6C78 = [12], R5C8 = 8, R1C7 = 9, R1C8 = 1 (step 23c), R1C456 = [863], R5C67 = [43], R4C4 = 3, R4C8 = 7, R89C8 = [59], R9C7 = 7, R89C1 = [75], R23C6 = [29], R2C8 = 6, R3C8 = 3, R2C2 = 3, clean-up: no 8 in R23C1 (step 25)

42. R3C4 = 4 (cage sum)

and the rest is naked singles

4 7 5 8 6 3 9 1 2
1 3 9 7 5 2 8 6 4
6 8 2 4 1 9 5 3 7
9 6 8 3 2 1 4 7 5
2 1 7 5 9 4 3 8 6
3 5 4 6 8 7 1 2 9
8 9 1 2 7 5 6 4 3
7 4 6 9 3 8 2 5 1
5 2 3 1 4 6 7 9 8

[Andrew]

Thanks J-C for uA97 V2. That was really neat to create a puzzle with 3 IOUs in R7, all with a difference of 7, so that 7 is locked for R7 and N8!

I must admit that I'd seen J-C's step 2d before I managed to solve this puzzle but, as explained in the comment before step 16, I hope I would have found it anyway. IMHO the overlap of R7C3467 and R7C456 is easier to spot than J-C's way. I was also prepared to do combination analysis on R7C3467 and R7C456, using the fact that R7C46 are common cells for both hidden cages, but that wasn't necessary.

After he had gone through J-C's walkthrough, Afmob suggested that this puzzle should be rated Hard 1.25. However, having solved it myself with help on that one step, I felt that it had a long solution and a fairly narrow solving path with many of my key steps the same as those used by J-C, so I'll rate uA97 V2 at 1.5.

Here is my walkthrough.

Prelims

a) R45C5 = {39/48/57}, no 1,2,6
b) R67C5 = {19/28/37/46}, no 5
c) R89C1 = {17/26/35}, no 4,8,9
d) R89C9 = {29/38/47/56}, no 1
e) R1C123 = {489/579/678}, no 1,2,3
f) 10(3) cage at R5C6 = {127/136/145/235}, no 8,9
g) 10(3) cage in N9 = {127/136/145/235}, no 8,9

1. 45 rule on C1234 2 innies R89C4 = 12 = {39/48/57}, no 1,2,6

2. 45 rule on C6789 2 innies R89C6 = 15 = {69/78}

3. 45 rule on C5 2 innies R89C5 = 6 = {15/24}

4. R8C456 = {189/279/459/468/567} (cannot be {369/378} because R8C5 only contains 1,2,4,5), no 3, clean-up: no 9 in R9C4 (step 1)

5. 45 rule on N8 3 innies R7C456 = 12 = {138/156/237/246} (cannot be {129/147/345} which clash with R89C5), no 9, clean-up: no 1 in R6C5
[I probably ought to have looked at R9C456 next and found that it cannot be {159/249} because of clashes between R89C5 and R9C45, thus locking 9 in R8C46. However I don’t think it makes much difference missing it. Not much more progress can be made in N8 until R7C456 is reduced to one combination.]

6. 45 rule on R89 2 innies R8C37 = 11 = {29/38/47/56}, no 1

7. 45 rule on C1 1 innie R1C1 = 1 outie R7C2 + 5, R1C1 = {6789}, R7C2 = {1234}

8. 45 rule on C9 1 outie R7C8 = 1 innie R1C9 + 7, R1C9 = {12}, R7C8 = {89}

9. R1C789 = {139/148/157/238/247/256} (cannot be {346} because R1C9 only contains 1,2)
9a. R1C9 = {12}, no 1,2 in R1C78

10. 45 rule on R1 3 innies R1C456 = 11 = {137/146/236/245} (cannot be {128} which clashes with R1C9), no 8,9

11. 45 rule on N7 2 innies R7C12 = 1 outie R7C4 + 7, IOU no 7 in R7C1
11a. Max R7C12 = 13 -> max R7C4 = 6
11b. Min R7C4 = 1 -> min R7C12 = 8, min R7C1 = 5 because R7C12 cannot be [44]

12. 45 rule on N9 2 innies R7C89 = 1 outie R7C6 + 7, IOU no 7 in R7C9
12a. Min R7C89 = 9 -> min R7C6 = 2
12b. Max R7C6 = 8 -> max R7C89 = 15, no 8,9 in R7C9

13. Hidden killer quad 6,7,8,9 in R123C5, R45C5 and R67C5 for C5 -> R123C5 must contain two of 6,7,8,9
13a. R123C5 = {179/269/278/368/467} (cannot be {359/458} which only contain one of 6,7,8,9), no 5

14. Combined cage R4589C5 = 18 = {1359/1458/2349/2457} (cannot be {1278/2358} which clash with R89C5)

15. R123C5 (step 13a) = {269/278/368/467} (cannot be {179} which clashes with R4589C5), no 1
15a. 1 in C5 locked in R789C5, locked for N8

[The next step was helped by J-C’s walkthrough (step 2d). I’d already made notes about the overlap of R7C3467 and R7C456. I just hadn’t yet(?) spotted the key point that it gives another IOU in R7. It may be harder to spot this because in both the original puzzle and V1.5, R7C37 = R7C5 which doesn’t give an IOU.]

16. 45 rule on R89 4 outies R7C3467 = 19, R7C456 = 12 (step 5) -> R7C37 = R7C5 + 7, IOU no 7 in R7C37

17. 7 in R7 locked in R7C56, locked for N8, clean-up: no 5 in R89C4 (step 1) , no 8 in R89C6 (step 2)
17a. R7C456 (step 5) = {237} (only remaining combination), locked for R7 and N8, clean-up: no 7,8 in R1C1 (step 7), no 2,4,6,9 in R6C5, no 9 in R8C4 (step 1), no 4 in R89C5 (step 3)

18. R8C456 (step 4) = {189/459} (cannot be {468} because 4,8 only in R8C4) -> R8C6 = 9, R9C6 = 6, clean-up: no 2 in R8C1, no 2 in R8C37 (step 6), no 5 in R8C9, no 2 in R9C9

19. Naked pair {15} in R89C5, locked for C5, clean-up: no 7 in R45C5
19a. 10(3) cage in N9 = {127/136/145/235}
19b. 4 of {145} must be in R9C78 (R9C78 cannot be {15} which clashes with R9C5), no 4 in R8C8

20. Naked pair {48} in R89C4, locked for C4

21. Killer pair 3,8 in R45C5 and R67C5, locked for C5

22. 6 in C5 locked in R123C5, locked for N2
22a. R123C5 = {269/467}

23. R1C456 (step 10) = {137/146/236/245}
23a. 7 of {137} must be in R1C5 -> no 7 in R1C46

24. 17(3) cage at R7C6 = {269/278/359/368/467} (cannot be {179} because 1,9 only in R7C7, cannot be {458} because R7C6 only contains 2,3,7), no 1
24a. 3 of {359/368} must be in R7C6 -> no 3 in R8C7, clean-up: no 8 in R8C3 (step 6)
24b. 9 of {359} must be in R7C7 -> no 5 in R7C7

25. 13(3) cage at R7C3 = {238/247/256/346} (cannot be {139} because 1,9 only in R7C3, cannot be {148/157} because R7C4 only contains 2,3), no 1,9

26. R1C123 = {489/579/678}
26a. R1C1 = {69} -> no 6,9 in R1C23

27. R7C89 = R7C6 + 7 (step 12)
27a. R7C6 = {237} -> R7C89 = 9,10,14 = [81/91/86/95], no 4 in R7C9

28. 17(3) cage at R7C6 (step 24) = {269/278/359/368/467}
28a. 45 rule on N9 4 innies R7C789 + R8C7 = 24 = {1689/4569} (cannot be {4578} because R78C7 cannot be [47] since there is no 6 in R7C6), no 7, clean-up: no 4 in R8C3 (step 6)
28b. 6,9 locked in R7C789 + R8C7, locked for N9, 9 locked for R7, clean-up: no 2 in R8C9, no 5 in R9C9
28c. 1 of {1689} must be in R7C9, 6 of {4569} must be in R78C7 because 17(3) cage doesn’t contain both 4 and 5 -> no 6 in R7C9
28d. 6 in N9 locked in R78C7, locked for C7
28e. 17(3) cage at R7C6 must contain 6 in R78C7 = {269/368/467}, no 5, clean-up: no 6 in R8C3 (step 6)
28f. 2 in N9 locked in 10(3) cage = {127/235}, no 4

29. 9 in R9 locked in R9C23
29a. 17(3) cage in N7 = {179/269/359}, no 4,8
29b. 6 of {269} must be in R8C2 -> no 2 in R8C2

30. R8C8 = 2 (hidden single in R8)

31. 4,8 in N7 locked in R7C123 + R8C3
31a. 45 rule on N7 4 innies R7C123 + R8C3 = 20 = {1478/3458}, no 6
31b. 3,7 only in R8C3 -> R8C3 = {37}, clean-up: no 6 in R8C7 (step 6)
31c. 8 of {3458} must be in R7C3 or cannot make total for 13(3) cage at R7C3 -> no 5 in R7C3
31d. R78C3 = [47/83] = 11 -> R7C4 = 2, clean-up: no 8 in R6C5

32. R7C7 = 6 (hidden single in R7), R7C8 = 9 (hidden single in R7)
32a. Naked pair {48} in R8C47, locked for R8, clean-up: no 3,7 in R9C9
32b. Naked pair {37} in R8C39, locked for R8, clean-up: no 1,5 in R9C1

33. Naked pair {37} in R67C5, locked for C5, clean-up: no 4 in R123C5 (step 22a), no 9 in R45C5
33a. Naked triple {269} in R123C5, locked for N2
33b. Naked pair {48} in R45C5, locked for N5

34. Hidden killer pair 6,9 in R4C4 and R56C4 for C4, R56C4 cannot contain both 6 and 9 -> R4C4 = {69}, R56C4 must contain 6 or 9
34a. 15(3) cage at R5C3 = {159/168/249/267/456} (cannot be {258/348/357} which don’t contain 6 or 9), no 3
34b. 6 or 9 must be in R56C4 -> no 6,9 in R5C3

35. R1C456 (step 10) = {146/245} (cannot be {236} because 2,6 only in R1C5) -> R1C6 = 4, R1C45 = [16/52], no 3 in R1C4

36. 3 in R1 locked in R1C78, locked for N3
36a. R1C789 (step 9) = {139/238}, no 5,6,7
36b. 7 in R1 locked in R1C23, locked for N1

37. 17(4) cage in R1C6 = {1349/1457/2348/2456}
37a. 3 of {2348} must be in R2C6 -> no 8 in R2C6

38. R3C6 = 8 (hidden single in C6)

39. 9 in C9 locked in R234C9 = {179/269/359}, no 4,8

40. Hidden killer pair 4,8 in R56C9 and R9C9 -> R56C9 must contain 4 or 8
40a. 24(4) cage at R5C9 = {1689/2589/4569} (cannot be {2679/3579} which don’t contain 4 or 8, cannot be {3489} because R7C9 only contains 1,5), no 3,7
40b. R7C9 = {15} -> no 1,5 in R56C9

41. Hidden killer pair 3,7 in R234C9 and R89C9 -> R234C9 must contain one of 3,7
41a. R234C9 (step 39) = {179/359} (cannot be {269} which doesn’t contain 3 or 7), no 2,6
41b. 3 of {359} must be in R4C9 -> no 5 in R4C9

42. 6 in C9 locked in R56C9, locked for N6
42a. 24(4) cage at R5C9 (step 40a) = {1689/4569} (cannot be {2589} which doesn’t contain 6), no 2

43. R1C9 = 2 (hidden single in C9), R1C5 = 6, R1C1 = 9
43a. Naked pair {38} in R1C78, locked for R1 and N3
43b. Naked pair {57} in R1C23, locked for R1 and N1 -> R1C4 = 1

44. 3 in C4 locked in R23C4, locked for N2
44a. 17(4) cage at R1C6 (step 37) = {1457} (only remaining combination), no 6,9
44b. Naked triple {157} in R2C678, locked for R2 -> R2C4 = 3, R2C9 = 9, R23C5 = [29]
44c. 1 locked in R2C78, locked for N3

45. R12C4 = [13] = 4 -> R2C23 = 12 = {48}, locked for N1, R2C1 = 6, clean-up: no 2 in R9C1

46. Naked pair {48} in R27C3, locked for C3

47. Naked pair {57} in R3C49, locked for R3 -> R3C78 = [46], R8C7 = 8, R1C78 = [38], R89C4 = [48], R9C9 = 4, R8C9 = 7, R8C3 = 3, R7C3 = 8 (step 31d), R7C1 = 5, R7C9 = 1, R7C2 = 4, R89C1 = [17], R8C2 = 6, R89C5 = [51], R9C78 = [53], R34C9 = [53], R3C4 = 7, R2C6 = 5, R2C23 = [84]

48. R7C12 = [54] = 9 -> R56C1 = 10 = {28}, locked for C1 and N4 -> R34C1 = [34], R45C5 = [84]

49. R7C89 = [91] = 10 -> R7C6 = 3 (step 12), R67C5 = [37]

50. 15(3) cage at R5C3 = {159} (only remaining combination) -> R5C3 = 1, R56C4 = {59}, locked for C4 -> R4C4 = 6, R3C23 = [12], R9C23 = [29], R4C3 = 5, R1C23 = [57], R6C3 = 6, R56C9 = [68], R56C1 = [82]

51. R3C67 = [84] = 12 -> R4C67 = 11 = [29]

and the rest is naked singles

9 5 7 1 6 4 3 8 2
6 8 4 3 2 5 1 7 9
3 1 2 7 9 8 4 6 5
4 7 5 6 8 2 9 1 3
8 3 1 9 4 7 2 5 6
2 9 6 5 3 1 7 4 8
5 4 8 2 7 3 6 9 1
1 6 3 4 5 9 8 2 7
7 2 9 8 1 6 5 3 4