Assassin 43

[rcbroughton]

Since the massively tricky 42 variant was broken open by Para this morning, thought I'd take a crack at 43.

Had to work a few combinations but when it fell, it fell fairly quickly.


0. Cage 24(3) at r1c1={789}
0a. Cage 9(3) at r1c2={126} {135} {234} - no 789
0b. Cage 12(2) at r2c3={39} {48} {57} - no 126
0c. Cage 7(3) at r3c2={124}
0d. Cage 8(2) at r3c3={17} {26} {35} - no 489
0e. Cage 21(3) at r6c1={489} {579} {678} - no 123
0f. Cage 8(3) at r7c1={125} {134} - no 6789
0g. Cage 12(2) at r7c6={39} {48} {57} - no 126
0h. Cage 26(4) at r8c5={2789} {3689} {4589} {4679} {5678} - no 1
0i. Cage 6(2) at r8c6={15} {24} - only values 1245

1. from 0 - naked triple {789} at r123c1 for c1, n1
1a. Only combinations [39] [48] [57] allowed in cage 12(2) at r2c3
1b. Only combinations [17] {26} {35} allowed in cage 8(2) at r3c3
1c. Only combinations 4{89} 5{79} 6{78} allowed in cage 21(3) at r6c1

2. 45 rule on c1234. Included cells r1c4 r9c4 equal 14(2)={59}/{68}

3. 45 rule on c6789. Included cells r1c6 r9c6 equal 7(2)=[16]/{25}/{34}
3a. 26(4) at r8c5 - only combinations with 2 is {2789} - 2 must be at r9c6 - can't have 2 at r89c5

4. from 0c - must use {124} in cage 7(3) at r3c2 - no 1,2,4 at r5c2

5. from 0f - must use 1 in cage 8(3) at r7c1 - no 1 at r45c1 r789c3 r89c2
5a. 13(3) at r5c1 - only combinations with 7 are {2[7]4} {57}1 - can't have 7 at r5c3
5b. 13(3) at r5c1 - only combinations with 9 is {139} - 1 must be at r5c3 - no 9 at r5c3
5c. 15(3) at r8c2 - no 1 - {249}/{357}/{456} blocked by 8(3) - only combinations {258} {267} {348} allowed - no 9

6. 6 locked in N4 for c1
6a. 13(3) at r5c1 - only combos with 5 are {57}1/[652] - 5 cannot be in r5c3

7. Must use {123} in either 12(3) at r7c9 or 12(3) at r1c9 - locked for c9

8. Must use {1234} in either 15(3) at r8c2 or 8(3) at r7c1 - locked for n7

9. 1 locked in c2 of cage 7(3) at r3c2 - no 1 at r12c2
9a. 9(3) at r1c2 now only {26}1 {234} {35}1 allowed - no 5,6 in r1c3

10. 45 rule on N1. Included cells r2c3 r3c2 r3c3 equal 12(3)=[345]/[426]/[516] - no 1,2 at r3c3
10a. 8(2) at r3c3 can only be {35}/[62] - no 6,7 at r3c4

11. 45 rule on N4. Included cells r46c3 minus excluded cells r37c2 equals 4
11a. min of excluded cells is 8, so min of included cells is 12 - no 1,2 in r46c3

12. 45 rule on N4. Included cells r4c123 r6c123 equal 32
12a. combinations are {125789} {234689} {245678} {145679}
12b. {234689} - {24} must be at r4c12
12c. {245678} - {24} must be at r4c12
12d. {145679} - {14} must be at r4c12
12e. no 4 at r4c3, r6c3 or r6c1

13. combos from step 12 remove combo {148} in 13(3) at r5c1
13a.combinations now {238} [391] [634] [571] - no 4 at r5c1, no 5 at r5c2

14. 45 rule on N7. Excluded cells r6c12 r78c4 equal 21
14a. only combinations with 9 are [59]{16} [59]{25} [59]{34} - no 9 at r78c4

15. Must use 7 in cage 21(3)=5{79}/6{78} at r6c1 - locked for c2
15a. add to 13a [571] eliminated from 13(3) - no 5 at r5c1

16. combo 32(6) from step 12 cannot contain {236} because of r5c1 - combo {234689} eliminated
16a. no 3 in 32(6) cells r46c3

17. Value 3 locked in row 5 of N4

18. Only combinations {258}/{348}/{26}7 in cage 15(3) at r8c2 - no 6 at r9c3

19. 45 rule on N9. Included cells r7c78 r8c7 equal 17(3)
19a. combinations are {39}5 {458} [764] [962] {78}2 {79}1 - no 1,2 r7c8

20. 45 rule on column 1. Included cells r456c1 equal 13(3)=[265]/[436] - no 2 at r5c1
20a. 13(3) at r5c1 {238} must now be [382] - no 3,8 at r5c3

21. 45 rule on column 1. Excluded cells r34567c2 r5c3 total 28(6)
-> r5c23 must be [91]/{38}/[82]/[34]
-> r67c2 must be {78} or {79}
-> r5c23 and r46c2 can't have 6, 24, 23 35,45 as it breaks 13(3) from step 20
21a. Only combos r5c23=[91] r67c2={78} r34c2=[12] or
21b. r5c23=[34] r67c2={79} r34c2={41}
21c. no 2 at r3c2
21d. no 4 at r4c2
21e. no 8 at r5c2
21f. no 2 from r5c3

22. 2 locked in row 4 of N4

23. innies in n1 = h12(3)=4{35} or 1{56} - no 4 at r2c3
23a. 12(2) r2c3 = [39]/[57] - no 8 at r2c4

24. Hidden triple {124} in r159c3 for c3

25. 9(3) at r1c2 - h12(3) from step 23 removes combo {135} - only left with {126} {234} - no 5

26. 15(3) at r8c2 = only left with {38}4/{58}2 - no 2,4,6 r89c2

27. 5 locked in column 3 of N1

26. Hidden single 5 at r6c1 for n4

remainder falls out of singles and cage sums


Regards
Richard

[Andrew]

Think in combinations if you want to survive this Assassin.

Had to work a few combinations but when it fell, it fell fairly quickly.

More than just a few combinations Richard. You got stuck into them pretty quickly. We know how much you like them.

I did more moves before starting to use the combinations so ended up with a considerably different solving path. Here is my walkthrough.
Thanks to Ed for the comments and corrections, which have been made in red.

1. R2C34 = {39/48/57), no 1,2,6

2. R3C34 = {17/26/35}, no 4,8,9

3. R7C67 = {39/48/57), no 1,2,6

4. R8C67 = {15/24}

5. R123C1 = {789}, locked for C1 and N1, clean-up: no 3,4,5 in R2C4, no 1 in R3C4

6. 9(3) cage in N1 = {126/135/234}

7. 7(3) cage in N14 = {124}
[I missed the fact that R3C2 “points” at R56C2 so there cannot be 1,2,4 in R56C2. Must train myself to look for those moves!]

8. 21(3) cage in N47 = {489/579/678}, no 1,2,3
8a. R6C1 = {456} -> R67C2 = {789}

9. R789C1 = 1{25/34}, 1 locked for C1 and N7

10. 26(4) cage in N8, no 1

11. R34C2 = 1{2/4}, 1 locked for C2

12. 9(3) cage in N1, 1 only in R1C3 -> no 5,6 in R1C3

13. 45 rule on C1234 2 innies R19C4 = 14 = {59/68}

14. 45 rule on C6789 2 innies R19C6 = 7 = [16]/{25}/{34}, no 7,8,9, no 6 in R1C6

15. 45 rule on C1 3 innies R456C1 = 13 = 6{25/34}, 6 locked for N4, no 2,4 in R5C1, no 4 in R6C1
[At this stage I ought also to have seen 45 rule on C9 3 innies R456C9 = 21 = {489/579/678}, no 1,2,3 which would probably have made progress in N6 a bit quicker]

16. 21(3) cage in N47 = {579/678} = 7{59/68}, 7 locked in R67C2, locked for C2

17. 45 rule on C12 3 outies R159C3 = 7 = {124}, locked for C3; clean-up: no 8 in R2C4, no 6,7 in R3C4

18. Naked triple {124} in R4C12 + R5C3, locked for N4

19. 13(3) cage in N4 = [382/391/634/652], no 5 in R5C1

20. 7 in N7 locked in R7C23 + R8C3
20a. 45 rule on N7 3 innies R7C23 + R8C3 = 22 = {679}, locked for N7
20b. 6 in N7 locked in R78C3, locked for C3, clean-up: no 2 in R3C4

21. 15(3) cage in N7 = {258/348}, no 2,4 in R89C2
[These 2,4 could also have been eliminated by killer pair 2/4 in R789C1 + R9C3]
21a. 8 in N7 locked in R89C2, locked for C2
21b. No [382] in 13(3) cage in N5 = [391/634/652]

22. R67C2 = {79}, locked for C2 -> R6C1 = 5

23. R5C2 = 3, R5C1 = 6 (naked singles) -> R5C3 = 4, R4C1 = 2, R4C2 = 1, R3C2 = 4

24. R89C2 = {58}, locked for C2, R9C3 = 2, R1C3 = 1 (naked singles), clean-up: no 5 in R1C6, no 6 in R9C6

25. R3C34 = {35}, locked for R3
25a. R23C3 = {35} -> R23C4 = 12 = {39/57}

26. R19C4 = {68} (cannot be {59} which clashes with R23C4), locked for C4

27. R5C789 = {179/278} = 7{19/28}, no 5, 7 locked for R5 and N6
27a. R5C456 = 5{19/28}, 5 locked for N5

28. 8 in R46C3 locked for 27(5) cage in N45 = 8{1279/1459/2359/3457}, no 9 in R5C4, no 7,9 in R6C4

29. 6 in R78C3 locked for 22(4) cage in N78 = 6{259/349/457} (cannot be {3568} because no 7,9), no 1,7,9 in R78C4

30. 1 in C4 locked in R56C4, locked for N5
30a. 27(5) cage in N45 = 8{1279/1459} = 18{279/459}, no 3, no 4 in R6C4

31. R24C4 = {79} (hidden pair in C4), no 4 in R4C4
31a. R4C34 + R6C3 = {789} -> R56C4 = {12}, locked for C4 and N5

32. 45 rule on N9 3 innies R7C78 + R8C7 = 17 = {179/269/278/359/458/467} (cannot be {368} which has no 1,2,4,5), no 1,2 in R7C8

33. 45 rule on R123 2 remaining innies R3C58 = 10 = {19/28}, no 6,7

34. 45 rule on R789 3 innies R7C258 = 18 = 7[29]/7{38}/7{56}/9[18]/9[27]/9{36}/9{45}, no 7,9 in R7C5
Ed pointed out that 7{38} and 9{45} clash with R7C67. This eliminates 4 from R7C58. I haven't included that elimination since it might affect the later steps which have already been posted is this walkthrough.

35. 4 in C4 locked in R78C4, locked for N8, clean-up: no 3 in R1C6 (moved from step 36),no 8 in R7C7, no 2 in R8C7
35a. 22(4) cage in N78 = 6{349/457} = 46{39/57}

36. Naked triple {345} in R78C4 + R9C6 for N8, clean-up: no 7,9 in R7C7, no 1 in R8C7

37. 26 cage in N8 must have R9C4 = {68}, R9C6 = {35}, only valid combinations {3689/5678} = 68{39/57}, 6,8 locked for N8, no 2, clean-up: no 4 in R7C7

38. R7C26 = {79}, locked for R7 -> R7C3 = 6

39. 8 in R7 locked in R7C89, locked for N9

40. 21(5) cage in C5 = {12369/12378/12459/12468/12567/13458} (cannot be {13467/23457 because no 3,4,5,6,7 in R37C5}, 1 locked for C5, no 9 in R3C5, no 8 in R46C5

41. 19(4) cage in N2 must have R1C4 = {68}, R1C6 = {24}, only valid combinations {2368/2458/2467} = 2{368/458/467}, no 9, 2 locked for N2

42. 1 in N2 locked in R2C6 + R3C56
42a. 45 rule on N2, R23C4 = 12 (step 25a) -> 3 innies R2C6 + R3C56 = 14 = 1{49/58/67}, no 3
42b. 4,5 in R2C6 + R3C56 only in R2C6, 6,7 only in R23C6 -> no 1,8,9 in R2C6

43. 1 in N2 locked in R3C56, locked for R3

44. 45 rule on N9 3 innies R7C78 + R8C7 = 17 = {458} (only valid combination), locked for N9 -> R7C8 = 8, R7C7 = 5, R8C7 = 4
[Forgot to do my usual clean-up this time; R7C6 and R8C6 are fixed in step 52.]

45. R7C4 = 4 (hidden single in N8) -> R9C1 = 4 (hidden single in N7)

46. 18(3) cage in N69, R7C8 = 8 -> R6C89 = 10 = {19/46}, no 2,3

47. 5 in N6 locked in R4C89, only valid combination for 16(3) cage in N36 = {259} -> R3C8 = 2, R4C89 = {59}, locked for R4 and N6, clean-up: no 1 in R6C89
47a. R6C89 = {46}, locked for R6 and N6

48. R4C4 = 7, R4C3 = 8, R6C3 = 9, R56C4 = {12} (step 34)
48a. R8C3 = 7, R7C2 = 9, R6C2 = 7, R7C6 = 7 (naked singles), R8C4 = 5 (cage sum) -> R9C6 = 3, R3C4 = 3, R3C3 = 5, R2C3 = 3, R2C4 = 9, R4C7 = 3, R8C2 = 8, R9C2 = 5, R6C5 = 3, R6C6 = 8 (naked singles)

49. R5C56 = {59} -> R5C4 = 1 (step 27a) -> R6C4 = 2, R6C7 = 1

50. 9 in N8 locked in R89C5, locked for C5 -> R5C5 = 5, R5C6 = 9

51. 21(5) cage in C5 = {13458} (only remaining combination) -> R3C5 = 8, R4C5 = 4, R7C5 = 1

52. R4C6 = 6, R1C4 = 6, R12C2 = [26], R1C6 = 4, R1C5 = 7, R2C5 = 2, R2C6 = 5, R3C6 = 1, R8C6 = 2, R78C1 = [31], R7C9 = 2, R5C8 = 7, R5C9 = 8, R5C7 = 2, R7C6 = 7, R9C4 = 8 (naked singles)

53. R1C17 = {89}, locked for R1

54. R2C17 = {78}, locked for R2

55. 12(3) cage in N9 R7C9 = 2 -> R89C9 = 10 = [37/91]

56. 16(3) cage in N3 = [934] (only valid combination)

and the rest is naked singles