Assassin 36

[sudokuEd]

Very enjoyable, difficult Killer. Thanks a lot Ruud.

Took a couple of goes. Had to use hidden cages, lots of combination conflicts and a few nice (small?) chains to make some decent inroads. Maybe the chains weren't necessary - but got things moving nicely .

[edit 9th Feb: Nasenbaer has kindly pointed out a nice move after step 12, and some other hidden singles missed. Thanks Peter]

Assassin 36

0. 27(4)n14 = 9{378/468/567}(no 1,2)

1."45"n1 -> r4c3 + 1 = r3c1 -> r3c1 = {4..9}, r4c3 = {3..8}

2. -> 9 required in 27(4)n12 only in n1:9 locked n1 -> no 8 r4c3 (step 1)

3. 8(3)n1 = 1{25/34}: 1 locked n1 = [4/5..]
3a. no 8 9(2)n1

4. 9(2)n1 = {27/36} ({45} blocked by 8(3)n1 step 3)

5. {5679} blocked from 27(4) by 9(2)n1 (leaves no 6 or 7 for 9(2))
5a. 27(4) = 89{37/46} (no 5)
5b. 8 locked for n1 -> no 7 r4c3(step 1)
5c. no 6 r3c1 (step 1)

Now some chains make a big impact
6. from step 1 with r4c3 = {346}
6a.r4c3 = 3 -> r3c1 = 4 and 9(2) = {27}: Blocked: (no 4 or 2 for 8(3)n1)
6b.r4c3 = 6 -> r3c1 = 7 : Blocked: no 6 or 7 for 9(2)n1
6c. -> r4c3 = 4, r3c1 = 5

7. 8(3)n1 = {134}:locked for n1

8. 9(2)n1 = {27}: locked for n1, c3

9. 27(4)n14 now 23(3) = {689}

10. r4c12 = 11(2) = {29/38}(no 1,6,7) = [2/8,3/9,8/9..]

11. 17(3)n4 = {179/269/368}:no 5 ({278/359} blocked by r4c12 (step 10))

12. 17(3)n4 = [8/9], r4c12 = [8/9]:Killer pair [89] for n4

(NB:"45" on c1 -> r1459c2 = 11(4) = {1235} -> r9c2 = 5, naked triple on {123} for c2, 4 locked for c1 in r12c1 and this Killer pair [89] would also work for c1. Very handy! Sadly, not in this walk-through)

13. "45" n4 -> r5c3 + r6c23 = h13(3) and must have 5 = 5{17/26} (no 3)

14. h13(3)n4 must have 2 or 7(step 13): only available in r6c2 -> r6c2 = {27}
14a. ->5 for n4 only in c3 ->locked for c3

15. "45" n3 -> r4c7 + 6 = r3c9
15a. r4c7 = {123}
15b. r3c9 = {789}

16. "45" n3 -> r4c789 = 14

17. combining steps 15 and 16 and some more short chains
17a. r4c7 = 1 -> r3c9 = 7 -> r4c89 = 13 = {58}
17b. r4c7 = 2 -> r3c9 = 8 -> r4c89 = 12 = {57} ({39} blocked by r4c12 (step 10))
17c. r4c7 = 3 -> r3c9 = 9 -> r4c89 = 11 = {56} ({38} blocked by r4c7)

18. -> r4c89 = 5{6/7/8}(no 3,9):5 locked for r4,n6
18a. no 4 r3c5

19. 38(6)n69 = {356789}(no1,2,4)
19a. from step 18 -> 5 required in 38(6) in n9:5 locked for n9
19b. 5 only in r78c7 for c7 -> 5 locked for 38(6)

20. "45" n2 -> r4c456 = h16(3) = {169/178/367}(no 2) ({268} blocked by r4c12:step 10)
20a. no 7 r3c5

21. 21(3)n2 = {489/579/679}
21a. 31(7)n25 = 12347{59/68}
21b. 2 required in 31(7) in n2 only: 2 locked n2
21c. no 7 r4c5

22. 9(2)c5 = {18/36} = [1/6..]

23. "45" c5 -> r12c5 = 7 = [43/52] ([61] blocked by 9(2)c5 step 22)

24. 21(3) n2 must have 4 or 5 but not both = {489/579} (no 6) = 9{48/57}
24a. -> r1c46 = {789}
24b. 9 locked in r1c46 n2,r1
24c. no 6 r2c7

25. 6 in r1 now in n3:locked for n3

26. 12(4)n3 = {1245} (no 3)
26a. no 9 r3c9 (step 15)
[missed these next two but do them now if you like
26b. r2c8 = 5 (single 12(4)
26c. r1c5 = 5 (single n2)]

27. 20(3)n36 = {578}
27a. -> no 7 or 8 in r56c9
[if you did 26bc then 27c.r4c9 = 5 (single 20(3))]

28. 6 in r4 only in n5:locked for n5
28a. (from step 20) h16(3)n5 must have 6 = 6{19/37} (no 8)
28b. no 1 r3c5

29. 1 in n2 only in 31(7) -> no 1 r4c46

30. should have done this step earlier
31(7)n25 - the {1234579} combo can only have {45} in n2: blocked by r1c5
30a. -> 31(7) = {1234678} (no 5,9}
30b. r1c5 = 5 (hidden single n2)
30c. r2c5 = 2 (step 23)

Now things go crazy
31. r12c2 = [27], r1c46 = {79}:locked for r1,n2, r12c7 = [69]

32. from step 28a. r4c456 = {367} only:locked for r4,n5

33. r4c89 = {58}:locked for r4,n6
33a. r3c9 = 7, r4c7 1 (single r4)

34. r4c12 = {29}:locked for n4

35. r3c78 = {24}:locked for r3,n3
35a. r2c8 = 5

36. 9(2)c5 = {36}:locked for c5

37. 13(2)c5 = {49} locked for c5,n8

38. r56c5 = {18} locked for c5,n5
38a. r7c5 = 7

39. 17(3)n4 = {368}:locked for n4

40. r56c3 = {15}:locked for c3

41. 3 in c3 in n7:3 locked for n7

42. 14(3)n45 = {149}
42a. r5c3 = 1
42b. r56c4 = {49}:locked for c4,n5

the rest goes on.

[Andrew]

Some neat moves there Ed! You solved the key area of N1 and R4C3 more directly than I did and also made use of some hidden cages that I never used. I saw the 16(3) one in R4C456 but didn't make any use of it as 16(3) isn't normally a particularly useful combination unless one knows that one cell has a small number. Bringing it into your walkthrough after 4,5 had already been eliminated from it certainly made it more useful. Don't think I even spotted the 14(3) one in R4C789.

A bit late but here's my walkthrough. I didn't get to solve the key area until a long way in because I was using more methodical moves first before I started looking for chains. However I did get that naked quad in C2 fairly early but it wasn't immediately as powerful as in Ed's comment because I hadn't yet solved the key area. There are more comments about Ed's moves after my steps 12 and 36.


Clean-up is used in various steps, using the combinations in steps 1 to 8 for further eliminations from these two cell cages; it is also used for the two cell split sub-cages that are produced by applying the 45 rule. In some of the later steps, clean-up is followed by further moves and sometimes more clean-up.

1. R12C3 = {18/27/36/45}, no 9

2. R12C7 = {69/78}

3. R34C5 = {18/27/36/45}, no 9

4. R78C4 = {17/26/35}, no 4,8,9

5. R78C6 = {18/27/36/45}, no 9

6. R89C5 = {49/58/67}, no 1,2,3

7. R9C34 = 10(2), no 5

8. R9C67 = 10(2), no 5

9. 8(3) cage in N1 = 1{25/34}, 1 locked for N1, clean-up: no 8 in R12C3
9a. No 4,5 in R12C3 because {45} would clash with 8(3) cage
9b. Killer pair 2/3 in 8(3) cage and R12C3, locked for N1

10. R1C456 = {489/579/678}, no 1,2,3

11. 20(3) cage in N36, no 1,2

12. 27(4) cage in N14 = 9{378/468/567}, no 1,2
[I missed the clash with the 9(2) cage that Ed used to eliminate the {5679} combination and therefore 5. A neat one using the combination of the nonet and the column!]

13. 12(4) cage in N36 = 12{36/45}, no 7,8,9
[I should then have spotted that 12(3) cage in N3 cannot be {129} -> no 9]

14. 31(7) cage in N25 = 12347{59/68}

15. 34(6) cage in N47 = 9{13678/14578/23578/24568/34567}, contains two of 1,2,3,4 and three of 5,6,7,8

16. 38(6) cage in N69 = {356789}

17. 45 rule on N1 1 innie R3C1 – 1 = 1 outie R4C3 -> no 9 in R4C3
17a. 9 in N1 must be locked in 27(4) cage -> no 8 in R4C3, R3C1 = {45678}

18. 45 rule on N3 1 innie R3C9 – 6 = 1 outie R4C7 -> R3C9 = {789}, R4C7 = {123}

19. 45 rule on C1 4 outies R1459C2 = 11 = {1235}, locked for C2

20. 45 rule on C5 2 innies R12C5 = 7 -> R1C5 = {456}, R2C5 = {123}, R1C46 = {789} (step 10)

21. 16(3) cage in C5 cannot contain more than one of 1,2,3, R2C5 = {123} -> 9(2) cage in C5 must contain one of 1,2,3 -> no {45} in R34C5
21a. 16(3) cage in C5 must contain one of 1,2,3, valid combinations are {169/178/259/268/349/358/367}

22. 45 rule on C123 2 innies R59C3 = 9, no 9, clean-up: no 4 in R5C9, no 1 in R9C4

23. 45 rule on C789 2 innies R59C7 = 11, no 1, clean-up: no 6 in R5C7, no 9 in R9C6

24. 1 in C7 locked in R34C7, locked for 12(4) cage

25. 1 in N9 locked in 15(4) cage

26. 45 rule on C9 4 outies R1459C8 = 14, no 9

27. 9 in C8 locked in R678C8, locked for 38(6) cage

28. 16(3) cage in N14, valid combinations {169/178/259/268/349/358/367/457}, no 1,2 in R4C1

29. 17(3) cage in N4, valid combinations {179/269/278/359/368/458}, no 1,2 in R56C1

30. 15(4) cage in N7, valid combinations {1239/1248/1257/1347/1356/2346}, contains at least two of 1,2,3

31. 15(4) cage in N9 must contain 1 (step 25) and at least one of 2,4, valid combinations 1{239/248/257/347}, no 6, contains two of 2,3,4,5 and one of 7,8,9

32. Time to try contradiction moves. Looks like there may be interactions between R4C3 and R12C3 so try R4C4 = 3/6/7 (no 4/5 in R12C3).
If R4C3 = 3 => R2C2 + R3C23 = {789} => R12C3 = {36} clashes with R4C3 -> no 3 in R4C3
If R4C3 = 6 => R2C2 + R3C23 = {489} (cannot be {678}) => 8(3) cage = {125} => R12C3 = {36} clashes with R4C3 -> no 6 in R4C3
If R4C3 = 7 => R2C2 + R3C23 = {569} => 8(3) cage = {134} => R12C3 = {27} clashed with R4C3 -> no 7 in R4C3

33. R4C3 = {45} -> R2C2 + R3C23 = {6789} from combinations in step 12
33a. 8(3) cage = 1{25/34} [4/5] -> R3C1 = {45}
33b. Remaining valid combinations for 27(4) cage are {4689/5679} = 69{48/57}, 6 locked for N1, clean-up: no 3 in R12C3 = {27}, locked for C3 and N1, clean-up: no 3,8 in R9C4
33c. R2C2 + R3C23 = {689} -> R4C3 = 4, clean-up: no 5 in R5C3, no 6 in R9C4
33d. 8(3) cage in N1 = {134} with 4 locked for C1 -> R3C1 = 5
33e. R4C12 = 11 = [83/92], 2 in N4 locked in R45C2, locked for C2
[Alternatively after step 33a I could have used step 17 to give R3C1 = 5, R4C3 = 4 directly but I’d forgotten about that one. Poor short term memory as one gets older! Fortunately the remaining sub-steps above are almost as quick and make the other related eliminations.]

34. 2 in N7 locked in 15(4) cage, valid combinations {1239/1257} = 12{39/57}, no 6,8, 1 locked for N7, clean-up: no 8 in R5C3, no 9 in R9C4

35. 34(6) cage in N47 must contain 4 in R78C2, 7 in R678C2 and 5 in R678C3, remaining valid combinations 9{14578/34567} = 4579{18/36}

36. 1 in C3 locked in R56C3, locked for N4
36a. Remaining valid combinations for 17(3) cage in N4 are {269/278/359/368}
[I missed the killer pair 8/9 in R4C1 and the 17(3) cage. No real problem as I then got them in a naked triple in step 38b!]

37. 8 in C1 locked in R456C1, locked for N4

38. 6 in C1 locked in R56C1, locked for N4, clean-up: no 3 in R9C3, no 7 in R9C4
38a. 17(3) cage in N4 = 6{29/38}, no 3,7 in R56C1, no 5 in R5C2
38b. R456C1 = {689}, locked for C1 and N4 -> R6C2 = 7

39. R45C2 = {23}, locked for C2 and N4 -> R1C2 = 1, R12C1 = {34}, locked for C1, R9C2 = 5, R5C3 = 1, R6C3 = 5, clean-up: no 8 in R8C5
39a. R56C4 = 13 = {49}/[58]/[76]
39b. 34(6) cage in N47 = {345679}, no 8
39c. R9C3 = 8 (hidden single in N7), R9C4 = 2, clean-up: no 6 in R78C4, no 7 in R78C6, no 5 in R8C5

40. 9 in N8 locked in R789C5, locked for C5

41. R9C8 = 1 (hidden single in C8), clean-up: no 9 in R9C7

42. 2 in N9 locked in R78C9, locked for C9
42a. 15(4) cage in N9 = 12{39/48/57}, no 4,7 in R78C9

43. R9C1 = 7 (naked single), clean-up: no 3 in R9C67 = {46}, locked for R9, R9C5 = 9, R8C5 = 4, R9C9 = 3, R9C67 = [64], clean-up: no 3 in R2C5, no 3,5 in R78C6
[I should also have included R5C7 = 7 in the clean-up after fixing R9C7 = 4. Just spotted that while checking the walkthrough before posting it.]
43a. R78C9 = {29} (step 42a), locked for C9 and N9 -> R6C8 = 9 (hidden single in C8), clean-up: no 4 in R5C4
43b. R78C6 = {18}, locked for C6 and N8, clean-up: no 7 in R78C4 = {35}, locked for C4 and N8, clean-up: no 8 in R6C4
43c. R7C5 = 7 (naked single), R56C5 = 9 = {36}/[81], clean-up: no 2 in R34C5

44. R1C5 = 5, R2C5 = 2 (hidden singles in C5)

45. R7C2 = 4 (hidden single in N7)

46. 3 in 31(7) cage (step 14) locked in R234C6, locked for C6

47. R1C5 = 5 -> R1C46 = 16 = {79}, locked for R1 and N2 -> R12C3 = [27], clean-up: no 6,8 in R2C7 -> R2C7 = 9, R1C7 = 6
47a. R23C6 = {34}, locked for C6 and N2, clean-up: no 6 in R4C5

48. 8 in R1 locked in R1C89, locked for N3 -> R1C9 = 8, R1C8 = 3, R2C9 = 1 (only possible combination)
48a. R3C9 = 7, R4C89 = 13 = [85] (only possible combination)

49. R3C7 = 2, R3C8 = 4, R2C8 = 5 (naked singles) -> R4C7 = 1, clean-up: no 8 in R3C5

50. R3C6 = 3 (naked single), R2C6 = 4, R12C1 = [43]
50a. No 5 in 31(7) cage -> no 9 (step 14) -> {1234678} -> R4C6 = 7, R4C4 = 6, R2C4 = 8, R3C4 = 1

and the rest is naked singles, simple elimination and cage sums