Assassin 45

[frank]

Tough one to get started.

My approach was to work with the hidden 21/3 cage in C8N6
and the 22/3 cage in N69. The hidden 21/3 cage cannot be
{678} because of the 6/2 cage, so must contain a 9. The 22/3
cage must contain a 9. This can't be R6C9. To show it can't
be R7C9 compute outies of N9 = 36, and find a contradiction
for the 14/3 cage. So R6C8=9. Thereafter it all falls out.
Be interested to see if there is a cleaner start than this.

[Para]

Hi all

Here's my walk-through. It looked to get stuck very shortly but then it was mostly big leaps till the end.

Frank, i think the step you are looking for is my steps 13 and 17. Step 13 is basically your first step. Step 17 is another way (not a contradiction move) of coming to your conclusion.

Walk-Through Assassin 45

1. R12C1 and R89C1 = {18/27/36/45} : no 9
2. 11(3) in R1C8 and R3C6 = {128/137/146/236/245}: no 9
3. R12C9, R5C12 and R89C9 = {29/38/47/56}: no 1
4. R5C89 = {15/24}: no 3,6,7,8 or 9
5. 22(3) in R6C8 = {589/679}:no 1,2,3 or 4; 9 locked in 22(3) cage -->> R4C9: no 9
6. 21(3) in R7C2 and R8C7 = {489/579/678}: no 1,2 or 3
7. 17(5) in R7C5 = {12347/12356}: no 8,9; 1,2 and 3 locked in 17(5) cage for N8
8. 10(3) in R8C3 = {127/136/145/235}: no 8,9
9. 45 test on N5: 2 outies: R5C37 = 13 = {49/58/67}: no 1,2 or 3
10. 45 test on C34: 2 innies: R28C4 = 5 = {14/23}
11. 45 test on C67: 2 innies: R28C6 = 10 = {37/46}/[82]/[91] -->> R2C6 = {346789}; R8C6 = {123467}
12. 45 test on R89: 3 outies: R7C258 = 12: min R7C2 = 4: max R7C58 = 8 -->> R7C8: no 8,9
13. 45 test on C9: 3 outies: R456C8 = 21 = {8[4]9}/{7[5]9} -->> R46C8 = {789}; R5C8 = {45}; 9 locked in R46C8 for C8 and N6
13a. Clean up: R5C9: no 4, 5; R5C3: no 4 (step 9)
13b. 11(2) in R5C1: no {56} clashes with R5C37 + R5C8: need at least one of 5 or 6
13c. R5C37: no {58}: R5C89 = {15}: no {58} or R5C89 = {24} -->> R5C12 = {38}: no {58}
14. 45 test C1: 3 outies: R456C2 = 11 = {128/137/146/236/245}: no 9
14a. Clean up: R5C1: no 2
15. 45 test R1234: 3 innies: R4C456 = 19 = {289/379/469/478}: no 1
16. 45 test R6789: 3 innies: R6C456 = 11 = {128/137/146/236/245}: no 9
17. 45 test on N9: 1 outie = 2 innies: R9C6 = R7C79 -->> min R7C79 = 6, so min R9C6 = 6; max R9C6 = 9, so max R7C79 = 9 -->> R9C6 = {6789}; R7C7 = {1234}; R7C9 = {5678}
17a. R6C8 = 9 (only 9 in 22(3) cage in R6C8)
17b. R67C9 = {58/67} -->> R12C9 and R89C9: no {56}
18. 16(3) in R3C9: no {169/259/349} because it needs one of {78} in R4C8: no 9
18a. 9 in C9 locked in 11(2) cages in R1C9 and R8C9: one must be {29} -->> 2 locked in C9 in R1289C9
18b. R5C89 = [51]
18c. Clean up: R7C9: no 8(step 17b)
18d. 16(3) in R3C9 = {358/367/457}: 7 or 8 in R4C8, so R34C9 no 7,8; only place for 5 is R3C9, so R3C9 no 4
19. 1 in N5 locked in R6
19a. 1 locked in R6C456 -->> R6C456 = 11 = {128/137/146}: no 5
19b. 5 in N5 locked in R4
19c. 5 locked in R4C456 -->> R4C456 = 19 = {568}: locked for R4 and N5
19d. R4C8 = 7
19e. Clean up: R6C456: no 2,4; R5C3: no 6; R3C9: no 3; R7C9: no 6
20. Naked triple: R6C456 = {137}-->> locked for R6 and N5
20a. Naked triple: R5C456 = {249} -->> locked for R5
20b. R5C3 = 7; R5C7 = 6; R6C9 = 8; R7C9 = 5; R34C9 = [63]
21. 17(4) in R4C6 = {128}6: no {137}: needs one of {58} in R4C6, no {245}: needs one of {137} in R6C6; -->> R4C6 = 8; R5C6 = 2; R6C6 = 1
21a. R6C4 = 3; R6C5 = 7
21b. 16(3) in R4C5 = [54]7
21c. R45C4 = [69]
21d. Clean up: R28C4 = {14}(step 10) -->> locked for C4
22. R9C4 = 5 (hidden)
22a. 17(5) in R7C5 = {12347}: no 6, locked for N8
22b. R8C6 = 7; R8C4 = 4; R7C4 = 8 (hidden singles); R2C4 = 1
22c. Naked triple {123} in R789C5 -->> locked for C5
22d. Naked pair {69} in R79C6 -->> locked for C6
22e. Naked pair {24} in R46C7 -->> locked for C7
23. 21(3) in R8C7 = [867] (R8C7 = 8, R9C67 = [67]
23a. R7C6 = 9
24. 13(3) in R7C8 = {346}-->> locked for C8 and N9
24a. R7C7 = 1; R6C7 = 4; R4C7 = 2
24b. 11(3) in R3C6 = [45] (only possible combination)
24c. R2C6 = 3; R1C6 = 5; R12C7 = [39]
24d. R12C9 = {47}
25. 17(3) in R3C3 = {19[7]}: no {278} needs 1,4 or 9 in R4C3
25a. R3C4 = 7; R34C3 = {19} -->> locked in C3
25b. R1C4 = 2
26. 15(3) in R1C3 = [85]2 -->> R12C3 = [85]
26a. R1C8 = 1
27. 9(2) in R12C1 = [72]
27a. R12C9 = [47]; R23C8 = [82]; R123C5 = [968]
27b. R123C2 = [643]

And from here on all singles and cage sums (ok to be honest it was singles and cage sums the last two steps as well).

greetings

Para

p.s. Now onto our Easter presents. Ruud's such a generous guy.

[Andrew]

Nice walkthrough Para. Your steps 13b and 13c were neat although there would be no need for contradiction moves if step 17 had been a bit earlier. Step 18a was another neat one. C9 was very useful in solving this Assassin, your step 18a was the one point about C9 that I didn't spot when I solved it.

Here is my walkthrough. I was pleased to be able to use the "odd number trick" in step 41, something I've only done once or twice before on this forum; don't think I've seen anyone else use it.

1. R12C1 = {18/27/36/45}, no 9

2. R12C9 = {29/38/47/56}, no 1

3. R5C12 = {29/38/47/56}, no 1

4. R5C89 = {15/24}

5. R89C1 = {18/27/36/45}, no 9

6. R89C9 = {29/38/47/56}, no 1

7. R123C8 = {128/137/146/236/245}, no 9

8. 11(3) cage in N256 = {128/137/146/236/245}, no 9

9. 22(3) cage in N69 = 9{58/67}, no 9 in R4C9

10. R789C2 = {489/579/678}, no 1,2,3

11. 10(3) cage in N78 = {127/136/145/235}, no 8,9

12. 21(3) cage in N89 = {489/579/678}, no 1,2,3

13. 17(5) cage in N8 = 123{47/56}, no 8,9, 1,2,3 locked for N8

14. 45 rule on N5 2 outies R5C37 = 13 = {49/58/67}, no 1,2,3

15. 45 rule on C2 3 innies R456C2 = 11 = {128/137/146/236/245}, no 9, clean-up: no 2 in R5C1

16. 45 rule on C8 3 innies R456C8 = 21 = {489/579/678}, no 1,2,3
16a. R5C8 = {45} -> R456C8 = {489/579} = 9{48/57}, 9 locked in R46C8 for C8 and N6, no 4,5,6 in R46C8
16b. Clean-up: no 4 in R5C3, no 4,5 in R5C9

17. 45 rule on N9 2 innies R7C79 = 1 outie R9C6, min R7C79 = 6 -> min R9C6 = 6, max R7C79 = 9 -> max R7C7 = 4, max R7C9 = 8

18. R6C8 = 9 (only remaining 9 in 22(3) cage), R67C9 = {58/67} [5/6, 7/8]
18a. R12C9 = {29/38/47} (cannot be {56} which clashes with R67C9) [2/3/4, 7/8/9]
18b. R89C9 = {29/38/47} (cannot be {56} which clashes with R67C9) [2/3/4, 7/8/9]
18c. Killer triple 7/8/9 in R12C9, R67C9 and R89C9 for C9

19. 16(3) cage in N36 = 7{36}/7{45}/8{26}/8{35}, no 1

20. R5C9 = 1 (hidden single in C9), R5C8 = 5, clean-up: no 4 in R3C9 (step 19), no 6 in R5C12, no 8 in R5C37, no 8 in R7C9

21. 45 rule on C1234 2 innies R28C4 = 5 = {14/23}

22. 45 rule on C6789 2 innies R28C6 = 10 = {19/28/37/46}, no 5, no 1,2 in R2C6

23. 45 rule on R89 3 outies R7C258 = 12, min R7C2 = 4 -> max R7C58 = 8, no 8

24. 45 rule on R1234 3 innies R4C456 = 19 = {289/379/469/478/568}, no 1
24a. 45 rule on R6789 3 innies R6C456 = 11
24b. 45 rule on N5 R5C456 = 15

25. 1 in R4 locked in R4C123, locked for N4
25a. R6C456 = 1{28/37/46}, no 5
25b. 5 in R6 locked in R6C123, locked for N4

26. 5 in N5 locked in R4C456 = {568} (only remaining combination), locked for R4 and N5, clean-up: no 2,4 in R6C456 (step 25a)

27. R4C8 = 7 (naked single) -> R34C9 = [54/63], clean-up: no 6 in R5C3, no 6 in R7C9

28. 9 in R4 locked in R4C13, locked for N4 -> R5C3 = 7, clean-up: no 4 in R5C12, no 2 in R5C2, R5C7 = 6 (step 14) -> R67C9 = [85] (step 18), R3C9 = 6 -> R4C9 = 3

29. R5C12 = {38}, locked for R5 and N4

30. R46C7 = {24}, locked for C7

31. R123C8 = {128} (only remaining combination), locked for C8 and N3, clean-up: no 9 in R12C9
31a. R12C9 = {47}, locked for C9 and N3
31b. R89C9 = {29}, locked for N9
31c. R789C8 = {346}(only remaining combination), locked for N9

32. R7C7 = 1 (hidden single in N9)
32a . R89C7 = {78} -> R9C6 = 6, clean-up: no 4 in R28C6, no 3 in R8C1Steps renumbered and combined. I found that I had two step 31s

33. 14(3) cage in N689 = [491] (only remaining combination) -> R6C7 = 4, R7C6 = 9, R4C7 = 2, clean-up: no 1 in R8C6

34. 17(5) cage in N8 (step 13) = {12347}, no 5, locked for N8 -> R79C4 = [85], R4C4 = 6, clean-up: no 4 in R8C1

35. 45 rule on N7 2 innies R7C13 – 5 = 1 outie R9C4, R9C4 = 5 -> R7C13 = 10 = {46}/[73], no 2, no 3 in R7C1

36. R9C4 = 5 -> R89C3 = 5 = {14/23}, no 6 [Typo corrected. Thanks Ed]

37. 9 in N7 locked in R89C2, locked for C2
37a. R789C2 = 9{48/57} [4/5], no 6, no 4,7 in R89C2

38. R89C1 = {18/27}/[63] (cannot be [54] which clashes with R789C2)

39. R456C2 (step 15) = [182] (only remaining combination) -> R5C1 = 3, clean-up: no 6 in R12C1, no 6 in R8C1

40. R9C2 = 9, R8C2 = 5, R89C9 = [92] (naked singles) -> R7C2 = 7, clean-up: no 2 in R8C1, no 3 in R8C3

41. R6C1 = 5 (only remaining odd number in the 13(3) cage in N47). There’s an idea for you Ruud! -> R7C1 = 6, clean-up: no 4 in R12C1

[Richard pointed out that R67C1 = [56] was the only remaining combination for this cage. Quite correct. However it was the remaining odd number that I spotted]
BTW I’ve now discovered that step 15 of my Assassin 41 walkthrough was a Swordfish and have edited that. Another idea for you Ruud!

42. R6C3 = 6 (naked single) -> R7C3 = 4, R7C8 = 3, R89C8 = [64], R7C5 = 2, R4C3 = 9, R4C1 = 4 (naked singles) -> R3C1 = 9, clean-up: no 1 in R89C3 = [23], no 3 in R2C4, no 8 in R2C6

43. R89C1 = {18}, locked for C1 -> R12C1 = {27}
[I first saw these as a naked quad in R1289C1 but hadn’t done all the eliminations in step 42 at that stage.]

44. R28C6 = {37}, locked for C6 -> R6C6 = 1, R6C45 = [37], R9C5 = 1, R8C456 = [437], R2C6 = 3, R89C1 = [18], R89C7 = [87] (naked singles), clean-up: R2C4 = 1

45. R4C3 = 9 -> R3C34 = 8 = [17]

46. R12C3 = {58}, locked for N1 -> R1C4 = 2

and the rest is naked singles although probably quicker if you use naked pairs also

[PsyMar]

My main strategy involved conflicting combinations between size-3 hidden cages in the central nonet for most of the grunge work.

[Andrew]

Use of Odd Numbers

In step 41 of my walkthrough I used the fact that there was only one remaining odd number candidate to fix that cell because the cage total was odd. BTW Richard has pointed out that it wasn't necessary there, since it was only remaining combination, but it was what I spotted and it may be a useful technique in future.

I've just realised that there is a corollary to this. If there is only one remaining odd number candidate in a cage and the cage total is even, that odd number candidate can be eliminated. This isn't as powerful as for the odd total cage but could still be a useful technique.

One can extend the original thought. If there is a cage with even total and one cell only contains odd numbers while there is only one remaining odd number in the other cells, then that remaining odd number can be fixed.

[rcbroughton]

Use of Odd Numbers

In step 41 of my walkthrough I used the fact that there was only one remaining odd number candidate to fix that cell because the cage total was odd. BTW Richard has pointed out that it wasn't necessary there, since it was only remaining combination, but it was what I spotted and it may be a useful technique in future.

I've just realised that there is a corollary to this. If there is only one remaining odd number candidate in a cage and the cage total is even, that odd number candidate can be eliminated. This isn't as powerful as for the odd total cage but could still be a useful technique.

One can extend the original thought. If there is a cage with even total and one cell only contains odd numbers while there is only one remaining odd number in the other cells, then that remaining odd number can be fixed.

I think there are a number of applications of this - all of these are a simplification of looking at valid combinations for a cage that might be easier to spot:

Using a convetion that {E} is a cell containing even candidates only, {O} is a cell containing odd candidates only and {OE} is a cell containing odd and even candidates, I can see the following possible cases and deductions, building on your original idea and corollaries:

Cage sum is Odd in an odd number of cells
1. {E}{OE} Remainder{OE} - no deduction
2. {O}{OE} Remainder{OE} - no deduction
3. {E}{OE} Remainder{E} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell (this is the elimination you used at step 41 - and as there was only one odd digit, the elimination placed it)
4. {O}{OE} Remainder{O} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell
5. {O}{OE} Remainder{E} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell
6. {E}{OE} Remainder{O} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell

Cage sum is Odd in an even number of cells
1. {E}{OE} Remainder{OE} - no deduction
2. {O}{OE} Remainder{OE} - no deduction
3. {E}{OE} Remainder{E} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell
4. {O}{OE} Remainder{O} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell
5. {O}{OE} Remainder{E} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell
6. {E}{OE} Remainder{O} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell

Cage sum is Even in an odd number of cells
1. {E}{OE} Remainder{OE} - no deduction
2. {O}{OE} Remainder{OE} - no deduction
3. {E}{OE} Remainder{E} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell
4. {O}{OE} Remainder{O} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell
5. {O}{OE} Remainder{E} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell
6. {E}{OE} Remainder{O} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell

Cage sum is Even in an even number of cells
1. {E}{OE} Remainder{OE} - no deduction
2. {O}{OE} Remainder{OE} - no deduction
3. {E}{OE} Remainder{E} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell
4. {O}{OE} Remainder{O} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell
5. {O}{OE} Remainder{E} - {OE} cell must be {O} all {E} can be eliminated from the {OE} cell
6. {E}{OE} Remainder{O} - {OE} cell must be {E} all {O} can be eliminated from the {OE} cell

Rgds
Richard