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PostPosted: Tue Jun 17, 2008 5:33 pm 
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Grand Master
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Joined: Thu Apr 24, 2008 4:27 pm
Posts: 791
Quick!! The Sudocue forums are back up! If you haven't already, I suggest you grab the V2 Assassins (and anything else you want from the forums) now!


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PostPosted: Wed Jun 18, 2008 11:43 pm 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Old SSv3.2 scores:
Killer rating table      
Rounded Score from SSv3.2
pg# on this thread
(E) = Easy (H) = Hard
======================================================================
|A ## Rate Score|A ## Rate Score|A ## Rate Score|
|----------------------+----------------------+----------------------|
|A.45 1.00|A.46Lite 0.95|A.48Hevvie 2.50 3.75|
|A.45EEggsv2 1.75 1.90|A.47 1.20| |
|A.45EEggsv3 1.65|A.47v1.5 0.85| |
|A.46 1.10|A.48 1.05| |
|====================================================================|
page #7
Old scores SSv3.3.30:
Killer rating table      
Rounded Score from SSv3.3.0
! = 0.10 change from previous version of score
pg# on this thread
(E) = Easy (H) = Hard
======================================================================
|A ## Rate Score|A ## Rate Score|A ## Rate Score|
|----------------------+----------------------+----------------------|
|A.45 0.95|A.46Lite 0.95|A.48Hevvie 2.50 !2.95|
|A.45EEggsv2 1.75 !2.05|A.47 !0.90| |
|A.45EEggsv3 H1.25!1.95|A.47v1.5 0.85| |
|A.46 1.25 !1.20|A.48 !1.20| |
|====================================================================|
page #7
Killer rating table
SudokuSolver Target range v3.6.3
Rating.....Score
0.50 = 0.85
0.75 = 0.90-0.95
1.00 = 1.00-1.20
1.25 = 1.25-1.45
1.50 = 1.50-1.70 (E) = Easy (H) = Hard
===========================================================================================
|A ## by Rate Score|A ## by Rate Score|A ## by Rate Score|
|-----------------------------+-----------------------------+-----------------------------|
|A.45 Ruud 1.15|A46Lite Ruud 1.05|A48Hev Ruud 2.50 3.75|
|EEggsv2 Ruud 1.75 1.90|A.47 Ruud 0.95| |
|EEggsv3 Ruud H1.25 1.75|A.47v15 Ruud 1.00| |
|A.46 Ruud 1.25 1.20|A.48 Ruud 1.15| |
|=========================================================================================|
page #7
Assassin 45 by Ruud (April 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2304:3329:3842:3842:6916:4357:4357:2823:2824:2304:3329:3842:6916:6916:6916:4357:2823:2824:3602:3329:4372:4372:6916:2839:2839:2823:4122:3602:3602:4372:6430:4127:4384:2839:4122:4122:2852:2852:6430:6430:4127:4384:4384:1579:1579:3373:3373:4655:6430:4127:4384:3635:5684:5684:3373:5431:4655:4655:4410:3635:3635:3389:5684:2367:5431:2625:4410:4410:4410:5445:3389:2887:2367:5431:2625:2625:4410:5445:5445:3389:2887:
Solution:
+-------+-------+-------+
| 7 6 8 | 2 9 5 | 3 1 4 |
| 2 4 5 | 1 6 3 | 9 8 7 |
| 9 3 1 | 7 8 4 | 5 2 6 |
+-------+-------+-------+
| 4 1 9 | 6 5 8 | 2 7 3 |
| 3 8 7 | 9 4 2 | 6 5 1 |
| 5 2 6 | 3 7 1 | 4 9 8 |
+-------+-------+-------+
| 6 7 4 | 8 2 9 | 1 3 5 |
| 1 5 2 | 4 3 7 | 8 6 9 |
| 8 9 3 | 5 1 6 | 7 4 2 |
+-------+-------+-------+
Quote:
frank: Tough one to get started...contradiction...Be interested to see if there is a cleaner start than this
Para: Frank, i think the step you are looking for is my steps 13 and 17. It looked to get stuck very shortly but then it was mostly big leaps till the end
Andrew: 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.
Walkthrough by 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.
Walkthrough by Andrew including odd number trick:
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
Discussion/application of odd numbers by Andrew & rcbroughton:
Andrew wrote:
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


Last edited by Ed on Sat Dec 27, 2008 10:28 pm, edited 1 time in total.

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PostPosted: Wed Jun 18, 2008 11:45 pm 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 45v2 (aka Easter Eggs V2) by Ruud (April 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2304:4097:4354:4354:6916:4101:4101:2567:4360:2304:4097:4354:6916:6916:6916:4101:2567:4360:4882:4097:2068:2068:6916:3607:3607:2567:4122:4882:4882:2068:5918:2335:6176:3607:4122:4122:2852:2852:5918:5918:2335:6176:6176:2091:2091:2605:2605:3375:5918:2335:6176:2355:4404:4404:2605:5175:3375:3375:9018:2355:2355:3901:4404:1343:5175:3649:9018:9018:9018:4165:3901:1863:1343:5175:3649:3649:9018:4165:4165:3901:1863:
Solution:
+-------+-------+-------+
| 3 4 2 | 6 8 5 | 7 1 9 |
| 6 5 9 | 7 1 2 | 4 3 8 |
| 8 7 1 | 3 9 4 | 2 6 5 |
+-------+-------+-------+
| 5 6 4 | 1 3 7 | 8 9 2 |
| 9 2 8 | 5 4 6 | 3 7 1 |
| 7 1 3 | 9 2 8 | 5 4 6 |
+-------+-------+-------+
| 2 9 6 | 4 5 3 | 1 8 7 |
| 1 3 5 | 8 7 9 | 6 2 4 |
| 4 8 7 | 2 6 1 | 9 5 3 |
+-------+-------+-------+
Quote:
sudokued: V2 looks like a real toughie. .. help needed now
Tag solution: sudokuEd, mhparker & PsyMar
Andrew in 2012: This was a hard variant, so I'm not surprised that there was a "tag" to solve it. I've only tried it now. There are differences in our solving paths but the key breakthrough areas are the same. I'll rate my walkthrough at 1.75.
Forum 2021 Revisit here
Tag Walkthrough:
Thanks for the challenge Ruud. V2 looks like a real toughie. Found lots of little ones that were very tasty and interesting. But help needed now.

Assassin 45V2
1. 17(2)n3 = {89}: locked for n3, c9

2. 9(3)n6 = {126/135/234}(no 789}

3. "45" c9: 3 outies r456c8 = 20 = h20(3) = {479/569/578}(no 123)
3a. {389} blocked since 8 or 9 must be in r45c7 for c7 since the only other place for 8 & 9 is r89c7, but a 16(3) cage cannot have both of 8 & 9

4. 8(2)n6 = [53/62/71]

5. 10(3)n3 = {127/136/145/235}

6. h20(3)n6 = [8/9], not both ->
6a. 15(3)n9 must have [8/9] = {159/168/249/258/348}(no 7)

7. 7(2)n9 = {16/25/34}(no 789}

8. r89c7 must have [8/9] (only other place in n9 besides 15(3))
8a. 16(3)n9 = {169/178/259/268/349/358}

9. 15(3)n9 must have only 1 of 1/2/3 (step 6a)
9a. -> 10(3)n3 must have 2 of 1/2/3
9b. -> 10(3)n3 = {127/136/235}(no 4) = [1/5...]
9c. {159} blocked from 15(3)n9 = {168/249/258/348}

10. 35(5)n8 = {56789}:locked n8
10a. max. r9c6 = 4 -> min. r89c7 = 12 -> 3 minimum in each cell
10b. no 1 or 2 r89c7

11. "45" r12: r3c258 = 22 = h22(3)r3 = {89}[5]/{679}(no 1234, no 5 r3c25)
11a. = 9{58/67}: 9 locked r3
11b. 10(3)n3: no 5,6,7 r12c8

12. 11(2)n4 (no 1)

13. "45"n5: r5c37 = 11 = h11(2)(no 1)

14. "45" r1234: r4c456 = 11 = h11(3) (no 9)

15. "45" r6789: r6c456 = 19 = h19(3) (no 1)

16. 9(3)n5 = {126/135/234}(no 789)

17. "45" r5: r5c456 = 15 = h15(3)r5
17a. {159} with 1 only in r5c5 (5 in r5c5 means 9(3) = {135}: 2 1's n5)
17b. {168} with 1 only in r5c5 (6 in r5c5 means 9(3) = {126}: 2 1's n5)
17c. {249} with 2 only in r5c5 (4 in r5c5 means 9(3) = {234}: 2 2's n5)
17d. {258}: blocked by 11(2) & h11(2)r5
17e. {267}: Blocked by 11(2) & h11(2)r5
17f. {348} with 3 only in r5c5 (4 in r5c5 means 9(3) = {234}: 2 3's n5)
17g. {357}: Blocked by 11(2) & h11(2)r5
17h. {456}

18. In summary: h15(3)r5 = {159/168/249/348/456}(no 7)
18a. r5c46: no 1,2,3

19.20(3)n7 = {389/479/569/578}(no 1,2)

20. 5(2)n7 = {14/23}

21. 9(2)n1 = {18/27/36/45}

22. 16(3)n1 = {169/178/259/268/349/358/457} ({367} blocked by 20(3)n7)

23. 19(3)n1: no 1

24. 10(3)n4 = {127/136/145/235}

25. "45"c1: r456c2 = 9 = h9(3) = {126/135/234}(no 7,8,9)
25a. no 234 r5c1
25b. 1 in h9(3) only in r6c2 -> no 5 or 6 r6c2

26."45" c34:r28c4 = 15 = h15(2) = {69/78}

27. 23(4) n5 has to fit in with this h15(2) & 9(3)n5
27a. {1589}: 8/9 must be in r5c3
27b. {1679}: 7 must be in r5c3
27c. {2489}: 8/9 must be in r5c3
27d. {2579}: blocked: 7/9 must be in r5c3 but {25}n5 clash with 9(3)
27e. {2678}: 6 must be in r5c3
27f. {3479}: 7/9 must be in r5c3
27g. {3569}: 3/6 must be in r5c3: (9(3) = [3/6])
27h. {3578}: 3/5/7 in r5c3
27i. {4568}: 6/8 must be in r5c3

28. Conclusion: no 2 or 4 r5c3
28a. no 9 or 7 r5c7 (h11(2)r5)

29. "45" c67: r28c6 = 11 (no 1)
29a. r2c6 = 2..6

30. "45" r89: r7c258 = 22 = h22(3)r7 = {589/679}(no 1..4)
30a. 9 locked r7

31. "45"n9: r9c6 + 7 = r7c79 = 8..11
31a. max. r7c7 = 6 -> min r7c9 = 2

mhparker
32. Innie/outie difference, c1: r345c1 - r6c2 = 21 -> no {1234} in r34c1, no 4 in r6c2

33. Innie/outie difference, r123: r3c1349 - r4c7 = 9 -> no 1 in r4c7
(reason: from step 32, r3c1 >= 5 -> r3c1349 >= 11)

34. Only 2 possible locations for digit 7 in n9:
(a) in r7c9 -> r789c7 = 16 (in total) -> r7c7 = {1234} (since r89c7 >= 12) -> no 1 in r6c7
(b) within 16/3 cage -> 16/3 = {178} -> r7c7 + r7c9 = 8 -> no 1 in r6c7
(reason: would imply r7c6 + r7c7 = 8, thus - whatever value r7c7 would take - r7c6 would clash with r7c9)
Summary: no 1 in r6c7

35. 1 in n6 now locked in c9 -> no 1 in r3c9; 7/2 cage at r89c9 = {25|34}

36. No 9 now possible in 15/3 cage in n9
({159} excluded due to 10/3(n3) cage, {249} excluded due to 7/2(n9) cage)
-> 8 in n9 locked in 15/3 cage (since we know it must contain one of {89}) -> 9 in n9 locked in r89c7 -> no 7 in r89c7 -> hidden single(n9) r7c9 = 7

Ed
Great work Mike. Love those sort of sneaky contradictions.

Here's some more: a bit greedy, but have V3 yet! That's me for the day. :D

37. r6c89 = [91]/{46}

38. 9 in c8 now in n6: no 9 r4c7
38a. h20(3) = {479/569}
38b. no 7 r4c8

39. 16(3)r3c9 = {169/259/349}
39a. r4c8 = 9
39b. r56c8 = [74/56]
39c. no 2 r5c9

40. r6c89 = {46}: locked for n6, r6
40a. no 7, 5 r5c3 (h11(2)r5)

41. "45" n9: r9c6 = r7c7 = {1234}
41a. 9(3)n6 = {135/234} = 3{15/24}

42. h22(3)r7 = {589}:Locked for r7

43. 6 r7 only in n7: locked for n7
43a. 20(3) = {389/479/578}

44. h19(3)r6 = {289/379} = 9{28/37}(no 5)
44a. 9 locked n5, r6

45. 2 6's n7
45a. 6 r7c1 -> r6c12 = {13} (cage combo) -> hidden triple {235} r6c1257
45b 6 r7c3 -> r6c3 = 3/5 (cage combinations 13(3)) -> naked triple {235} r6c357
45c. hidden triple {235} in r6c12357 for r6
45d. and no 2 r6c123

46. 19(3)n1 = {478/568} = 8{47/56}(no 2,3)
46a. 8 locked c1
46b. no 3 r5c2
46c. no 1 r12c1

47. h9(3) n4 = [621/423] ({135} blocked by 1 and 3 only in r6c2
47a. r5c2 = 2, r5c1 = 9
47b. r4c2 = {46}

48. h11(2)r5 = {38}/[65] = [3/5..]
48a. [53] blocked from 8(2)n6
48b. r5c89 = [71]

49. r6c8 = 4 (h20(3)n6), r6c9 = 6

50. 10(3)n3 = 3{16/25}
50a. 3 locked n3, c3

51. 1 in n5 locked in h11(3) = 1{28/37/46}(no 5)
51a. 1 locked r4

52. 8(3)n1 = 1{25/34}
52a. 1 locked r3
52b. no 5 r3c34

53. 5 in n5 only in r5 in h15(3)n5 = {456}: locked r5, n5


PsyMar
54. combinations for 19/3 in r34 -> no 6 in r34c1
55. combinations for h22/3 in r3c258 -> no 6 in r3c25
56. r4c2 = hidden single 6
57. outies of c1 = r6c2 = 1
58. 19/3 in c12 = {568} -> 58 pair in c1, elim from rest of c1
59. 9/2 in n1 = {27|36}
60. r4c3 = hidden single 4
61. 10/3 in c12 = [316|712]
62. 4 of c1 locked in 5/2 in n7 -> 5/2 in n7 = [14], elim from rest of n7
63. combinations for 8/3 in r34 = {134} -> r3c34 = {13} pair -> elim from rest of r3
64. 7 of r4 locked in n5 -> elim from rest of n5
65. r6c46 = naked pair (89) -> elim from rest of n5/r6
66. innies of r6789 = r6c456 = 19/3 -> r6c5 = 2
67. 20/3 in n7 = {389|578} = {8...} -> elim 8 from rest of n7 and c2
68. combinations for 14/3 in r89 = {239|257|347} -> no 1 in r9c4 (cannot be 149 as only 1 and only 4 both in r9c4)
69. r4c456 = naked triple {137} -> elim from rest of r4
70. combinations for 16/3 in c89 = {259} -> {25} pair in r34c9 -> elim from rest of c9
71. r89c9 = naked pair {34} -> elim from rest of n9
72. r9c1469 = naked quad {1234} -> elim from rest of r9
73. 2 of r9 locked in n8 -> elim from rest of n8

Edit: Here's the rest.
74. combinations for 14/3 in r34 = {248|257} -> no 6
75. r3c8 = hidden single 6
76. r12c8 = {13} pair -> elim from rest of c8/n3
77. r79c8 = naked pair {58} -> elim from rest of n9 -> r8c8 = 2 -> r7c7 = 1
78. outies of n9 = r9c6 = 1 -> r89c1 = [14] -> r89c9 = [43] -> r9c4 = 2
79. combination for 9/3 in c67 = [531] -> r34c9 = [52] && r7c4 = 4 -> r34c1 = [85] && r4c7 = 8 -> 23/4 in c34 = [1859] && 24/4 in c67 = [7638] && 9/3 in c5 = [342] -> r3c34 = [13]
80. innies of c1234 = r28c4 = 15/2 = {78} -> r1c4 = 6
81. innies of c6789 = r28c6 = 11/2 = [29] -> r3c67 = [42] && r89c7 = [69] -> r1c6 = 5
82. r9c5 = hidden single 6
83. r7c2 = hidden single 9 -> r3c2 = 7 -> r3c5 = 9
84. combinations: 9/2 in n1 = [36] -> naked singles and last-digit-in-cage moves solve the puzzle
2012 Walkthrough by Andrew:
Prelims

a) R12C1 = {18/27/36/45}, no 9
b) R12C9 = {89}
c) R5C12 = {29/38/47/56}, no 1
d) R5C89 = {17/26/35}, no 4,8,9
e) R89C1 = {14/23}
f) R89C9 = {16/25/34}, no 7,8,9
g) 10(3) cage at R1C8 = {127/136/145/235}, no 8,9
h) 8(3) cage at R3C3 = {125/134}
i) 19(3) cage at R3C1 = {289/379/469/478/568}, no 1
j) 9(3) cage at R4C5 = {126/135/234}, no 7,8,9
k) 10(3) cage at R6C1 = {127/136/145/235}, no 8,9
l) 9(3) cage at R6C7 = {126/135/234}, no 7,8,9
m) 20(3) cage at R7C2 = {389/479/569/578), no 1,2
n) 35(5) cage at R7C5 = {56789}

Steps resulting from Prelims
1a. Naked pair {89} in R12C9, locked for C9 and N9
1b. Naked quint {56789} in 35(5) cage at R7C5, locked for N8
1c. Max R9C6 = 4 -> min R89C7 = 12, no 1,2

2. 45 rule on N5 2 outies R5C37 = 11 = {29/38/47/56}, no 1

3. 45 rule on C34 2 innies R28C4 = 15 = {69/78}

4. 45 rule on C67 2 innies R28C6 = 11 = {29/38/47/56} -> R2C6 = {23456}

5. 45 rule on R12 3 outies R3C258 = 22 = {589/679}, 9 locked for R3
5a. 5 of {589} must be in R3C8 -> no 5 in R3C25
5b. 9 in C1 only in R45C1, locked for N4, clean-up: no 2 in R5C1, no 2 in R5C7 (step 2)
5c. 10(3) cage at R1C8 = {127/136/145/235}
5d. R3C8 = {567} -> no 5,6,7 in R12C8

6. 45 rule on R89 3 outies R7C258 = 22 = {589/679}, 9 locked for R8

7. 45 rule on C2 3 innies R456C2 = 9 = {126/135/234}, no 7,8, clean-up: no 3,4 in R5C1
7a. 1 of {126/135} must be in R6C2 -> no 5,6 in R6C2

8. 45 rule on C8 3 innies R456C8 = 20 = {389/479/569/578}, no 1,2, clean-up: no 6,7 in R5C9
8a. 3 of {389} must be in R5C8 -> no 3 in R46C8

9. 45 rule on R1234 3 innies R4C456 = 11 = {128/137/146/236/245}, no 9
9a. 6 of {146/236} must be in R4C46 (cannot be in R4C5 which would clash with 9(3) cage at R4C5 = 6{21}), no 6 in R4C5

10. 45 rule on R6789 3 innies R6C456 = 19 = {289/379/469/478/568}, no 1
10a. 2,3 of {289/379} must be in R6C5 -> no 2,3 in R6C46

11. Killer quad 1,2,3,4 in R12C1, R67C1 and R89C1, locked for C1

12. Hidden killer pair 8,9 in 15(3) cage at R7C8 and 16(3) cage at R8C7 for N9, neither cage can contain both of 8,9 -> each cage must contain one of 8,9 -> 15(3) cage = {159/168/249/258/348}, no 7, 16(3) cage = {169/178/259/268/349/358}
12a. R456C8 (step 8) = {479/569/578} (cannot be {389} which contains both of 8,9), no 3, clean-up: no 5 in R5C9
12b. 15(3) cage = {168/249/258/348} (cannot be {159} which clashes with R456C8)
12c. 9 of {249} must be in R7C8 -> no 9 in R89C8
12d. 10(3) cage at R1C8 = {127/136/235} (cannot be {145} which clashes with R456C8), no 4

13. 16(3) cage at R8C7 (step 12) = {169/178/259/268/349/358}, 15(3) cage at R7C8 (step 12b) = {168/249/258/348}
13a. Consider placements for 7 in N9
R7C9 = 7
or 7 in 16(3) cage = {178}, 8 locked for N9 => 15(3) cage = {249}, locked for N9 => R89C9 = {16} => R7C79 = {35}
-> R7C9 = {357}
13b. 45 rule on N9 2 innies R7C79 = 1 outie R9C6 + 7
13c. R9C6 = {1234} -> R7C79 = 8,9,10,11 = [17/35/53/27/37/47] (other permutations blocked by step 13a), no 6 in R7C7

14. 17(3) cage at R6C8 = {179/278/359/368/458/467} (cannot be {269} because R7C9 only contains 3,5,7)
14a. 8,9 of {179/269/278/359/368/458} must be in R6C8, 7 of {467} must be in R7C9 -> no 5,7 in R6C8, no 7 in R6C9
14b. R456C8 (step 12a) = {479/569/578}
14c. 8 of {578} must be in R6C8 -> no 8 in R4C8

15. 45 rule on C1 3 innies R345C1 = 1 outie R6C2 + 21
15a. Max R345C1 = 24 -> max R6C2 = 3

16. 45 rule on R5 3 remaining innies R5C456 = 15 = {159/168/249/258/267/348/357/456}
16a. 1 of {159/168} must be in R5C5 (cannot be [159/951] which clash with 9(3) cage at R4C5 = {135}, cannot be [168/861] which clash with 9(3) cage = {126}, CCC) -> no 1 in R5C46
16b. 3 of {348/357} must be in R5C5 (cannot be [348/843] which clash with 9(3) cage at R4C5 = {234}, cannot be [357/753] which clash with 9(3) cage = {135}, CCC) -> no 3 in R5C46

17. 45 rule on N1 2 innies R3C13 = 1 outie R1C4 + 3
17a. Min R3C13 = 6 -> min R1C4 = 3

18. 45 rule on N9 3(1+2) outies R6C7 + R79C6 = 1 innie R7C9 + 2
18a. Min R6C7 + R79C6 = 6 (cannot be [121/212] because no 6 in R7C7, cannot be [131] when R7C9 = 3) -> no 3 in R7C9, clean-up: no 5 in R7C7 (step 13c)
18b. 17(3) cage at R6C8 (step 14) = {179/278/359/458/467} (cannot be {368} because R7C9 only contains 5,7)
18c. R7C9 = {57} -> no 5 in R6C9
18d. Killer pair 5,7 in R7C258 and R7C9, locked for R7

19. 10(3) cage at R6C1 = {127/136/145/235}
19a. 5,7 of {127/145/235} must be in R6C1 -> no 2,4 in R6C1

20. 16(3) cage at R3C9 = {169/259/349/367/457}
20a. 9 of {349} must be in R4C8, 4 of {457} must be in R34C9 (R34C9 cannot be {57} which clashes with R7C9) -> no 4 in R4C8

21. 16(3) cage at R3C9 = {169/259/349/367/457}
21a. Consider permutations for R456C8 (step 12a) = {479/569/578}
R456C8 = {479} => R4C8 = 9 => 16(3) cage = {169/259/349}
or R456C8 = [569/956] (cannot be [659] => R5C9 = 3 => no remaining combination for 16(3) cage) => 16(3) cage = {169/259/349/457}
or R456C8 = [578] => 16(3) cage = {457}
or R456C8 = [758] => R5C9 = 3 => 16(3) cage = {457}
-> 16(3) cage = {169/259/349/457}
21b. 9 of {169} must be in R4C8 -> no 6 in R4C8

22. 16(3) cage at R3C9 (step 21a) = {169/259/349/457}
22a. Consider placements for R4C8
22b. R4C8 = 5 => R3C8 = {67} => R3C258 (step 5) = {679}, locked for R3 => no 7 in R3C9
or R4C8 = 7 => no 7 in R3C9
or R4C8 = 9 => 16(3) cage = {169/259/349}, no 7 in R3C9
-> no 7 in R3C9

23. R7C79 (step 13c) = [17/35/27/37/47]
23a. Consider placements for 3 in N9
3 in R789C7, locked for N9 => 3 in C8 only in 10(3) cage at R1C8 = {136/235} => R456C8 (step 12a) = {479/578} (cannot be {569} which clashes with 10(3) cage), 7 locked for N6 => R7C9 = 7 (hidden single in C9)
or 3 in 15(3) cage at R7C8 + R89C9, locked for N9 => R7C79 = [17/27/47] => R7C9 = 7
-> R7C9 = 7

24. R7C258 (step 6) = {589} (only remaining combination), locked for R7
24a. 6 in R7 only in R7C13, locked for N7

25. 16(3) cage at R3C9 (step 21a) = {169/259/349/457}
25a. 7,9 only in R4C8 -> R4C8 = {79}
25b. R456C8 (step 12a) = {479/569/578}
25c. 4 of {479} must be in R6C8, 9 of {569} must be in R4C8 -> no 9 in R6C8
25d. 5 of {569} must be in R5C8 -> no 6 in R5C8, clean-up: no 2 in R5C9

26. R7C9 = 7 -> R6C89 = 10 = [46/64/82], no 1,3 in R6C9

27. 9(3) cage at R6C7 = {126/135/234}
27a. 5,6 of {126/135} must be in R6C7 -> no 1 in R6C7

28. Max R7C34 = 10 -> min R6C3 = 3
28a. 1 in R6 only in R6C12, locked for N4 and 10(3) cage at R6C1, no 1 in R7C1
28b. 10(3) cage = {127/136/145}
28c. 2 of {127} must be in R7C1 -> no 2 in R6C2

29. 8(3) cage at R3C3 = {125/134}, 1 locked for R3

30. 14(3) cage at R3C6 = {239/248/257/347/356} (cannot be {149} because 1,9 only in R4C7, cannot be {158/167} because R3C67 = {58/67} because R3C1 + R3C258 contain four of 5,6,7,8,9 in R3), no 1 in R4C7
30a. 1 in N6 only in R45C9, locked for C9, clean-up: no 6 in R89C9

31. 16(3) cage at R3C9 (step 21a) = {169/259/349} (cannot be {457} which clashes with R89C9) -> R4C8 = 9, R34C9 = {25/34}/[61], no 6 in R4C9
31a. Killer triple 1,2,3 in R34C9, R5C9 and R89C9, locked for C9
31b. R7C9 = 7 -> R6C89 = 10 = {46}, locked for R6 and N6, clean-up: no 3 in R3C9, no 5,7 in R5C3 (step 2)

32. 8 in C7 only in R45C7, locked for C7
32a. 8 in R45C7, CPE no 8 in R4C6

33. R5C1 = 9 (hidden single in C1), R5C2 = 2
33a. R456C2 (step 7) = {126/234}, R6C2 = {13} -> R4C2 = {46}
33b. 19(3) cage at R3C1 = {478/568}, 8 locked for C1, clean-up: no 1 in R12C1
33c. R4C2 = {46} -> no 6 in R34C1

34. 10(3) cage at R6C1 (step 28b) = {127/136/145}
34a. 2,4,6 only in R7C1 -> R7C1 = {246}

35. 9 in R6 only in R6C46 -> R6C456 (step 10) = {289/379}, no 5

36. R5C456 (step 16) = {168/348/456} (cannot be {357} which clashes with R5C8), no 7
36a. R5C456 = {168/456} (cannot be {348} which clashes with 9(3) cage at R4C5 = [432], CCC), no 3, 6 locked for R5 and N5, clean-up: no 5 in R5C7 (step 2)
36b. 7 in R5 only in R5C78, locked for N6

37. R4C2 = 6 (hidden single in N4), R6C2 = 1 (step 33a), R34C1 = 13 = {58}, locked for C1, clean-up: no 4 in R12C1

38. 1 in C1 only in R89C1 = {14}, locked for C1 and N7

39. 9(3) cage at R4C5 = {126/135/234}
39a. 4 of {234} must be in R5C5 -> no 4 in R4C5

40. 45 rule on R789 3 remaining outies R6C137 = 15 = {357} (cannot be {258} because R6C1 only contain 3,7), locked for R6 -> R6C5 = 2
40a. Naked pair {89} in R6C46, locked for N5
40b. 9(3) cage at R4C5 = {126/234}, no 5
40c. R4C5 = {13} -> R5C5 = {46}
40d. Naked triple {456} in R5C456, locked for R5 and N5 -> R5C8 = 7, R5C9 = 1, clean-up: no 6 in R3C9 (step 31)
40e. Naked triple {137} in R4C456, locked for R4, clean-up: no 4 in R3C9 (step 31)

41. Naked pair {25} in R34C9, locked for C9
41a. Naked pair {34} in R89C9, locked for C9 and N9 -> R6C9 = 6, R6C8 = 4
41b. 16(3) cage at R8C7 = {169/259}, no 3,4

42. R3C258 (step 5) = {679} (only remaining combination, cannot be {589} which clashes with R3C1) -> R3C8 = 6, R12C8 = 4 = {13}, locked for C8 and N3
42a. Naked pair {79} in R3C25, locked for R3

43. Naked triple {258} in 15(3) cage at R7C8, locked for N9 -> R7C7 = 1, R89C7 = {69}, R9C6 = 1 (cage sum), R89C1 = [14], R89C9 = [43]
43a. R9C4 = 2 -> R89C3 = 12 = [39/57/75], no 8, no 9 in R8C3
43b. Naked pair {34} in R7C46, locked for R7

44. R4C3 = 4 (hidden single in R4), R3C34 = 4 = {13}, locked for R3

45. 14(3) cage at R3C6 = {248} (only remaining combination), no 5

46. 7 in C7 only in R12C7 -> 16(3) cage at R1C6 = {457} (only remaining combination), no 7 in R1C6
46a. 2 in N3 only in R3C79, locked for R3

47. 20(3) cage at R7C2 = {389/578}, 8 locked for C2 and N7

48. 16(3) cage at R1C2 = {349/457}
48a. Killer pair 3,7 in R12C1 and 16(3) cage, locked for N1 -> R3C3 = 1, R3C4 = 3, R7C4 = 4, R7C6 = 3, R6C7 = 5, R34C9 = [52], R4C7 = 8, R3C67 = [42], R1C6 = 5, R5C456 = [546], R2C6 = 2, R5C37 = [83], R6C46 = [98], R4C456 = [137], R8C6 = 9, R89C7 = [69], clean-up: no 7 in R1C1, no 3 in R8C3 (step 43a)

49. Naked pair {57} in R89C3, locked for C3 and N7 -> R9C2 = 8, R6C3 = 3, R7C3 = 6 (cage sum), R12C3 = [29], R1C4 = 6 (cage sum)

and the rest is naked singles.

I'll rate my walkthrough for A45 V2 at 1.75. I used several forcing chains and some combination analysis.


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PostPosted: Wed Jun 18, 2008 11:47 pm 
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Posts: 1044
Location: Sydney, Australia
Assassin 45v3 (aka Easter Eggs V3) by Ruud (Apr 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2048:3585:4610:4610:5892:3077:3077:4359:3336:2048:3585:4610:5892:5892:5892:3077:4359:3336:3602:3585:3860:3860:5892:3095:3095:4359:4122:3602:3602:3860:5406:2335:7712:3095:4122:4122:3364:3364:5406:5406:2335:7712:7712:1835:1835:2861:2861:4911:5406:2335:7712:2099:3380:3380:2861:5687:4911:4911:6970:2099:2099:4157:3380:2111:5687:2113:6970:6970:6970:5957:4157:2119:2111:5687:2113:2113:6970:5957:5957:4157:2119:
solution:
+-------+-------+-------+
| 7 8 3 | 9 6 1 | 4 2 5 |
| 1 4 6 | 2 5 3 | 7 9 8 |
| 9 2 5 | 8 7 4 | 3 6 1 |
+-------+-------+-------+
| 4 1 2 | 6 3 9 | 5 8 7 |
| 8 5 9 | 1 2 7 | 6 3 4 |
| 6 3 7 | 5 4 8 | 2 1 9 |
+-------+-------+-------+
| 2 6 8 | 4 9 5 | 1 7 3 |
| 3 9 4 | 7 1 2 | 8 5 6 |
| 5 7 1 | 3 8 6 | 9 4 2 |
+-------+-------+-------+
Quote:
mhparker: (no twist in the tail in this one)
Andrew in 2012: Much easier than the V2. This was an enjoyable variant. :D I'll rate my walkthrough at Hard(?) 1.25.
Walkthrough by mhparker:
By the way, here's my walkthrough for V3. Haven't proof read it yet, so let me know of any significant mistakes...


Walkthrough - Assassin 45 V3

1. Preliminaries:

a) 8/2 at R1C1: no 4,8,9
b) 13/3 at R1C9: no 1,2,3
c) 9/3 at R4C5: no 7,8,9
d) 30/4 at R4C6 = {6789} -> no 6,7,8,9 in R5C45
e) 13/2 at R5C1: no 1,2,3
f) 7/2 at R5C8: no 7,8,9
g) 11/3 at R6C1: no 9
h) 19/3 at R6C3: no 1
i) 8/3 at R6C7 = {1(25|34)}
j) 22/3 at R7C2 = {(67|58)9} -> no 9 elsewhere in C2,N7
k) 8/2 at R8C1: no 4,8
l) 8/3 at R8C3 = {1(25|34)} -> no 1 in R9C1 -> no 7 in R8C1
m) 23/3 at R8C7 = {689} -> no 6,8,9 in R9C89 -> no 2 in R8C9
n) 8/2 at R8C9: no 4,8,9

2. Outies N5: R5C37 = 15/2 = {69|78} -> 13/2 at R5C1 = [94]|{58}

3. Innies C34: R28C4 = 9/2 -> no 9

4. Innies C67: R28C6 = 5/2 = {14|23}

5a. Outies C1: R456C2 = 9/3 -> no 7,8 -> 13/2 at R5C1 = [94]|[85]
5b. R46C2 = {(1|2)3} -> no 3 elsewhere in C2 or N4, no 1 in R34C1

6a. Innies R5: Split cage R5C456 = 10/3 -> no 8,9 in R5C6, no 4,5 in R5C45
6b. 1 locked in R5C45 -> no 1 elsewhere in R5 or N5 -> 7/2 at R5C8 = {25|34}

7. Outies R89: R7C258 = 22/3 = {(67|58)9} -> no 9 elsewhere in R7 -> no 2 in R6C3

8a. Naked quad on {6789} in C6 at R4569C6 -> no 6,7,8,9 in R13C6
8b. 7 in C6 now locked in N5 -> no 7 in R46C4 or R5C7 -> no 8 in R5C3 (step 2)

9. Naked triple on {689} in C7 at R589C7 -> no 6,8,9 in R1234C7

10a. 21/4 at R4C4 cannot contain both of {67} due to R5C6
10b. 9/3 at R4C5 must contain 2 of {123} -> forms naked killer triple on {123} with R5C4 -> no 2,3 in R46C4
10c. 21/4 at R4C4 must contain at least one of {45} in C4 -> no 4,5 in R28C4 (step 3)

11. Innies N2: R13C46 = 22/4 -> min. R13C4 = 13 -> no 1,2,3,4,5 -> no 9 in R34C3

12. Innies N9: R789C7+R7C9 = 21/4 -> can only contain 2 of {6789} (in R89C7) -> no 6,7,8 in R7C9

13. Outies N1: R13C4(min. 13)+R4C123(R4C23 min. 3) = 24/5 -> max. R4C1 = 8 -> no 9 in R4C1

14. Innie/outie difference N3: R3C79 - R1C6 = 3 -> max. R3C79 = 8 -> no 8,9 in R3C9

--- Now comes the complicated bit that most automated solvers can't do... ---

15a. 7 in C7 locked in either of 12/3 cages at R1C6 or R3C6
15b. Only one of these 2 12/3 cages can contain a 7, since 7 not available in R13C6 = {(14|23)7}
15c. {689} not available -> the other 12/3 cage must be {345}
15d. The only other location for the second 5 in C67 is within 8/3 cage at R6C7 = {125} (no 3,4)

16a. N3: 17/3 and 13/2 cages must each contain one of {89} -> 13/2 <> {67} -> no 6,7 in R12C9
16b. 6 in N3 locked in R123C8+R3C9 -> no 6 in R4C8

17. 17/3 at R1C8: {359} and {458} both blocked due to 13/2 at R1C9 -> no 5 in R123C8

18a. Hidden 15/4 cage(N3) at R123C7+R3C9 also cannot contain a 5, as both {1257} and {1356} blocked by 8/3 at R6C7
-> 5 in N3 locked in 13/2 at R1C9 = {58} -> no 5,8 elsewhere in C9,N3 -> no 3 in R89C9
18b. Hidden 15/4 cage(N3) = {(17|26)34} -> 3,4 locked -> no 3,4 in 17/3 cage at R1C8

--- end of complicated bit ---

19a. 9 in N3 locked in 17/3 in C8 -> no 9 elsewhere in C8
19b. 9 in C9 locked in R46C9 -> no 9 in R5C7 -> no 6 in R5C3 (step 2)
19c. 9 in N9 locked in R89C7 -> no 9 in R9C6 -> 9 in N8 locked in C5 -> no 9 elsewhere in C5
19d. 9 in N2 locked in R13C4 -> no 9 elsewhere in C4, no 7,8 in R4C1 (step 13)

20. 6 in R5 locked in R5C67 -> no 6 in R46C6

21a. {34} in C6 locked in R1238
21b. R28C6 must contain exactly 1 of {34} (step 4) -> R13C6 must contain exactly 1 of {34}
-> 23/5 at R1C5 cannot contain both of {34} -> must contain 5 in C5 (since 9 not available)
-> no 5 elsewhere in C5 or N2

22a. Naked quad on {1234} in R1238C6 -> no 1,2,3,4 elsewhere in C6 -> R7C6 = 5
22b. Naked pair on {12} in R67C7 -> no {12} elsewhere in C7

23a. Hidden single in C7 at R4C7 = 5
23b. 7/2 at R5C8 = {34} -> no 3,4 elsewhere in R5 or N6 -> R5C12 = [85], R5C67 = [76] -> R5C3 = 9

24. Hidden single in C1 at R3C1 = 9 -> no 6 in R4C1

25. Hidden single in C4,N5 at R6C4 = 5 -> R45C4 = [61], R5C5 = 2

26. Hidden single in C4 at R1C4 = 9

27. Hidden single in C6 at R9C6 = 6

28. Hidden single in C8 at R2C8 = 9

29a. 22/3 at R789C2 = {679} -> no 6,7 elsewhere in C2 or N7
29b. 14/3 at R123C2 = {248} -> no 2,4,8 elsewhere in C2 or N1
29c. 8/2 at R8C1 = {35} -> no 3,5 elsewhere in C1 or N7
29d. 8/2 at R1C1 = {17} -> no 1,7 elsewhere in C1 or N1

30. Hidden single in C1 at R6C1 = 6

31. Hidden single in C3 at R7C3 = 8

... and so on (no twist in the tail in this one)
2012 Walkthrough by Andrew:
Prelims

a) R12C1 = {17/26/35}, no 4,8,9
b) R12C9 = {49/58/67}, no 1,2,3
c) R5C12 = {49/58/67}, no 1,2,3
d) R5C89 = {16/25/34}, no 7,8,9
e) R89C1 = {17/26/35}, no 4,8,9
f) R89C9 = {17/26/35}, no 4,8,9
g) 9(3) cage at R4C5 = {126/135/234}, no 7,8,9
h) 11(3) cage at R6C1 = {128/137/146/236/245}, no 9
i) 19(3) cage at R6C3 = {289/379/469/478/568}, no 1
j) 8(3) cage at R6C7 = {125/134}
k) 22(3) cage at R7C2 = {589/679}
l) 8(3) cage at R8C3 = {125/134}
m) 23(3) cage at R8C7 = {689}
n) 30(4) cage at R4C6 = {6789}

Steps resulting from Prelims
1a. 22(3) cage at R7C2 = {589/679}, 9 locked for C2 and N7, clean-up: no 4 in R5C1
1b. 8(3) cage at R8C3 = {125/134}, CPE no 1 in R9C1, clean-up: no 7 in R8C1
1c. 23(3) cage at R8C7 = {689}, CPE no 6,8,9 in R9C89, clean-up: no 2 in R8C9
1d. 30(4) cage at R4C6, CPE no 6,7,8,9 in R5C45

2. Naked quad {6789} in R4569C6, locked for C6, 7 also locked for N5 and 30(5) cage at R4C6
2a. Naked triple {689} in R589C7, locked for C7

3. 45 rule on N5 2 outies R5C37 = 15 = [69/78/96]
3a. R5C12 = [58/85/94] (cannot be {67} which clashes with R5C37), no 6,7
3b. Killer pair 8,9 in R5C12 and R5C37, locked for R5
3c. Killer pair 6,7 in R5C37 and R5C6, locked for R5, clean-up: no 1 in R5C89
3d. 1 in R5 only in R5C45, locked for N5
3e. Hidden killer triple 1,2,3 in R5C45 and R5C89 for R5, R5C89 contains one of 2,3 -> R5C45 = {123}
3f. Killer triple 1,2,3 in 9(3) cage at R4C5 and R5C4, locked for N5

4. 45 rule on C34 2 innies R28C4 = 9 = {18/27/36/45}, no 9

5. 45 rule on C67 2 innies R28C6 = 5 = {14/23}

6. 45 rule on R89 3 outies R7C258 = 22 = {589/679}, 9 locked for R7

7. 45 rule on C2 3 innies R456C2 = 9 = {135/234} (cannot be {126} because R5C2 doesn’t contain 1,2,6), no 6,7,8, 3 locked for C2 and N4, clean-up: no 5 in R5C1
7a. R5C2 = {45} -> no 4,5 in R46C2
7b. Max R4C2 = 3 -> min R34C1 = 11, no 1 in R34C1

8. 45 rule on N3 2 innies R3C79 = 1 outie R1C6 + 3
8a. Max R1C6 = 5 -> max R3C79 = 8, no 8,9 in R3C9

9. Hidden killer pair 8,9 in 17(3) cage at R1C8 and R12C9, neither can contain both of 8,9 -> 17(3) cage and R12C9 must each contain one of 8,9
9a. R12C9 = {49/58} (cannot be {67} which doesn’t contain 8 or 9), no 6,7
9b. 17(3) cage = {179/269/278/368} (cannot be {359/458} which clash with R12C9, cannot be {467} which doesn’t contain 8 or 9), no 4,5

10. 45 rule on N9 4 innies R7C79 + R89C7 = 21 cannot contain more than two of 6,7,8,9 -> no 6,7,8 in R7C9

11. 7 in C7 only in 12(3) cages at R1C6 and R3C6, one must be {147/237} and the other must be {345}
11a. Only other cage containing 5 in C67 is 8(3) cage at R6C7 -> 8(3) cage = {125}

12. 45 rule on R5 3 remaining innies R5C456 = 10 = {127/136}
12a. 9(3) cage at R4C5 = {135/234} (cannot be {126} which clashes with R5C456, CCC), no 6, 3 locked for C5 and N5

13. 6 in R5 and 6 in N5 only in 21(4) cage at R4C4 and 30(4) cage at R4C6 -> both these cages must contain 6
13a. 21(4) cage = {1569/2469/2568} (cannot be {3468/3567} because R5C4 only contains 1,2), no 7, clean-up: no 8 in R5C7 (step 3)
[Cracked …]

14. Naked pair {69} in R5C37, locked for R5 -> R5C1 = 8, R5C2 = 5, R5C6 = 7, clean-up: no 2 in R5C89, no 8 in 22(3) cage at R7C2
14a. Naked pair {34} in R5C89, locked for R5 and N6
14b. Naked pair {12} in R5C45, locked for N6
14c. 30(4) cage at R4C6 = {6789}, 8 locked for C6 and N5
14d. 8 in C7 only in R89C7, locked for N9

15. Naked triple {679} in 22(3) cage at R7C2, locked for C2 and N7, clean-up: no 1,2 in R89C1
15a. Naked pair {35} in R89C1, locked for C1 and N7

16. R7C3 = 8 (hidden single in N7) -> R6C3 + R7C4 = 11 = [47/65/74/92], no 2 in R6C3, no 3,6 in R7C4
16a. R7C258 (step 6) = {679} (only remaining combination), locked for R7, clean-up: no 4 in R6C3

17. R7C9 = 3 (hidden single in R7), R5C89 = [34], clean-up: no 9 in R12C9, no 5 in R89C9
17a. Naked pair {58} in R12C9, locked for C9 and N3

18. 9 in N3 only in 17(3) cage at R1C8, locked for C8
18a. 9 in N9 only in R89C7, locked for C7 and 23(3) cage at R8C7 -> R5C7 = 6, R5C3 = 9, R9C6 = 6, clean-up: no 2 in R7C4 (step 16)
18b. Naked triple {456} in R467C4, locked for C4, clean-up: no 3 in R28C4 (step 4)

19. R3C1 = 9 (hidden single in N1), R4C12 = 5 = [23/41]
19a. 14(3) cage at R1C2 = {248} (only remaining combination), locked for C2 and N1, clean-up: no 6 in R12C1
19b. Naked pair {17} in R12C1, locked for C1 and N1
19c. Naked pair {13} in R46C2, locked for N4

20. R6C1 = 6 (hidden single in C1), R6C3 = 7, R7C4 = 4 (cage sum), R46C4 = [65], R5C4 = 1 (cage sum), R5C5 = 2, clean-up: no 8 in R28C4 (step 4), no 1 in R2C6 (step 5)

21. Naked pair {27} in R28C4, locked for C4 -> R9C4 = 3, R89C3 = 5 = {14}, locked for C3 and N7 -> R4C3 = 2, R4C1 = 4, R46C5 = [34], R46C2 = [13], R7C1 = 2, R89C1 = [35], R3C4 = 8, R3C3 = 5 (cage sum), R1C4 = 9, clean-up: no 2 in R2C6 (step 5)

22. 45 rule on N8 1 remaining innie R7C6 = 5 -> R67C7 = [21], clean-up: no 7 in R89C9
22a. R89C9 = [62], R7C8 = 7, R89C8 = [54], R4C8 = 8, R4C6 = 9, R34C9 = [17]

23. R4C7 = 5 -> R3C67 = 7 = {34}, locked for R3 -> R3C2 = 2, R3C8 = 6, R3C5 = 7, R28C4 = [27]

24. Naked pair {34} in R23C6, locked for C6 -> R1C6 = 1

25. 45 rule on N2 1 remaining innie R3C6 = 4

and the rest is naked singles.

I'll rate my walkthrough at Hard(?) 1.25, based on steps 11 and 13; maybe step 13 is technically the more difficult.


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PostPosted: Wed Jun 18, 2008 11:57 pm 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 46 by Ruud (Apr 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3584:4609:4609:4609:3076:4101:4101:4101:2824:3584:3584:4107:4107:3076:4622:4622:2824:2824:5138:5138:5138:4107:3076:4622:4888:4888:4888:5138:2076:2076:9758:9758:9758:3361:3361:4888:3364:3364:2854:2854:9758:2089:2089:1323:1323:4909:1070:1070:9758:9758:9758:2867:2867:5685:4909:4909:4909:6201:2874:3899:5685:5685:5685:3135:3135:6201:6201:2874:3899:3899:4678:4678:3135:3657:3657:3657:2874:3917:3917:3917:4678:
Solution:
+-------+-------+-------+
| 5 7 3 | 8 4 6 | 9 1 2 |
| 1 8 9 | 2 7 3 | 6 4 5 |
| 6 4 2 | 5 1 9 | 8 3 7 |
+-------+-------+-------+
| 8 2 6 | 3 5 7 | 4 9 1 |
| 4 9 5 | 6 8 1 | 7 2 3 |
| 7 3 1 | 4 9 2 | 5 6 8 |
+-------+-------+-------+
| 3 5 4 | 9 2 8 | 1 7 6 |
| 2 1 8 | 7 6 4 | 3 5 9 |
| 9 6 7 | 1 3 5 | 2 8 4 |
+-------+-------+-------+
Quote:
Ruud, in lead-in: This Assassin is at the upper boundary of my “acceptable” difficulty range. It requires several advanced moves
Andrew: It certainly was! At least a V1.5 in my opinion. After doing Assassin 47V1.5, I'll reassess Assassin 46 as a V2
PsyMar in A47 thread: I never was able to finish (I think a lot of other people couldn't either judging by the lack of forum activity!)
Andrew (in 2012): Ed suggested to me that a new walkthrough was needed for Assassin 46 because both of the walkthroughs posted in 2007 were too hard for the SSscore. Neither of the original walkthroughs used the interactions between two overlapping hidden cages in N8.
Rating 1.25.
Walkthrough by mhparker including Nice Loop(x-cycle) & Nishio:
Well, after gobbling up the second easter egg all to myself, I didn't really want to be the first to post a walkthrough for this one. However, I strongly suspect that everyone else's walkthrough will be quite different, so there's plenty room for more...

Anyhow, here's how I did it:

Edit: Modified to incorporate Andrew's suggestions


Walkthrough - Assassin 46

1. 11/3 at R1C9: no 9
2. 8/2 at R4C2: no 4,8,9
3. 13/2 at R4C7: no 1,2,3
4. 13/2 at R5C1: no 1,2,3
5. 11/2 at R5C3: no 1
6. 8/2 at R5C6: no 4,8,9
7. 5/2 at R5C8 = {14|23}

8a. 4/2 at R6C2 = {13} -> no 1,3 elsewhere in R6 or N4
8b. 8/2 at R4C2 = {26} -> no 2,6 elsewhere in R4 or N4
8c. 13/2 at R4C7 = {49|58) (no 7)
8d. 13/2 at R5C1 = {49|58) (no 7)
8e. (Cleanup) R5C4: no 5,8,9
8f. (Cleanup) R6C78: no 8

9. 24/3 at R7C4 = {789} -> no 7,8,9 in R8C56
10. 11/3 at R7C5: no 9

11a. Innie R5: R5C5 = 8
11b. 13/2 at R5C1 = {49} -> no 4,9 elsewhere in R5 or N4
11c. 5/2 at R5C8 = {23} -> no 2,3 elsewhere in R5 or N6
11d. 11/2 at R5C3 = [56]
11e. (Cleanup) R6C78: no 9

12a. Innies N5: R5C46 = 7/2 -> R5C67 = [17]
12b. 11/2 at R6C7 = {56} -> no 5,6 elsewhere in R6 or N6

13. 13/2 at R4C7 = {49} -> no 4,9 elsewhere in R4 or N6

14. Naked single R6C9 = 8 -> R6C1 = 7, R4C9 = 1, R4C1 = 8

15. Innies C5: R46C5 = 14/2 = [59]

16a. Innies C1234: R46C4 = 7/2 = [34] -> R46C6 = [72]
16b. (Cleanup) 7 no longer available for 18/3 at R2C6 -> no 2 in R2C7

17. Innies R1: R1C159 = 11/3 -> no 9 in R1C1

18. Innies N1: R1C23+R2C3 = 19/3 -> no 1

19. Outies N1: R123C4 = 15/3 = {(19|28)5} (3,4,6 not available) -> no 5 elsewhere in C4 or N2

20. 7 in N2 now locked in 12/3 at R1C5 = {(14|23)7} (no 6) -> 11/3 at R7C5 = {(14|23)6} (no 7)

21. 6 in N2 now locked in C6 -> no 6 in R789C6

22. Innies R7: R7C456 = 19/3 -> no 1 in R7C5

23. Innies N7: R8C3+R9C23 = 21/3 -> no 1,2,3 in R9C23

24. Innies R3: R3C456 = 15/3 -> no 1 in R3C4 (both {59} and {68} unavailable for R3C56)

25. I/O diff. N1: R2C3 = R1C4 + 1 -> no 9 in R1C4, no 4,7,8 in R2C3
26. I/O diff. N3: R2C7 = R1C6 -> no 1,5 in R2C7
27. I/O diff. N7: R8C3 = R9C4 + 7 -> R8C3 = {89}, R9C4 = {12}
28. I/O diff. N9: R9C6 = R8C7 + 2 -> no 4,5,8,9 in R8C7, no 9 in R9C6

29. 18/3 at R1C2 = {89}[1] | {79}[2] | {49|67}[5] | {37|46}[8] -> no 2 in R1C23, no 5 in R1C2

30a. From list in step 29, we can exclude {67}[5], as this would place a 6 in R2C3 (step 25),
thus clashing with R1C23
30b. Examining the remaining possibilities listed in step 29, we can see that either R1C23
contains a 9, or R1C4 contains an 8 -> R2C3 = 9 (step 25)
30c. In either case listed in step 30b, we can observe that the h19/3(N1) innie cage at
R1C23+R2C3 must contain a 9 -> no 9 elsewhere in N1

31. 11/3 at R1C9: No 1 available in R12C9 -> R12C9 must sum to at least 5 -> no 7,8 in R2C8

32. Nishio: if R9C6 = 8, then 8 in N9 would be locked in R8(C8) -> no way to place 8 in 24/3 at R7C4
-> original hypothesis (R9C6 = 8) must be wrong -> no 8 in R9C6 -> no 6 in R8C7 (step 28)
Note: Another, more formalized, way of expressing this is as the following discontinuous 5-node

Nice Loop (x-cycle):
R9C6-8-R9C78=8=R8C8-8-R8C34=8=R7C4-8-R9C6 => R9C6<>8,
where the first link (R9C6-R9C78) is actually a strong link (R9C6 and R9C78 belong to the same 15/3 cage)
being used as a weak inference. So who said Nishio was just trial and error?!

33a. Hidden triple on {789} in N8 at R78C4+R7C6 -> R7C6 = {89}
33b. 15/3 at R7C6 = {(25|34)8} | {(15|24)9} -> R8C6 = {45}
33c. Min. of R7C4 + R7C6 = 15 -> R7C5 = max. 4 (see step 22) -> no 6 in R7C5

34. Nishio: From step 26, R2C7 = R1C6 -> these cells cannot contain any digit not present as candidate in R3C123
(otherwise there would be nowhere to place that digit in N1) -> no 8,9 in R1C6 or R2C7

35a. 19/4 at R3C7 cannot contain both of {89}
35b. 16/3 at R1C6 cannot contain both of {89}
35c. Therefore, R3C789 and R1C78 must each contain one of {89} -> 18/3 cage at R1C2 cannot contain both of {89}
-> no 1 in R1C4, no 2 in R2C3 (see step 25)
35d. 2 no longer available in h19/3(N1), which we know must already contain a 9 (see step 30c)
-> cannot also contain an 8 ({289} not available) -> no 8 in R1C23

36. Hidden single in N1 at R2C2 = 8 -> R12C1 = {15|24}

37a. Common Peer Elimination (CPE): R3C4 can see all cells with candidate 9 in R2 (R2C346) -> no 9 in R3C4
37b. Both {69} and {78} now impossible for R2C3+R3C4 -> no 1 in R2C4

38a. Hidden single in C4 at R9C4 = 1 -> R8C3 = 8 (step 27)
38b. R78C4 = {79}, locked for C4 and N8

39. Naked single in N8 at R7C6 = 8 -> no 1 in R8C7 (step 33b) -> no 3 in R9C6 (step 28)

40a. Naked pair on {45} in C6 and N8 at R89C6 -> no 4,5 elsewhere in C6 or N8
40b. Cleanup: no 4 in R2C7 (step 26)

41. 11/3 at R7C5 = {236}, locked for C5 -> 12/3 at R1C5 = {147}

42. Split innie h11/2 cage at R7C45 = [92] -> R8C4 = 7

43a. 18/3 cage at R8C8: no 1 in R8C8 (due to candidate 8 not being available)
43b. 1 in N9 now locked in R7 -> no 1 in R7C123
43c. R7C789 = 14/3 = {167} (8,9 not available), locked for R7 and N9 -> R7C123 = 12/3 = {345}, locked for N7

44. R9C23 = 13/2 = {67}, locked for R9 and N7 -> 12/3 at R8C1 = {129}

45. Hidden single in C1 at R3C1 = 6 -> R3C23 = {15|24}

46. Hidden single in C1 at R7C1 = 3 -> R7C23 = [54]

47. 5 in N1 locked in R12C1 = {15}, locked for N1 -> R3C3 = 2 -> R3C2 = 4 (last digit in cage)

The rest is pretty straightforward now
Hard-work Walkthrough by Andrew:
Ruud wrote:
This Assassin is at the upper boundary of my “acceptable” difficulty range. It requires several advanced moves.

It certainly was! At least a V1.5 in my opinion. After doing Assassin 47V1.5, I'll reassess Assassin 46 as a V2.

Mike found some very good advanced moves. I particularly liked his steps 32, 34 and 37a.

Step 32 was a nice contradiction move; I must admit I don't know enough about advanced techniques to follow the more formalised presentation that followed.

Step 34 was a logical beauty. I don't understand why it was called Nishio. It's pure logic.

Step 37a is so obvious when it's pointed out but one needs to know enough to look for it.


In the Assassin 47 thread, PsyMar wrote "Much easier than last week's, which I never was able to finish (I think a lot of other people couldn't either judging by the lack of forum activity!)"

I must admit I wasn't sure whether to post my walkthrough, although I was thinking of doing so just to show the sort of hard work that I did because I didn't have the ability to see Mike's really clever moves. PsyMar's comment decided it for me; the thread needed another walkthrough.

In contrast my walkthrough has one extremely heavy step where I consider the various combinations in N9 and their interactions between each other and with N7 and N8. I hope that doesn't put you off working through it. There is a summary of the results of that step for those who don't want to work through it.

1. R4C23 = {17/26/35}, no 4,8,9

2. R4C67 = {49/58/67}, no 1,2,3

3. R5C12 = {49/58/67}, no 1,2,3

4. R5C34 = {29/38/47/56}, no1

5. R5C67 = {17/26/35}, no 4,8,9

6. R5C89 = {14/23}

7. R6C23 = {13}, locked for R6 and N4, clean-up: no 5,7 in R4C23, no 8 in R5C4
7a. Naked pair {26} in R4C23, locked for R4 and N4, clean-up: no 7 in R4C78, no 7 in R5C12, no 5,9 in R5C4

8. R6C78 = {29/47/56}, no 8

9. 11(3) cage in N3 = {128/137/146/236/245}, no 9

10. 24(3) cage in N78 = {789}, no 7,8,9 in R8C56

11. R789C5 = {128/137/146/236/245}, no 9

12. 38(7) cage in N5 = 789{1256/1346/2345}, 7,8,9 locked for N5, clean-up: no 4 in R5C3, no 1 in R5C7

13. 45 rule on R5 1 innie R5C5 = 8, clean-up: no 5 in R5C12, no 3 in R5C4
13a. Naked pair {49} in R5C12, locked for R5 and N4, clean-up: no 7 in R5C3, no 2 in R5C4, no 1 in R5C89
13b. R5C34 = [56] (naked singles)
13c. Naked pair {23} in R5C89, locked for R5 and N6, clean-up: no 9 in R6C78
13d. R5C67 = [17] (naked singles), clean-up: no 4 in R6C78
13e. Naked pair {56} in R6C78, locked for R6 and N6, clean-up: no 8 in R4C78
13f. Naked pair {49} in R4C78, locked for R4 and N6
13g. R46C9 = [18] (naked singles)
13h. R46C1 = [87] (naked singles)

14. 11(3) cage in N3, min R12C9 = 5 -> max R2C8 = 6

15. 45 rule on R1 3 innies R1C159 = 11 = {137/146/236/245}, no 9

16. 45 rule on N7 R7C123 = 12, 3 remaining innies R8C3 + R9C23 = 21 = {489/579/678}, no 1,2,3
16a. R9C234 = {149/158/167/248/257/347/356} -> R9C4 = {123}

17. 45 rule on N7 3 remaining outies R789C4 = 17 = {179/278} = 7{19/28} [1/2, 8/9], no 3, 7 locked for C4, N8 and 24(3) cage, no 7 in R8C3
17a. R9C234 = {149/158/167/248/257} [1/2, 4/5/6, 7/8/9]

18. 45 rule on N3 1 outie R1C6 = 1 innie R2C7 -> no 7 in R1C6, no 1 in R2C7

19. 45 rule on N1 1 innie R2C3 – 1 = 1 outie R1C4 -> no 4,9 in R1C4, no 1,7,8 in R2C3

20. 45 rule on N1 3 remaining outies R123C4 = 15 = {159/249/258/348} [1/2/3, 4/5, 8/9]
20a. Killer pair 8/9 in R123C4 and R78C4 for C4
20b. 45 rule on C4 2 remaining innies R46C4 =7 = [34/52]

21. 45 rule on N9 1 outie R9C6 – 2 = 1 innie R8C7, no 5,8,9 in R8C7, no 2,9 in R9C6

22. 45 rule on R89 3 outies R7C456 = 19 = {289/379/469/478/568}, no 1, no 2,3 in R7C6

23. 45 rule on N3 3 remaining outies R123C6 = 18

24. 45 rule on N9 3 remaining outies R789C6 = 17

25. 45 rule on C6 2 remaining innies R46C6 = 9 = [54/72], no 3 in R4C6, no 9 in R6C6
25a. R6C5 = 9 (hidden single in R6)

26. From steps 20b and 25, R4C456 = [357] (cannot be [5n5]) -> R6C46 = [42], clean-up: no 4 in R2C3, no 2 in R2C7
[Alternatively, and much easier to follow, 45 rule on N5 remembering that R46C4 = 7 (step 20b) and R46C9 = 9 (step 25) -> R4C5 = 5 -> R4C46 = [37] -> R6C46 = [42] followed by the same clean-up. The first version was how I did it; I only saw the easier way while checking this walkthrough.]

27. R789C5 = {146/236} = 6{14/23}, 6 locked for C5 and N8, clean-up: no 4 in R8C7
27a. R123C5 = 7{14/23}

28. 5 in C4 locked in R123C4, locked for N2, clean-up: no 5 in R2C7
28a. R123C4 = 5{19/28}
28b. R123C6 = 6{39/48}
28c. R789C6 = 5{39/48}

29. 15(3) cage in N89 = {159/249/258/348/456} (cannot be {168} because 1,6 in same cell), no 3 in R8C6

30. 45 rule on N1 3 innies R1C23 + R2C3 = 19 = {289/379/469/568} (cannot be {478} because no 4,7,8 in R2C3), no 1

31. 45 rule on R12 3 outies R3C456 = 15 = {159/168/249/258/267/348/357/456}, no 1 in R3C4

32. R1C234 = {189/279/378/459/468} (cannot be {567} because R1C23 + R2C3 cannot be {667}), no 2,5 in R1C23
32a. R1C23 + R2C3 (step 30) = {289/379/469} (cannot now be {568}) = 9{28/37/46}, 9 locked for N1

33. R9C23 = {4/5/6, 7/8/9}, R8C3 = {89} -> R7C123 and 12(3) cage must contain 1,2,3, two of 4/5/6 and one of 7/8/9
R7C123 = 12 = {129/138/156/246/345}
12(3) cage = {129/138/237/246/345} (cannot be {147/156} which clash with all combinations for R7C123)

34. R9C678 = 15 with R9C6 = {3458} and R8C7 2 less than R9C6, valid combinations = {159/258/348/357/456} (cannot be {168/249} which would make R8C7 + R9C78 {166/229}, cannot be {267} because no 2,6,7 in R9C6; note that {357} can only be [357], not [537] which would make R8C7 + R9C78 [337])

[Mike’s step 35 could have been used here. It looks as if Mike had probably made more eliminations from R1C6 and R2C7 but it works here because R1C6 = R2C7 so whatever is in R1C78 + R2C7 must be in R1C678. To avoid having to look at his walkthrough for this step, it is
"35a. 19/4 at R3C7 cannot contain both of {89}
35b. 16/3 at R1C6 cannot contain both of {89}
35c. Therefore, R3C789 and R1C78 must each contain one of {89} -> 18/3 cage at R1C2 cannot contain both of {89}
-> no 1 in R1C4, no 2 in R2C3 (see step 25)
35d. 2 no longer available in h19/3(N1), which we know must already contain a 9 (see step 30)
-> cannot also contain an 8 ({289} not available) -> no 8 in R1C23" Note that the step references are for Mike's walkthrough, not for this one.

I'm not sure why I missed that step. It’s the sort of step that I often use so I should have seen it.]

Now the hard work; interactions between the cages in N9, N7 and N8. If you don’t want to work through the details there is a summary after this step.

35. R7C789 = {167/239/257/347/356} (cannot be {149} which clashes with all combinations for R7C123)
18(3) cage = {279/369/459/468/567} (cannot be {189} because 1,8 only in R8C8, cannot be {378} which clashes with all combinations for R7C789)

35a. If R7C789 = {167} => 18(3) cage = {459} => R8C7 + R9C78 = {238}
R7C123 = {345} => R7C456 = [928] => R89C6 = {45} are consistent

35b. If R7C789 = {239} and 18(3) cage = {468} => R8C7 + R9C78 = [157] => R9C678 = [357]
If R7C789 = {239} and 18(3) cage = {567} => R8C7 + R9C78 => 1{48} => R9C678 = 3{48}
For both these cases R7C789 = {239} => R7C123 = {156} => R7C456 = [748] -> no valid combinations in R789C6 with R7C6 = 8 and R9C6 = 3 -> R7C789 cannot be {239} -> no 9 in R7C789 -> 18(3) cage cannot be {567} which clashes with all remaining combinations for R7C789

35c. If R7C789 = {257} and 18(3) cage = {369} => R8C7 + R9C78 = 1{48} => R9C678 = 3{48}
If R7C789 = {257} and 18(3) cage = {468} => R8C7 + R9C78 = 3{19} (cannot be 1{39} which would make R9C678 = 3{39}) => R9C678 = 5{19}
For both these cases R7C789 = {257} => R7C123 = {138} => R7C456 = [964] => R8C6 = 5 => R9C6 = 8 (step 24) which clashes with both these cases -> R7C789 cannot be {257} -> no 2 in R7C789

35d. R7C789 = {347} clashes with all combinations for the 18(3) cage -> R7C789 cannot be {347}

35e. If R7C789 = {356} => 18(3) cage = {279} => R8C7 + R9C78 = 1{48} => R9C678 = 3{48}
R7C789 = {356} => R7C123 = {129} => R7C456 = [748] -> no valid combinations for R789C6 with R7C6 = 8 and R9C6 = 3 -> R7C789 cannot be {356}

Summary of step 35
R7C789 = {167}, locked for R7 and N9
18(3) cage in N9 = {459}, locked for N9
R8C7 = {23} -> R9C6 = {45}(step 21)
R7C123 = {345}, locked for R7 and N7

Amazing that step 35 reduced each of those 3-cell cages to a sole combination!

36. R7C5 = 2, R9C4 = 1 (naked singles), clean-up: no 2 in R2C3

37. R89C6 = {45}, locked for C6 and N8 -> R7C6 = 8 (step 24) -> R7C4 = 9, R8C34 = [87], clean-up: no 4,8 in R2C7

38. R89C5 = {36}, locked for C5

39. 7 in R9 locked in R9C23 -> R9C234 = {67}1, no 9, naked pair {67} locked for R9 and N7

40. R89C5 = [63] (naked singles)

41. Naked pair {28} in R9C78, locked for R9 and N9 -> R9C1 = 9, R8C7 = 3 -> R89C6 = [45] (step 21 or cage sums) -> R9C9 = 4, R5C12 = [49], clean-up: no 3 in R1C6

42. 6 in C1 locked in R123C1, locked for N1, clean-up: no 5 in R1C4

43. 9 in N1 locked in R12C3
If R1C3 = 9 => R1C6 <>9 => R2C7 <>9 (step 18)
If R2C3 = 9 => R2C67 <>9
-> no 9 in R2C7 = 6 -> R1C6 = 6 (step 18)
43a. Naked pair {39} in R2C36, locked for R2

44. R6C78 = [56], R7C789 = [176] (naked singles)

45. R3C1 = 6 (hidden single in N1) -> R3C23 = 6 = {24}/[51], no 3,7, no 1 in R3C2

46. R1C6 = 6 -> R1C78 = 10 = {28}/[91], no 3,4,5, no 9 in R1C8

47. R1C234 = {37}8/[792], no 4,8 in R1C23

48. R2C2 = 8 (hidden single in N1) -> R12C1 = 6 = {15}, locked for C1 and N1

47. R7C123 = [354], R8C12 = [21], R6C23 = [31], R1C2 = 7, R9C23 = [67], R4C23 = [26], R3C23 = [42] (naked singles)

48. R3C5 = 1 (hidden single in R3) -> R12C5 = [47]

49. R3C456 = 15 (step 31) -> R3C4 = 5, R3C6 = 9 (only remaining permutation) -> R2C36 = [93], R1C3 = 3, R12C4 = [82], R12C9 = [25], R12C1 = [51], R1C78 = [91], R2C8 = 4, R3C789 = [837], R4C78 = [49], R5C89 = [23], R8C89 = [59] and R9C78 = [28] (naked singles)
Andrew's 2012 walkthrough:
Ed suggested that the archive comments about the originally posted walkthroughs are inconsistent with the SS score and that it should be solved again and another walkthrough posted.

Prelims

a) R4C23 = {17/26/35}, no 4,8,9
b) R4C78 = {49/58/67}, no 1,2,3
c) R5C12 = {49/58/67}, no 1,2,3
d) R5C34 = {29/38/47/56}, no 1
e) R5C67 = {17/26/35}, no 4,8,9
f) R6C89 = {14/23}
g) R6C23 = {13}
h) R6C78 = {29/38/47/56}, no 1
i) 11(3) cage at R1C9 = {128/137/146/236/245}, no 9
j) 24(3) cage at R7C4 = {789}
k) 11(3) cage at R7C5 = {128/137/146/236/245}, no 9

Steps resulting from Prelims
1a. Naked pair {13} in R6C23, locked for R6 and N4, clean-up: no 5,7 in R4C23, no 8 in R5C4, no 8 in R6C78
1b. Naked pair {26} in R4C23, locked for R4 and N4, no 7 in R4C78, no 7 in R5C12, no 5,9 in R5C4
1c. Naked triple {789} in 24(3) cage at R7C4, CPE no 7,8,9 in R8C56

2. 45 rule on R5 1 innie R5C5 = 8, clean-up: no 5 in R5C12, no 3 in R5C4
2a. Naked pair {49} in R5C12, locked for R5 and N4, clean-up: no 7 in R5C3, no 2,7 in R5C4, no 1 in R5C89
2b. R5C34 = [56], clean-up: no 2,3 in R5C67

3. 45 rule on N5 1 remaining innie R5C6 = 1, R5C7 = 7, clean-up: no 4 in R6C78
3a. Naked pair {23} in R5C89, locked for N6, clean-up: no 9 in R6C78
3b. Naked pair {56} in R6C78, locked for R6 and N6, clean-up: no 8 in R4C78

4. Naked pair {49} in R4C78, locked for R4 and N6 -> R46C9 = [18], R46C1 = [87]

5. 45 rule on C1234 2 innies R46C4 = 7 = [34/52]
5a. 45 rule on C6789 2 innies R46C6 = 9 = [72/54]
5b. R6C5 = 9 (hidden single in N5)
5c. 45 rule on C5 1 remaining innie R4C5 = 5, R46C4 = [34], R46C6 = [72]

6. 45 rule on N7 1 innie R8C3 = 1 remaining outie R9C4 + 7 -> R8C3 = {89}, R9C4 = {12}
6a. 24(3) cage at R7C4 = {789}, 7 locked for C4 and N8
6b. 5 in C4 only in R123C4, locked for N2

7. 7 in C5 only in 12(3) cage at R1C5 = {147/237}, no 6
7a. 11(3) cage at R7C5 = {146/236}, 6 locked for N8

8. 45 rule on N1 1 innie R2C3 = 1 remaining outie R1C4 + 1 -> R1C4 = {1258}, R2C3 = {2369}

9. 45 rule on N3 1 remaining outie R1C6 = 1 innie R2C7 -> R2C7 = {34689}

10. 45 rule on N9 1 remaining outie R9C6 = 1 innie R8C7 + 2 -> R8C7 = {1236}, R9C6 = {3458}

11. 45 rule on R1 3 innies R1C159 = 11 = {137/146/236/245}, no 9

12. 45 rule on N9 3 remaining outies R789C6 = 17 = {359/458}
12a. 45 rule on R89 3 outies R7C456 = 19 = {289/379/469/478/568}, no 1
12b. R7C456 = {289/469/478/568} (cannot be {379} = [739] which clashes with R789C6 = {359}, CCC), no 3
12c. R7C456 = {289/469/568} (cannot be {478} = [748] which clashes with R789C6 = {458}, CCC), no 7
12d. 6 of {469} must be in R7C5 -> no 4 in R7C5
[That just leaves a little analysis in step 20, the rest is straightforward.]

13. 24(3) cage at R7C4 = {789} -> R8C4 = 7
13a. Naked pair {89} in R7C4 + R8C3, CPE no 8,9 in R7C123

14. 7 in N7 only in 14(3) cage at R9C2, locked for R9
14a. 14(3) cage = {167/257} (cannot be {347} because R9C4 only contains 1,2), no 3,4,8,9
14b. R9C4 = {12} -> no 1,2 in R9C23

15. 7 in N9 only in R7C89 -> 22(4) cage at R6C9 contains 7,8 = {1678/2578/3478}, no 9
15a. 9 in N8 only in R7C46 -> R7C456 (step 12c) = {289/469}, no 5

16. Hidden killer pair 8,9 in 12(3) cage at R8C1 and R8C3 for N7, R8C3 = {89} -> 12(3) cage must contain one of 8,9 -> 12(3) cage = {129/138}, no 4,5,6, 1 locked for N7

17. 19(4) cage at R6C1 = {3457} (only remaining combination, cannot be {2467} which clashes with R7C5), 3,4,5 locked for R7 and N7
17a. 12(3) cage at R8C1 (step 16) = {129} (only remaining combination), locked for N7 -> R8C3 = 8, R7C4 = 9, R7C6 = 8, R7C5 = 2 (step 15a), R9C4 = 1, clean-up: no 2 in R2C3 (step 8), no 8 in R2C7 (step 9)
17b. 11(3) cage at R7C5 (step 7a) = {236} (only remaining combination), 3 locked for C5 and N8
17c. Naked pair {67} in R9C23, locked for R9 -> R89C5 = [63]

18. 45 rule on R9 2 remaining innies R9C19 = 13 -> R9C1 = 9, R9C9 = 4, R9C6 = 5, R8C6 = 4, R8C7 = 3 (cage sum), R5C12 = [49], clean-up: no 3 in R1C6, no 4 in R2C7 (both step 9)
18a. R8C89 = {59} (hidden pair in N9)

19. Naked triple {369} in R2C367, locked for R2

20. 45 rule on N1 3 innies R1C23 + R2C3 = 19 = {379/469} (cannot be {289} because 18(3) cage at R1C2 cannot be [828], cannot be {478} because R2C3 only contains 3,6,9, cannot be {568} because 5,8 only in R1C2), no 1,2,5,8, 9 locked for N1

21. R2C2 = 8 (hidden single in N1), R12C1 = 6 = {15}, locked for C1 and N1 -> R7C123 = [354], R8C12 = [21], R3C1 = 6
[Clean-ups and routine placements omitted from this stage.]

22. R1C23 + R2C3 (step 20) = {379} (only remaining combination) -> R1C2 = 7, R12C3 = {39}, locked for N1 -> R3C23 = [42]

23. 45 rule on R12 3 outies R3C456 = 15 = {159/357} -> R3C4 = 5, R2C4 = 2, R2C3 = 9 (cage sum), R2C67 = [36], R3C6 = 9, R3C7 = 8, R3C89 = {37}, locked for R3 and N3 -> R12C9 = [25], R3C5 = 1

and the rest is naked singles.

I'll rate this walkthrough at 1.25. I used interactions between two overlapping hidden cages in N8.


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PostPosted: Thu Jun 19, 2008 12:15 am 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 46v2 Lite by Ruud (April 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3840:3073:3073:3073:3844:4869:4869:4869:4104:3840:3840:3595:3595:3844:4366:4366:4104:4104:4626:4626:4626:3595:3844:4366:4888:4888:4888:4626:1308:1308:8734:8734:8734:4385:4385:4888:2084:2084:4390:4390:8734:1833:1833:2347:2347:4653:3374:3374:8734:8734:8734:2099:2099:5685:4653:4653:4653:3641:5690:2619:5685:5685:5685:5183:5183:3641:3641:5690:2619:2619:2118:2118:5183:3401:3401:3401:5690:3917:3917:3917:2118:
Solution:
+-------+-------+-------+
| 1 2 7 | 3 5 6 | 9 4 8 |
| 8 6 9 | 4 2 7 | 1 5 3 |
| 3 4 5 | 1 8 9 | 2 6 7 |
+-------+-------+-------+
| 6 3 2 | 7 1 5 | 8 9 4 |
| 7 1 8 | 9 4 2 | 5 3 6 |
| 5 9 4 | 6 3 8 | 7 1 2 |
+-------+-------+-------+
| 2 8 3 | 5 6 1 | 4 7 9 |
| 4 7 1 | 8 9 3 | 6 2 5 |
| 9 5 6 | 2 7 4 | 3 8 1 |
+-------+-------+-------+
Quote:
Ruud, lead-in: this time I have a V2 for Assassin 46 which is slightly easier than V1. It does not require detection of conflicting combinations...Still a very enjoyable puzzle... I think. :roll:
mhparker: One should never underestimate any of Ruud's killers, particularly when he posts comments on them with the rolling eyes emoticon!
Ruud, lead-in: Being accused several times of posting V2's which are way too difficult
Para: Generally you open with the message that your V2 was too difficult to put one the site. We never complain :D (we wouldn't dare to :evil: :twisted:). We are usually just stunned for a moment. But we never refused a challenge, don't think we ever will..Just messing around. ;)
mhparker: this particular killer one of the most interesting I have ever seen
Walkthrough by mhparker without conflicting combinations:
One should never underestimate any of Ruud's killers, particularly when he posts comments on them with the rolling eyes emoticon!

Ruud didn't just post a puzzle for us to solve. He was also giving us a test to see what we made of it, and we've all been sleeping...

All this stuff about V2's being way too difficult may well be true, but I strongly suspect that it's only half the story. Therefore, this post will also only cover half the story. The rest will come in a second post, so stay tuned!

In the meantime, here's the walkthrough:

1. 19/3 at R1C6: no 1
2. 5/2 at R4C2 = {14|23}
3. 17/2 at R4C7 = {89} -> no 8,9 elsewhere in R4 or N6
4. 8/2 at R5C1: no 4,8,9
5. 17/2 at R5C3 = {89} -> no 8,9 elsewhere in R5
6. 7/2 at R5C6: no 7 (8,9 already eliminated)
7. 9/2 at R5C8 = {27|36|45} (no 1)
8. 13/2 at R6C2: no 1,2,3
9. 8/2 at R6C7: no 4 (8,9 already eliminated)
10. 22/3 at R7C5 = {(58|67)9} -> no 9 elsewhere in C5 or N8
11. 10/3 at R7C6: no 8,9
12. 20/3 at R8C1: no 1,2
13. 8/3 at R8C8 = {1(25|34)} -> no 1 elsewhere in N9

14. Innies N5: R5C46 = 11/2 = [83|92] -> R5C7 = {45}
15. Innie R5: R5C5 = 4 -> R5C7 = 5 -> R5C6 = 2 -> R5C4 = 9 (step 14) -> R5C3 = 8
16. 1 in R5 locked in 8/2 at R5C1 = {17} -> no 1,7 elsewhere in R5 or N4
17. 9/2 at R5C8 = {36} -> no 3,6 elsewhere in N6
18. 13/2 at R6C2 = {49} (only combo possible) -> no 4,9 elsewhere in R6 or N4
19. 8/2 at R6C7 = {17} (only combo possible) -> no 1,7 elsewhere in R6 or N6

20. Naked Single (NS) at R6C9 = 2 -> R4C9 = 4

21. 2 in C5 locked in 15/3 at R1C5 -> {2(58|67}) (no 1,3) -> no 2 elsewhere in N2

22. Hidden Single (HS) in C5 at R4C5 = 1
23. HS in C5 at R6C5 = 3

24. Outie R123: R4C1 = 6 -> R6C1 = 5

25a. Outies N1: R123C4 = 8/3 = {134} (2 unavailable) -> no 1,3,4 elsewhere in C4 or N2
25b. 14/3 at R2C3 = {(19|37)4} -> no 2,5,6 in R2C3
Edit: Should have noticed here that 4 is locked in R23C4, allowing elimination of 4 in R2C3 and R1C4, making step 42 unnecessary.

26a. Outies N9: R789C6 = 8/3 = {134} (2 unavailable)
26b. 10/3 at R7C6 = {136} (only permutation possible) -> R8C7 = 6, R78C6 = {13}, R9C6 = 4 (step 26a)
26c. R9C78 = {29|38} (no 5,7)

27a. 7 in N9 locked in R7 -> no 7 elsewhere in R7
27b. Split 20/3 cage at R7C789 = {(49|58)7} (no 3)

28. Innies R7: R7C456 = 12/3 = {1(29|56)} ({138} unavailable, since {13} only in R7C6)
-> R7C6 = 1, no 8 in R7C45, R8C6 = 3

29a. 3 in R7 locked in N7 -> no 3 elsewhere in N7
29b. Split 13/3 cage at R7C123 = {(28|46)3} (no 9)

30. 20/3 at R8C1 = {(49|58)7} -> no 7 elsewhere in N7

31. 13/3 at R9C2 = {(17|26)5} (no 8,9) (3,4 unavailable) -> no 5 elsewhere in R9

32a. 14/3 at R7C4: no 9 in R8C3 (due to {13} unavailable in R78C4)
32b. 9 in N7 now locked in 20/3 at R8C1 = {479} -> no 4 elsewhere in N7
32c. 4 in R7 now locked in R7C78 -> no 4 in R8C8
32d. 8/3 at R8C8 = {125} (4 no longer available) -> no 2,5 elsewhere in N9
32e. NS at R9C9 = 1 -> R8C9 = 5, R8C8 = 2
32f. NS at R8C3 = 1
32g. Split 11/2 cage at R9C78 = {38} (no 9) -> no 8 elsewhere in R9 or N9
32h. Split 20/3 cage at R7C789 = {479} -> no 9 in R7C5

33. Innie N8: R9C4 = 2

34. R9C23 = {56} -> no 6 in R7C23

35. <step deleted>

36. Innies R9: R9C15 = 16/2 = {79}

37. Naked Pair (NP) on {23} in C3 at R47C3 -> no 2,3 elsewhere in C3

38a. Split 12/3 cage at R3C123: no 8 (due to {138} being unavailable in R3C3)
38b. 12/3 cage at R1C2: no 8 (due to {138} being unavailable in R1C3)
38c. 8 in N1 now locked in 15/3 cage at R1C1 = {(16|25|34)8} (no 7,9)

39. 7 in C3 locked in N1 -> no 7 elsewhere in N1
40. 8 in C9 locked in N3 -> no 8 elsewhere in N3

41. 19/3 at R1C6: no 5 (would imply {568}, impossible since R1C7 has none of these digits as candidates)

42. Innie/outie difference (i/o diff.) N1: R2C3 = R1C4 + 6 -> R1C4 = {13}, R2C3 = {79}
43. I/o diff. N3: R1C6 = R2C7 + 5 -> R2C7 = {1234}

44. 12/3 at R1C2: no 1 in R1C2 (would force 3 into R1C4, requiring 8 in R1C3 - unavailable)

45a. Innies R3: R3C456 = 18/3 -> no 2 in R3C5 (since max. sum of R3C46 = 13)
45b. Similarly, no 5 in R3C6 (since max. sum of R3C45 = 12)

46a. Innies R1: R1C159 = 14/3 -> no 8 in R1C15 (due to {1245} unavailable in R1C9)
46b. 8 in N1 now locked in R2C12 -> no 8 elsewhere in R2

47. 15/3 cage at R1C5: no 5 in R3C5 (due to {3489} unavailable in R12C5)

48. Innies N1: R1C23+R2C3 = 18/3 -> no 9 in R1C2 (since sum of R12C3 >= 11)

49. 12/3 at R1C2: no 3 in R1C2 (would imply {138}, impossible since R1C3 has none of these digits as candidates)

50. Split 12/3 cage at R3C123: no 9 in R3C12 (would require either of {12} in R3C3 - unavailable)

51. 9 in C1 now locked in R89C1 -> no 9 in R8C2

52. HS in C2 at R6C2 = 9 -> R6C3 = 4

53. Split 12/3 cage at R3C123: no 5 in R3C2 (would force R3C3 to {79} - cage sum already reached)

54. 16/3 cage at R1C9: no 9 in R2C8 (sum of R12C9 >= 9 -> R2C8 <= 7)

55. 15/3 at R1C1: min. sum of R2C12 = 11 (due to R1C1 <= 4) -> no 1,2 in R2C12

56a. Split 15/3 at R3C789: no 9 in R3C78 (due to {1245} unavailable in R3C9)
56b. Split 15/3 at R3C789: no 1 in R3C8 (since none of {5689} in R3C7)
56c. Split 15/3 at R3C789: no 3 in R3C8 (since {45} unavailable in both of R3C79)

57. I/o diff. C9: R357C9 - R2C8 = 17 -> no 7 in R2C8 (since. max. sum of R357C9 = 23 -> R2C8 <= 6)

--- Now for the hard-to-find move that cracks the puzzle... ---

58. 16/3 at R1C9 = {169|178|349|358|367} -> must have one of {13}
Innies N3: R1C78+R2C7 = 14/3 = {149|167|239|347} (5,8 unavailable) -> must have one of {13}
-> 16/3 at R1C9+R2C89 and 14/3 at R1C78+R2C7 form killer pair on {13} -> no 1,3 elsewhere in N3 (i.e., in R3C79).

59a. Split 15/3 at R3C789 must contain a digit under 5 -> R3C7 = 2
59b. Split cage R3C89 = 13/2 = {58|67} -> no 9 in R3C9

60. HS in R2 at R2C5 = 2

61. Split 12/3 cage at R3C123 = {(17|35)4} -> no 4 elsewhere in R3 or N1, no 9 in R3C3

62. HS in R3 at R3C6 = 9 -> R2C67 = [53|71]

63. HS in C4 at R2C4 = 4

64a. R2C7 now {13} -> R1C6 = {68} (step 43)
64b. 19/3 at R1C6: no 3 in R1C78 (since {79} both unavailable in R1C6)

65. R3C45 = [18|36] (innies, see step 45a)

66. NP on {68} in N2 at R1C6 and R3C5 -> no 6 in R1C5

67. 15/3 at R1C1 = {(16|25)8} (no 3) -> R2C1 = 8

68. HS in C2 at R7C2 = 8

69. 3 in N1 locked in split 12/3 cage at R3C123 = {345} -> R3C12 = {34}, R3C3 = 5

--- and the puzzle now ends with an avalanche of naked singles... ---
Spectacular.. Unique Rectangles (UR) and deadly patterns by mhparker:
I said above that I wanted to make a second post on this puzzle.

Before I start, I want to say that my walkthrough for this Lite version deliberately avoided using conflicting combinations, because Ruud assured us that the puzzle can be completed without them. Furthermore, although there's nothing spectacular in it (it's all text book stuff, as would be done by an automated solver), it's main purpose is to be used as a basis for what I want to say right now, which is what makes this particular killer one of the most interesting I have ever seen.

Let's start with the puzzle state immediately after step 57 in my walkthrough, which is as far as SumoCue v1.30 gets with this puzzle:

Code:
.-------.-----------------------.-------.-----------------------.-------.
| 1234  | 2456    5679    13    | 2567  | 6789    23479   34679 | 36789 |
|       '-------.---------------:       :---------------.-------'       |
| 348     34568 | 79      134   | 2567  | 5679    1234  | 13456   3679  |
:---------------'-------.       |       |       .-------'---------------:
| 1234    1234    579   | 134   | 678   | 6789  | 1237    567     36789 |
|       .---------------+-------'-------'-------+---------------.       |
| 6     | 23      23    | 57      1       57    | 89      89    | 4     |
:-------'-------.-------'-------.       .-------'-------.-------'-------:
| 17      17    | 8       9     | 4     | 2       5     | 36      36    |
:-------.-------'-------.-------'       '-------.-------'-------.-------:
| 5     | 9       4     | 68      3       68    | 17      17    | 2     |
|       '---------------+-------.-------.-------+---------------'       |
| 238     238     23    | 56    | 56    | 1     | 479     479     79    |
:---------------.-------'       |       |       '-------.---------------:
| 479     47    | 1       78    | 789   | 3       6     | 2       5     |
|       .-------'---------------:       :---------------'-------.       |
| 79    | 56      56      2     | 79    | 4       38      38    | 1     |
'-------'-----------------------'-------'-----------------------'-------'


From this position, there is namely a spectacular sequence of moves based on Unique Rectangles (UR) and deadly patterns, which are not mentioned in most Killer solving guides.

By the way, for anyone not familiar with UR-based techniques, you can find more information here:

http://www.sudocue.net/guide.php#UR

However, please bear in mind that all discussions relating to UR on the web are based on regular (vanilla) Sudoku. For Killer Sudoku, there is an important extra requirement, namely:

All (pre-defined) cages (if any) overlapping with the cells of the deadly pattern must contain exactly two UR cells of opposite polarity

Otherwise, you can't (in general) exchange the two digits of the deadly pattern to get an alternate solution, because then the cage constraints (such as the cage sums) will not be met.

UR Type 1 ("Unique Corner")

Refer now to the above candidate grid, in particular the four cells R47C23. All apart from one of the cells contains only the candidates {23}. Only R7C2 has a single extra candidate, 8. In order to avoid the so-called deadly rectangle (where the two participating digits can be freely interchanged in the solution to provide an equally valid alternate solution), R7C2 must take the value of the extra candidate, so we can directly place an 8 there, which creates a hidden single in C1 at R2C1 = 8. After making both of these moves, we end up with the following grid:

Code:
.-------.-----------------------.-------.-----------------------.-------.
| 1234  | 2456    5679    13    | 2567  | 6789    23479   34679 | 36789 |
|       '-------.---------------:       :---------------.-------'       |
| 8       3456  | 79      134   | 2567  | 5679    1234  | 13456   3679  |
:---------------'-------.       |       |       .-------'---------------:
| 1234    1234    579   | 134   | 678   | 6789  | 1237    567     36789 |
|       .---------------+-------'-------'-------+---------------.       |
| 6     | 23      23    | 57      1       57    | 89      89    | 4     |
:-------'-------.-------'-------.       .-------'-------.-------'-------:
| 17      17    | 8       9     | 4     | 2       5     | 36      36    |
:-------.-------'-------.-------'       '-------.-------'-------.-------:
| 5     | 9       4     | 68      3       68    | 17      17    | 2     |
|       '---------------+-------.-------.-------+---------------'       |
| 23      8       23    | 56    | 56    | 1     | 479     479     79    |
:---------------.-------'       |       |       '-------.---------------:
| 479     47    | 1       78    | 789   | 3       6     | 2       5     |
|       .-------'---------------:       :---------------'-------.       |
| 79    | 56      56      2     | 79    | 4       38      38    | 1     |
'-------'-----------------------'-------'-----------------------'-------'


From here, we can independently make a second UR move, as follows:

UR Type 4 ("Unique Pair")

Referring to the above candidate grid, in particular the four cells R89C15, we can see that the digit 9 has only two candidate positions in R8, and thus must be located in either of R8C15, the two top corners of the UR. Therefore, neither of these corners may contain the digit 7, because this would form the deadly rectangle on {79} in R89C15. Therefore, we can eliminate candidate 7 from R8C15, giving the following grid:

Code:
.-------.-----------------------.-------.-----------------------.-------.
| 1234  | 2456    5679    13    | 2567  | 6789    23479   34679 | 36789 |
|       '-------.---------------:       :---------------.-------'       |
| 8       3456  | 79      134   | 2567  | 5679    1234  | 13456   3679  |
:---------------'-------.       |       |       .-------'---------------:
| 1234    1234    579   | 134   | 678   | 6789  | 1237    567     36789 |
|       .---------------+-------'-------'-------+---------------.       |
| 6     | 23      23    | 57      1       57    | 89      89    | 4     |
:-------'-------.-------'-------.       .-------'-------.-------'-------:
| 17      17    | 8       9     | 4     | 2       5     | 36      36    |
:-------.-------'-------.-------'       '-------.-------'-------.-------:
| 5     | 9       4     | 68      3       68    | 17      17    | 2     |
|       '---------------+-------.-------.-------+---------------'       |
| 23      8       23    | 56    | 56    | 1     | 479     479     79    |
:---------------.-------'       |       |       '-------.---------------:
| 49      47    | 1       78    | 89    | 3       6     | 2       5     |
|       .-------'---------------:       :---------------'-------.       |
| 79    | 56      56      2     | 79    | 4       38      38    | 1     |
'-------'-----------------------'-------'-----------------------'-------'


As if that's not enough, we can perform yet another UR move here!:

"Killer" UR

This is a variant that is particular to Killer Sudoku, and which I've never seen documented anywhere.

Take a look at the 12/3 cage at R1C2 in the above grid. Note that, because of the candidates remaining in R1C23, if R1C2 were 5, this would force R1C3 to 6 (i.e., R1C234 = [561]). Similarly, if R1C2 were 6, this would force R1C3 to 5 (i.e., R1C234 = [651]). Because of the additional links on 5 and 6 between R1C2 and R9C2 and between R9C2 and R9C3, it can be seen that placing either of {56} in R1C2 would force a deadly rectangle on {56} in R19C23. Therefore we can eliminate both 5 and 6 from R1C2. Thereafter, we can remove candidate 6 from R1C3, because no possible permutations now use it (obsolete candidate).

Additionally, this opens up the following moves:
    1. Hidden single in C3 at R9C3 = 6 -> R9C2 = 5
    2. Hidden single in C2 at R2C2 = 6 -> R1C1 = 1 (last digit in cage)
    3. Naked single at R1C4 = 3
    4. R23C4 = {14} -> R2C3 = 9
After performing these additional moves, the candidate grid now becomes:

Code:
.-------.-----------------------.-------.-----------------------.-------.
| 1     | 24      57      3     | 2567  | 6789    2479    4679  | 6789  |
|       '-------.---------------:       :---------------.-------'       |
| 8       6     | 9       14    | 257   | 57      1234  | 1345    37    |
:---------------'-------.       |       |       .-------'---------------:
| 234     234     57    | 14    | 678   | 6789  | 1237    567     36789 |
|       .---------------+-------'-------'-------+---------------.       |
| 6     | 23      23    | 57      1       57    | 89      89    | 4     |
:-------'-------.-------'-------.       .-------'-------.-------'-------:
| 7       17    | 8       9     | 4     | 2       5     | 36      36    |
:-------.-------'-------.-------'       '-------.-------'-------.-------:
| 5     | 9       4     | 68      3       68    | 17      17    | 2     |
|       '---------------+-------.-------.-------+---------------'       |
| 23      8       23    | 56    | 56    | 1     | 479     479     79    |
:---------------.-------'       |       |       '-------.---------------:
| 49      47    | 1       78    | 89    | 3       6     | 2       5     |
|       .-------'---------------:       :---------------'-------.       |
| 79    | 5       6       2     | 79    | 4       38      38    | 1     |
'-------'-----------------------'-------'-----------------------'-------'


Unique "Swordfish"

Last but not least, take a look at the 6 cells R3C12, R4C23 and R7C13. These form a swordfish pattern, which, if every cell were to contain the same two candidates ({23} in this case), would be just as deadly as the deadly rectangle. This means that R3C12 cannot be {23}, ruling out the combination {237} for the split 12/3 cage at R3C123 (because neither of the digits {23} are available in the remaining cell R3C3). Since the only other combination for these 3 cells is {345}, this means that we can immediately eliminate the candidate 2 from R3C12 and place a "5" in R3C3.

After applying these UR-based techniques (the last one being optional), the puzzle becomes easily solvable via singles.

The sheer number and type of UR-based moves makes this puzzle for me a case study in uniqueness tests as applied to Killer Sudoku.
UR Type 2 & 3 by mhparker:
Wow - very informative Mike. Amazing to find so many of these moves in the one puzzle. (And yes - I was snoozing in LoL land :oops: )

Another example of the Killer UR elimination: which absolutely busts a very difficult puzzle wide open is in my walk-through for Assassin 44V2 step 40a : a move which Para saw as a short-cut to solve that puzzle. Richard and Para also saw the Type 1 UR in Assassin 42V2 step 50. [edit: links updated]

These techniques are clearly very powerful. Thanks again for the tut.

Ed


mhparker
Thanks for the links, Ed.

In addition to the UR's discussed above and in the other walktroughs, we need to keep a lookout for the Type 2's ("Unique Side"), where the same extra candidate appears on two corners, allowing for elimination of that candidate digit in all common peers.

Type 3's ("Unique Subset") are also conceivably useful, especially in the simplest case where only two extra candidate digits are involved, allowing them to form a killer pair with another peer cell, cage or split cage. Triples and beyond are probably not worth bothering about at first, unless we trip over them in a real example.

The swordfish pattern (abb. US?) is something that's probably quite rare, so I suspect we don't need to look out for it. But I was fascinated to see it for real, so couldn't resist mentioning it! Nevertheless, I should have referred to it as the deadly swordfish instead - sounds much better! :)
Walk-through by Andrew with some conflicting combinations:
Like PsyMar I've only come across one sudoku that had multiple solutions. It was a vanilla sudoku in a newspaper, only the 3rd one that I did, some months before I came across any sudoku websites. It had 4 solutions which they eventually admitted after initially only giving one solution.

Here is my walkthrough for Assassin 46 Light. Fairly similar to the route that Mike took. The main differences are that I used the innie/outie differences for N1, N3, N7 and N9 earlier and more often and I wasn't quite as rigorous in responding to Ruud's "It does not require detection of conflicting combinations." as you will see in the brief discussion after step 30.

1. R4C23 = {14/23}

2. R4C78 = {89}, locked for R4 and N6

3. R5C12 = {17/26/35}, no 4,8,9

4. R5C34 = {89}, locked for R4

5. R5C67 = {16/25/34}, no 7

6. R5C89 = {27/36/45}, no 1

7. R6C23 = {49/58/67}, no 1,2,3

8. R6C78 = {17/26/35}, no 4

9. R1C678 = {289/379/469/478/568}, no 1

10. 20(3) cage in N7 = {389/479/569/578}, no 1,2

11. R789C5 = 9{58/67}, 9 locked for C5 and N8

12. 10(3) cage in N89 = {127/136/145/235}, no 8,9

13. 8(3) cage in N9 = 1{25/34}, 1 locked for N9

14. 45 rule on R5 1 innie R5C5 = 4, clean-up: no 3 in R5C67, no 5 in R5C89

15. 45 rule on N5 2 innies R5C46 = 11 = [92] (only remaining combination) -> R5C3 = 8, R5C7 = 5, clean-up: no 3,6 in R5C12, no 7 in R5C89, no 5 in R6C23, no 3 in R6C78
15a. R5C89 = {36}, locked for N6, clean-up: no 2 in R6C78
15b. R5C12 = {17}, locked for N4, clean-up: no 4 in R4C23, no 6 in R6C23
15c. R6C23 = {49}, locked for N4 and R6
15d. R4C23 = {23}, locked for R4 and N4
15e. R46C1 = {56}, locked for C1
15f. R6C78 = {17}, locked for R6 and N6

16. R46C9 = [42] (naked singles)
16a. 8(3) cage in N9 = 1{25/34} (step 13), 2,4 only in R8C8 -> no 1,3,5 in R8C8
16b. 1 in N9 locked in R89C9, locked for C9

17. 45 rule on R123 2 outies R4C19 = 10 -> R4C1 = 6 -> R6C1 = 5

18. 45 rule on N1 3 outies R123C4 = 8 = 1{25/34}, 1 locked for C4 and N2

19. 45 rule on N3 3 outies R123C6 = 22 = 9{58/67}

20. 45 rule on N7 3 outies R789C4 = 15

21. 45 rule on N9 3 outies R789C6 = 8 = {134}, locked for C6 and N8

22. 1,4 in C4 locked in R123C4 = {134}, locked for C4 and N2, no 2,5

23. R4C5 = 1, R6C5 = 3 (hidden singles in C5)
23a. 45 rule on C1234 2 innies R46C4 = 13 = [58/76]
23b. 45 rule on C6789 2 innies R46C6 = 13 = [58/76]

24. 45 rule on N1 1 innie R2C3 – 6 = 1 outie R1C4 -> R1C4 = {13}, R2C3 = {79}

25. 45 rule on N3 1 outie R1C6 – 5 = 1 innie R2C7 -> R1C6 = {6789}, R2C7 = {1234}

26. 45 rule on N7 1 outie R9C4 – 1 = 1 innie R8C3 -> R8C3 = {14567}

27. 45 rule on N9 1 innie R8C7 – 2 = 1 outie R9C6 -> R8C7 = {36}, R9C4 = {14}

28. 3 in N8 locked in R78C6 -> no 3 in R8C7
28a. 10(3) cage in N89 = {13}6 -> R78C6 = {13}, R8C7 = 6, R9C6 = 4 (step 27), clean-up: no 7 in R9C4

29. R9C6 = 4 -> R9C78 = 11 = {29/38}, no 5,7

30. 7 in N9 locked in R7C789, locked for R7, R7C789 = 7{49/58}, no 3

At this stage one could use 45 rule on R9 3 innies R9C159 = 17 = {179/368} (cannot be {359} which clashes with R9C78) … but Ruud said “It does not require detection of conflicting combinations.” (I hope that steps 42, 46b and 56a are not considered to be conflicting combinations; I didn’t feel that they were although maybe 56a is) so

31. 45 rule on R89 3 outies R7C456 = 12 = {129/156} (cannot be {138} because 1,3 only in R7C6), no 8 = [291]/{56}1 -> R7C6 = 1, R8C3 = 3

32. 3 in R7 locked in R7C123, locked for N7, R7C123 = 3{28/46}, no 9

33. 20(3) cage in N7 = {479/578} = 7{49/58}, 7 locked for N7, clean-up: no 8 in R9C4
33a. 5 only in R8C2 -> no 8 in R8C2

34. R9C234 = {256} (only remaining combination), locked for R9, clean-up: no 9 in R9C78

35. R9C78 = {38}, locked for R9 and N9 -> R9C9 = 1, R8C9 = 5, R8C8 = 2, clean-up: no 6 in R9C4
35a. No 8 in R8C1 (step 33)

36. R7C789 = {479}, locked for R7 -> no 6 in R7C123 (step 32), no 2 in R7C4 (step 31)

37. R8C3 = 1 (hidden single in R8) -> R9C4 = 2 (step 26)

38. Naked pair {23} in R47C3, locked for C3

39. 8 in C9 locked in R123C9, locked for N3

40. 7 in C3 locked in R123C3, locked for N1

41. R1C678 = {289/379/469/478} (cannot be {568} because no 5,6,8 in R1C7) [2/3/4], no 5

42. 45 rule on N1 3 innies R1C23 + R2C3 = 18 and remembering R2C3 – 6 = R1C4 (step 24)
42a. R1C234 = {129/156/237/345} (cannot be {138} because no 1,3,8 in R1C3, cannot be {147} because R1C23 + R2C3 would then be {477}, cannot be {246} because no 2,4,6 in R1C4), no 1,3,8,9 in R1C2

43. 16(3) cage in N3 = {169/178/349/358/367} (cannot be {457} because 4,5 only in R2C8), no 9 in R2C8

44. 45 rule on R12 3 outies R3C456 = 18 = {189/369/378/459/468} (cannot be {279} because no 2,7,9 in R3C4, cannot be {567} because no 5,6,7 in R3C4), no 2, no 5 in R3C6

45. R3C789 = {159/168/258/267/357} [1/2/3], no 9 in R3C7, no 1,3,9 in R3C8

46. R1C6 – 5 = R2C7 (step 25)
46a. R1C678 = {289/379/469/478} (step 41)
46b. R1C678 cannot be [946] because R1C78 + R2C7 would then be [464] -> no 6 in R1C8

47. R1C678 has 2/3/4 in R1C78 (step 41), R3C789 has 1/2/3 (step 45), R2C7 = {1234} -> 16(3) cage in N3 (step 43) = {169/178/358/367} (cannot be {349}), no 4

48. 4 in N3 locked in R1C78 + R2C7
48a. 45 rule on N3 3 innies R1C78 + R2C7 = 14 = 4{19/37}, no 2 -> no 7 in R1C6 (step 25)
48b. R1C678 = {379/469/478}

49. R3C7 = 2 (hidden single in N3) -> R3C89 = 13 = [58]/{67}, no 3,9

50. 1 in N3 locked in R2C78, locked for R2

51. R3C123 = {147/345} = 4{17/35} (cannot be {138} because no 1,3,8 in R3C3), no 8,9, 4 locked for R3 and N1

52. R2C4 = 4 (hidden single in N2), clean-up: no 9 in R1C6 (step 25)

53. R3C6 = 9 (hidden single in R3)
53a. Naked pair {68} in R16C6, locked for C6
53b. R3C456 (step 44) = [189/369], no 5,7 in R3C5

54. R123C5 = 2{58/67}, 2,5,7 only in R12C5 -> no 6,8 in R12C5

55. 4 in N3 locked in R1C78 = 4{69/78}, no 3

56. R1C78 must contain 7/9 (step 55), R3C89 must contain 7/8 (step 49) -> 16(3) cage in N3 must contain 7/8/9 = {169/358/367} (cannot be {178})
56a. No 7 in R2C8 because R12C9 = {36} would clash with R5C9

57. 8 in N1 locked in 15(3) cage = {168/258}, no 3,9
57a. 5,6 only in R2C2 -> no 2,8 in R2C2
57b. 8 in N1 locked in R12C1, locked for C1

58. R7C2 = 8 (hidden single in R7)

59. Naked pair {56} in R29C2, locked for C2 -> R1C2 = 2, R2C1 = 8, R1C1 = 1, R2C2 = 6 (step 57), R9C23 = [56]

60. Naked pair {34} in R3C12, locked for R3 -> R3C3 = 5 (step 51), R3C4 = 1, R1C4 = 3, R12C3 = [79] (cage sums)

60. R2C5 = 2, R2C6 = 7 (hidden singles in N2)

61. R1C78 = {49} -> R1C6 = 6

and the rest is naked singles


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PostPosted: Thu Jun 19, 2008 12:39 am 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 47 by Ruud (Apr 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:5376:2305:2305:2305:5380:4101:4101:4101:4616:5376:5376:4619:5380:5380:5380:3855:4616:4616:2834:5376:4619:4885:5654:2071:3855:4616:5402:2834:2834:4619:4885:5654:2071:3855:5402:5402:6948:6948:6948:4885:5654:2071:3114:3114:3114:6948:2350:2350:2350:5654:5938:5938:5938:3114:3894:3894:4920:4920:1082:2875:2875:2109:2109:3135:3135:3135:4920:1082:2875:5445:5445:5445:2376:2376:8522:8522:8522:8522:8522:847:847:
Solution:
+-------+-------+-------+
| 6 3 4 | 2 5 7 | 1 8 9 |
| 2 8 7 | 1 6 9 | 5 4 3 |
| 1 5 9 | 8 4 3 | 7 2 6 |
+-------+-------+-------+
| 4 6 2 | 5 9 1 | 3 7 8 |
| 3 9 8 | 6 7 4 | 2 5 1 |
| 7 1 5 | 3 2 8 | 6 9 4 |
+-------+-------+-------+
| 8 7 6 | 9 1 2 | 4 3 5 |
| 9 2 1 | 4 3 5 | 8 6 7 |
| 5 4 3 | 7 8 6 | 9 1 2 |
+-------+-------+-------+
Quote:
CathyW: I managed to solve it Very Happy - took about 30 minutes after using a few higher level standard sudoku techniques
PsyMar: I used a couple fancy techniques, but not many
Andrew: About the same level as Assassin 46 Light..I managed to solve it by standard killer techniques although it did need a bit more thought at one stage
mhparker: (this) Assassin (is a) One-Trick Ponie
Andrew: I felt that (it was) still (an) Assassin but at the easy end of the difficulty range...must have taken me at least two hours
Walkthrough by PsyMar:
Here's a walkthrough for version 1. I used a couple fancy techniques, but not many. I'll tackle 1.5 in a bit.

Note: This walkthrough should work without any preliminary steps.
Note 2: Numbers between {} are unordered; numbers between [] are left to right first, then top to bottom.
i.e. theoretically a cage shaped
Code:
   9
7 563
824 5

would be [975638245].
(There's been some confusion on that at least once in the past, so I just thought I should make it part of a standard walkthrough header.)
Note 3: () surround my comments, | means or, ! means not (most commonly != means not equal)
Note 4. .. indicates a range of values
Everything else is pretty standard.

1. sole combination for 3/2 in n9 = {12} pair -> elim 1|2 from rest of n9 & r9
2. sole combination for 8/2 in n9 = {35} pair (not 17 or 26 due to step 0a) -> elim 3|5 from rest of n9 & r7
3. sole permutation for 4/2 in n8 = [13]
4. sole permutation 11/3 in r34 = [245]
5. sole combination for 21/3 in n9 = {678} triple -> elim 6|7|8 from rest of r8 and n9
6. combinations for 12/3 in n7 = {129} triple -> elim 1|2|9 from rest of n7 and r8 -> r8c4 = 4 (naked single)
7. r7c4 = 9 (hidden single) -> r7c3 = 6
8. sole combination for 9/2 in n7 = {45} pair -> elim 4|5 from rest of n7/r9
9. r9c3 = 3 and r9c7 = 9 (hidden singles)
10. sole combination for 23/3 in r6 = {689} triple -> elim 6|8|9 from rest of r6
11. sole combination for 19/3 in c4 = {568} triple -> elim 5|6|8 from rest of c4 -> r9c4 = 7 (naked single)
12. innies of r1234 = r2c4 = 1
13. combinations for 15/3 in n7 = {258|357} (cannot be 168 or 267 due to r68c7; these would cause four of {678} in the column, which can't happen) -> no 1 or 6; elim 5 from rest of c7; forms killer triple on {678} with r68c7 -> elim 6|7|8 from r15c7
14. r1269c6 = naked quad {6789} in c6 -> elim from rest of c6
15. 7 of c6 locked in n2 -> elim from rest of n2
16. combinations for 9/3 in r1 = {126|135|234} -> no 7..9; forms killer triple {123} with r1c7 -> no 1|2|3 in r1c15689
17. combinations for 9/3 in r6 = {135|234} (not 126 due to 23/3 in r6 eliming 6) -> no 6..9; elim 3 from rest of r6
18. combinations for 18/3 in c3 = {189|279} (cannot be 459 as then r168c3 all set to {12} -- can't have 3 choices from two values in one column) -> no 4|5; elim 9 from rest of c3; forms killer pair {12} with r8c3 -> elim 1|2 from r156c3
19. combinations for 27/4 in n4 = {3789|4689|5679} -> no 1|2; elim 9 from rest of n4
20. 9 of c3 locked in n1 -> elim from rest of n1
21. outies-innies of c89 = r7c58-r8c16 = -7; r7c58 >= 7 -> r8c16 >= 14 && r8c6 <= 9 -> r8c1 >= 5 -> r8c1 != 4
22. 12/4 in n6 = {1236|1245} -> no 7..9; elim 1|2 from rest of n6
23. r16c3 = {45} naked pair -> elim 4|5 from r5c3
24. By combinations, only one of {456} in 9/3 in r1; this is in r1c3; thus no 4|5|6 in r1c2
25. By combinations, only one of {456} in 9/3 in r6; this is in r6c3; thus no 4|5|6 in r6c2
26. combinations for 21/3 in r34 = {489|579|678} -> no 1..3
27. outies of n36 = r16c6 = 15/2 = [69|78|96]
28. Outies-innies of n6 = r3c9+r6c6-r4c7 = 11; r6c6+r3c9 <= 17 -> r4c7 <= 6 -> r4c7 = {35} -> r6c6+r3c9 = 14|16 - r3c9+r6c6 = [59|68|79|86] -> r3c9 = {5678}
29. innies of c6789 = r29c6 = 15/2 = [78|96]
30. combinations for 21/4 in n2 = {1479|1569|1578} -> no 2
31. combinations for 9/3 in r1 = {135|234} -> elim 3 from rest of r1
32. r16c24 = x-wing on 3 -> elim from rest of c2
33. 4 of n3 locked in 18/4 in n3 -> 18/4 in n3 = {1458|1467|2349|3456} (cannot be 2457 as this shares two digits with all combinations for 15/3 in c7, and can see all but one digit of said cage)
34. 12/4 in n6 forms killer pair {35} with r4c7 -> elim 5 from r4c89
35. 7 of n6 locked in r4; elim from rest of r4
36. 7 of n6 locked in 21/3 of r34; elim from r3c9; 21/3 of r34 must be {579|678} -> no 4
37. 4 of n6 locked in 12/4 -> 12/4 in n6 = {1245} quad -> elim from rest of n6 -> r4c7 = 3 (naked single)
38. sole combination for 15/3 in c7 = {357} -> elim 5|7 from rest of c7 and n3
39. sole combination for 18/3 in n3 = {2349} quad -> elim {2349} from rest of n3 -> r1c7 = 1 -> r5c7 = 2 (naked singles)
40. sole combination for 21/3 in r34 = {678} -> no 9
41. r6c8 = 9 (hidden single)
42. r4c5 = 9 (hidden single)
43. r3c3 = 9 (hidden single)
44. r16c24 = Unique Rectangle {23} -> r6c2 = 1 -> a jillion naked singles and last-digit-in-cage moves
45. r1c6 = 7 (hidden single) -> about a dozen naked singles and LDIC moves
46. r5c5 = 7 (hidden single) -> naked singles and last digits solve it.
634257189
287169543
159843726
462591378
398674251
715328694
876912435
921435867
543786912


Much easier than last week's, which I never was able to finish (I think a lot of other people couldn't either judging by the lack of forum activity!)
Walkthrough by Andrew:
Nice puzzle Ruud. About the same level as Assassin 46 Light.

I'm impressed that Cathy solved it in about 30 minutes. She said this was "after using a few higher level standard sudoku techniques in addition to the killer ones". Guess I'd better learn them. Must find time to study Andrew Stuart's book, I've only glanced at bits of it so far.

I managed to solve it by standard killer techniques although it did need a bit more thought at one stage.

1. R7C12 = {69/78}

2. R78C5 = {13}, locked for C5 and N8

3. R7C89 = {17/26/35}, no 4,8,9

4. R9C12 = {18/27/36/45}, no 9

5. R9C89 = {12}, locked for R9 and N9, clean-up: no 6,7 in R7C89, no 7,8 in R9C12
5a. R7C89 = {35}, locked for R7 and N9
5b. R78C5 = [13] (naked singles)

6. R1C234 = {126/135/234}, no 7,8,9

7. 11(3) cage in N14 = {128/137/146/236/245}, no 9

8. R345C4 = {289/379/469/478/568}, no 1

9. R345C6 = 1{25/34}, 1 locked for C6

10. 21(3) cage in N36 = {489/579/678}, no 1,2,3

11. R6C234 = {126/135/234}, no 7,8,9

12. R6C678 = {689}, locked for R6
12a. R6C234 = {135/234} = 3{15/24}, 3 locked for R6

13. 7 in R6 locked in R6C159
13a. 45 rule on R6 3 innies R6C159 = 13 = 7{15/24}

14. 11(3) cage in N89 = {245} (only remaining combination) -> R8C6 = 5, R7C67 = [24]

15. R345C6 = {134}, locked for C6

16. R8C789 = {678} (only remaining combination), locked for R8 and N9 -> R9C7 = 9

17. R9C34567 = 789{36/45}, 3,5 only in R9C3 -> no 4,6,7,8 in R9C3

18. R8C123 = {129} (only remaining combination), locked for R8 and N7 -> R8C4 = 4, clean-up: no 6 in R7C12

19. R7C12 = {78}, locked for R7 and N7 -> R7C34 = [69], clean-up: no 3 in R9C12

20. R9C12 = {45}, locked for R9 -> R9C3 = 3

21. 27(4) cage in N4 must contain 9 = 9{378/468/567}, no 1,2, 9 locked in R5C123 for R5 and N4
21a. R6C159 = 7{15/24} (step 13a), 1 only in R6C9 -> no 5 in R6C9

22. 12(4) cage in N6 must contain 1,2 = 12{36/45}, no 7,8,9, 1,2 locked for N6

23. 7 in N6 locked in R4C789, locked for R4

24. 7 in N4 locked in 27(4) cage = 79{38/56}, no 4

25. 45 rule on R1 3 innies R1C159 = 20 = {389/479/569/578}, no 1,2

26. 45 rule on C123 2 remaining outies R16C4 = 5 = {23}, locked for C4

27. R2C4 = 1 (hidden single in C4)

28. R345C4 = {568} (only remaining combination), locked for C4 -> R9C4 = 7

29. 7 in N5 locked in R56C5, locked for C5

30. 21(4) cage in N2 = 1{479/569/578}, no 2
30a. 7 only in R2C6 -> no 8 in R2C6

31. 45 rule on C789 2 remaining outies R16C6 = 15 = {69}/[78], no 8 in R1C6

32. 2,7 in C5 locked in R3456C5 = 27{49/58}, no 6

33. 45 rule on C5 1 remaining outie R2C6 – 1 = 1 innie R9C5 -> R2C6 = {79}

34. 45 rule on N6 2 outies R3C9 + R6C6 – 11 = 1 innie R4C7, max R3C9 + R6C6 = 18 (they can both be 9) -> no 8 in R4C7, min R4C7 = 3 -> min R3C9 + R6C6 = 14 -> no 4 in R3C9

35. 45 rule on N6 3 innies R4C789 – 10 = 1 outie R6C6, R6C6 = {689} -> R4C789 = 16, 18 or 19 and must contain 7 (step 23)
35a. 12(4) cage in N6 = 12{36/45} (step 22) and R6C78 = {689} -> R4C789 must contain either {457} (with 4 in R4C89) or {37} and one of 6,8,9
35b. R4C789 cannot be {457} (with 4 in R4C89) because this doesn’t provide any valid combinations for 21(3) cage in N36 -> R4C789 = 3{67/78/79} -> R4C7 = 3, R4C89 = {67/78/79}, no 4,5


36. 12(4) cage in N6 = {1245}, no 6
36a. 5 in N6 locked in R5C789, locked for R5

37. 21(4) cage in N36 = {579/678} = 7{59/68}
37a. 7 locked in R4C89 -> no 7 in R3C9
37b. 5 only in R3C9 -> no 9 in R3C9

38. R4C7 = 3 -> R23C7 = 12 = {57}, locked for C7 and N3

39. Naked pair {68} in R68C7, locked for C7

40. R3C9 + R6C6 – 11 = R4C7 (step 34), R4C7 = 3 -> R3C9 + R6C6 = 14 -> no 9 in R6C6
40a. R6C67 = {68} -> R6C8 = 9

41. R4C5 = 9 (hidden single in R4)

42. R1C234 = {135/234} (cannot be {126} which clashes with R1C7), = 3{15/24}, no 6, 3 locked for R1
[PsyMar also spotted killer pair {12} in R1C234 and R1C7, locked for R1. I should have seen that but my next step produced the same result.]

43. R1C678 = [628/718/916] -> R1C8 = {68}

44. Naked triple {678} in R148C8, locked for C8

45. Naked pair {68} in R1C8 + R3C9, locked for N3

46. 9 in N3 locked in 18(4) cage = {2349}, locked for N3 -> R1C7 = 1, R5C7 = 2

47. R1C234 = {234}, no 5, {234} locked for R1, 4 locked in R1C23 for N1 -> R1C9 = 9

48. R1C159 = 20 (step 20), R1C9 = 9 -> R1C15 = 11 = {56}, locked for R1 -> R1C68 = [78], R2C6 = 9, R3C9 = 6, R4C89 = [78]

49. 21(4) cage in N2, R2C4 = 1, R2C6 = 9 -> R12C5 = 11 = {56}, locked for C5 and N2 -> R345C4 = [856], R9C56 = [86], R6C67 = [86], R8C789 = [867]

50. 27(4) cage in N4 (step 24) = {3789}, no 5 -> R6C1 = 7, R7C12 = [87]
50a. R6C159 (step 13a) = 7{15/24} -> R6C59 = [24], R35C5 = [47], R3C6 = 3, R6C4 = 3, R1C234 = [342]

51. 6 in R4 locked in R4C12, 11(3) cage in N14 = {146} (only remaining combination) -> R3C1 = 1, R4C12 = {46}, locked for R4 -> R45C6 = [14], R4C3 = 2
51a. R23C3 = 16 = [79]

52. R2C2 = 8 (hidden single in N1)

and the rest is naked singles

Now to try v1.5
Walkthrough by CathyW w- multi-colouring:
OK - here's my walkthrough in 30 steps. It took a lot longer the 2nd time with keeping track of what I'd done!

1. 3(2) in r9c89 -> 1,2 not elsewhere in r9, N9
-> 8(2) in N9 = {35} -> r7c5=1, r8c5=3.
-> 9(2) in r9c12 = {36/45}

2. 23(3) in r6c678 -> 6,8,9 not elsewhere in r6
-> 9(3) in r6c234 = {135/234} -> r6c19 <>3

3. 8(3) in c4 must have 1, 1 not elsewhere in c4

4. Innies N7: r79c3=9: 27/45/63

5. Innies N9: r79c7=13: 49/67/76
-> 21(3) in r8c789 must have 8, 8 not elsewhere in r8

6. Innies r6: r6c159=13 -> r6c9 <>5

7. Innies r1: r1c159=20 -> r1c159 <>1 or 2

8. Innies r8: r8c46=9: {27/45} -> Killer combination with 21(3) -> r8c123 can't be 4 or 7.

9. Outies N1+N4=5 -> r16c4 = {14/23}

10. Outies N3+N7=15 -> r16c7 = 69/96/78

11. Innies c1234: r2c4+r9c34=11 -> r2c4={123}, r9c3={35}, r9c4={4567}

12. Innies c6789: r2c6+r9c67=24 -> r2c6={789}, r9c6={6789}, r9c7={79} -> r7c7<>7

13. r7c7=4 (can't be 6 as 11(3) must be {245}) -> r9c7=9, r7c6=2, r8c6=5, r7c3=6, r9c3=3, r8c4=4, r7c4=9

14. Naked Triple {123} in c4. Since r16c4=5, must be {23} -> r2c4=1 -> r9c4=7 -> 19(3) in c4 = {568}

15. 12(4) in N6 must have 1 and 2 -> r4c7<>1 or 2.

16. 27(4) in N4 must have 9 -> r4c3, r5c5 <>9

17. r16c6=15 (from step 10) -> r29c6=15 -> r2c6<>8

18. 7 locked to r12c6 -> r123c5<>7

19. Simple colouring on 3s -> r3c2<>3

20. 7 locked to r4c789 -> r4c1235<>7

21. 18(3) in r234c3 = {189/279/459} -> 9 not elsewhere in N1, c3.
-> From possible combinations r23c3<>2

22. 27(4) in N4 = {3789/4689/5679}
-> If 4689, r6c1=4 -> r5c123<>4
-> If 3789, r5c12 = {39}; If 4689, r5c12 = {69}; If 5679, r5c12 = {69} -> r5c12 = {369}

23. (Perhaps could have done this earlier) Innies c3: r1568c3 = 18 -> r1c3={45}, r5c3={78}, r6c3={45}, r8c3={12} -> r16c3 naked pair, 45 not elsewhere in c3 -> 18(3) = {189/279}

24. 9(3) in r1 and r6 = {135/234} -> r1c2<>6, r1c1789<>3; r16c2<>4 or 5. x-wing on 3 in r16c24 -> r245c2<>3.

25. Innies c7: r1568c7 = 17 = {1268/1367} -> r1c7={12}, r5c7={123}, r6c7={68}, r8c7={678} -> r1c8 = {56789}; r3c7<>1.

26. 15(3) in c7 must have 5 -> {258/357}, cannot have 6.

27. Innies c5: r129c5 = 19 = {469/568} -> r345c5<>6

28. Multi-colouring on 1s -> r1c7=1
Image
-> Singles: r1c3=4, r6c3=5, r6c4=3, r6c2=1, r1c4=2, r1c2=3, r3c6=3
-> Naked Pair (14) in N5 -> r456c5<>4
-> Hidden single in r4 -> r4c6=1, r5c6=4.

29. 21(4) in N1 = {2568}, cannot have 7 -> r2c3={79}, r3c1={17}, r3c3={179}; -> 8 locked to r45c3 -> r4c12<>8.

30. 11(3) in r3c1, r4c12 = {146} -> r3c1=1 ...

Straightforward from here with naked and hidden singles and locked candidates.

Multi-colouring is indeed single digit technique linking conjugate pairs in two chains - in the above scenario blue coloured cells can 'see' both pink and amber coloured cells therefore blue cannot be true and the green coloured cell must be true. I haven't got to grips with using Sumocue (I mostly use JSudoku if not solving on paper) so not sure about the pattern hints comparison - the chains of conjugate pairs don't necessarily form a pattern.

Edit: Sorry if some of the eliminations I made are not clear enough in this walkthrough. I'll try to be more explicit next time
One-trick pony Walkthrough by mhparker:
Andrew wrote:
It could be posted now in tiny text. You've got me intrigued Mike.

OK, here's my walkthrough. I've included a candidate grid at the point of interest for convenience.

Hope I haven't set people's hopes too high! :)

Walkthrough - Assassin 47

1. 9/3 at R1C2: no 7,8,9

2. 11/3 at R3C1: no 9

3. 19/3 at R3C4: no 1

4. 8/3 at R3C6 = {1(25|34)} -> no 1 elsewhere in C6

5. 21/3 at R3C9: no 1,2,3

6. 27/4 at R5C1 = {(378|468|567)9} -> no 9 elsewhere in N4

7. 12/4 a R5C7 = {12(36|45)} -> no 1,2 elsewhere in N6

8. 9/3 at R6C2: no 7,8,9

9a. 23/3 at R6C6 = {689} -> no 6,8,9 elsewhere in R6
9b. 9 in N4 locked in R5 -> no 9 elsewhere in R5
9c. 9/3 at R6C2 = {3(15|24)} -> no 3 elsewhere in R6

10. 7 in N6 locked in R4 -> no 7 elsewhere in R4

11. 7 in N4 locked in 27/4 at R5C1 = {(38|56)79} (no 4)

12. 15/2 at R7C1 = {69|78}

13. 19/3 at R7C3: no 1

14. 4/2 at R7C5 = {13} -> no 1,3 elsewhere in C5 or N8

15. 11/3 at R7C6: no 7,9 ({137} no longer available due to step 14)

16. 8/2 at R7C8: no 4,8,9

17. 21/3 at R8C7: no 1,2,3

18. 33/5 at R9C3 = {(36|45)789} -> no 7,8,9 elsewhere in R9

19. 3/2 at R9C8 = {12} -> no 1,2 elsewhere in R9 or N9

20. Hidden Single (HS) in R7 at R7C5 = 1 -> R8C5 = 3

21. 8/2 at R7C8 = {35} (only remaining combination) -> no 3,5 elsewhere in R7 or N9

22. 11/3 at R7C6 = {245} (only remaining combination) -> R8C6 = 5, R7C6 = 2, R7C7 = 4

23. 873 at R3C6 = {134} -> no 3,4 elsewhere in C6

24. (Sterile) Naked Quad on {6789} in R7 at R7C1234
-> sum of R7C1234 = 30 -> sum of R7C34 = 15 -> R8C4 = 4

25. 1,2 in R8 locked in 12/3 at R8C1 = {129} -> no 9 elsewhere in R8 or N7

26. 15/2 at R7C1 = {78} -> no 7,8 elsewhere in R7 or N7

27. Naked Single (NS) at R7C3 = 6 -> R7C4 = 9

28. 21/3 at R8C7 = {678} -> no 6,7,8 elsewhere in N9 -> R9C7 = 9

29. Innie N7: R9C3 = 3

30. 18/3 at R2C3 = {(18|27|45)9} (3,6 unavailable) -> 9 locked in R23C3
-> no 9 elsewhere in C3 or N1

31. 19/3 at R3C4 = {568} (only remaining combination - 4,9 unavailable)
-> no 5,6,8 elsewhere in C4 -> R9C4 = 7

32a. Innie C1234: R2C4 = 1
32b. Split cage 20/3 at R12C5: no 2

33a. 7 in N5 locked in R56 innies (R5C456+R6C5) = 19/4 -> no 7 elsewhere in C5

At this point, the grid is as follows:

Code:
.-----------.-----------------------------------.-----------.-----------------------------------.-----------.
| 12345678  | 123456      1245        23        | 45689     | 6789        1235678     123456789 | 123456789 |
|           '-----------.-----------.-----------'           '-----------.-----------.-----------'           |
| 2345678     2345678   | 245789    | 1           45689       6789      | 235678    | 23456789    23456789  |
:-----------.           |           :-----------.-----------.-----------:           |           .-----------:
| 12345678  | 12345678  | 1245789   | 568       | 245689    | 34        | 1235678   | 123456789 | 456789    |
|           '-----------:           |           |           |           |           :-----------'           |
| 1234568     1234568   | 12458     | 568       | 245689    | 134       | 35678     | 456789      456789    |
:-----------------------'-----------:           |           |           :-----------'-----------------------:
| 356789      356789      578       | 568       | 245678    | 134       | 12356       123456      123456    |
|           .-----------------------'-----------:           :-----------'-----------------------.           |
| 57        | 12345       1245        23        | 2457      | 689         68          689       | 1245      |
:-----------'-----------.-----------------------+-----------+-----------------------.-----------'-----------:
| 78          78        | 6           9         | 1         | 2           4         | 35          35        |
:-----------------------'-----------.           |           |           .-----------'-----------------------:
| 129         129         12        | 4         | 3         | 5         | 678         678         678       |
:-----------------------.-----------'-----------'-----------'-----------'-----------.-----------------------:
| 45          45        | 3           7           68          68          9         | 12          12        |
'-----------------------'-----------------------------------------------------------'-----------------------'



--- the next move is the key one for this puzzle, as it both ---
--- breaks the gridlock and provides for some rapid placements, ---
--- without requiring any heavy computational work ---

33b. This hidden 19/4 R56 innie cage cannot also contain an 8 (would imply {1378},
impossible because only 1 innie cell (R5C6) has either of {13}) -> no 8 in R5C45
33c. 8 in R5 now locked in 27/4 in N4 = {3789} -> no 8 elsewhere in N4
33d. 3 in 27/4 at R5C1 locked in R5C12 -> no 3 elsewhere in R5 or N4

34. NS at R6C1 = 7

35. NS at R5C3 = 8

36. 15/2 at R7C12 = [87]

37. HS in R5 at R5C5 = 7

38. HS in R6 at R6C4 = 3

39. NS at R1C4 = 2

40a. HS in C6/N2 at R3C6 = 3
40b. 4 in 8/3 at R3C6 locked in N5 at R45C6 -> no 4 elsewhere in N5

41a. HS in R4 at R4C7 = 3
41b. 7 in N6 now locked in R4C89 -> 21/3 at R3C9 = {7(59|68)} (no 4), no 7 in R3C9
41c. Split 12/2 cage at R23C7 = {57} (4,9 unavailable) -> no 5,7 elsewhere in C7 or N3

42. HS in C7 at R5C7 = 2

43. HS in C7 at R1C7 = 1

44. HS in R1 at R1C6 = 7 -> R1C8 = 8 (last digit in cage)

45. 6 in N4 locked in R4C12 -> no 6 elsewhere in R4, 11/3 cage at R3C1 = {146} (3 unavailable), no 6 in R3C1

46. 7 in C3/N1 locked in 18/3 at R2C3 = {79}[2]

47. NS at R8C3 = 1

48. HS in C5 at R6C5 = 2

49. Innie R6: R6C9 = 4 -> Split 6/2 cage at R5C89 = {15} (no 6) -> no 1,5 elsewhere in R5 or N6

From now on, the puzzle can be completed via Singles only


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PostPosted: Thu Jun 19, 2008 12:46 am 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 47 v1.5 by Ruud (April 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:4608:5633:5633:5633:3332:3333:3333:3333:4616:4608:4608:2827:3332:3332:3332:4111:4616:4616:4626:4608:2827:5653:4118:5655:4111:4616:3098:4626:4626:2827:5653:4118:5655:4111:3098:3098:3876:3876:3876:5653:4118:5655:5930:5930:5930:3876:3630:3630:3630:4118:4402:4402:4402:5930:3126:3126:3640:3640:3130:2875:2875:1597:1597:3903:3903:3903:3640:3130:2875:5189:5189:5189:1352:1352:7242:7242:7242:7242:7242:3151:3151:
Solution:
+-------+-------+-------+
| 3 8 9 | 5 6 7 | 4 2 1 |
| 5 6 7 | 4 2 1 | 9 3 8 |
| 2 4 1 | 9 3 8 | 5 6 7 |
+-------+-------+-------+
| 7 9 3 | 6 8 5 | 2 1 4 |
| 8 2 4 | 7 1 9 | 3 5 6 |
| 1 5 6 | 3 4 2 | 7 8 9 |
+-------+-------+-------+
| 9 3 5 | 8 7 6 | 1 4 2 |
| 6 7 2 | 1 5 4 | 8 9 3 |
| 4 1 8 | 2 9 3 | 6 7 5 |
+-------+-------+-------+
Quote:
CathyW: Managed to do 1.5 as well - amazing!!
mhparker:once it was in motion there was no stopping it... This is in contrast to the most difficult Assassins, where one has to be thankful for small mercies in being able to find any way at all of shaving off the next candidate
Jean-Christophe: The difficulty was to spot the starting place, then all the rest flows quite easilly
Andrew: It was no harder than Assassin 47, in fact probably slightly easier because there wasn't a difficult key move
mhparker: it shows that the observation that Andrew made about the V1.5 being no harder than the V1 (and possibly even easier) is correct. It also tells us - as we suspected - that both of these Assassins were One-Trick Ponies
Andrew: "One-Trick Ponies" wasn't a suggestion that these puzzles were too easy to be Assassins. I felt that they were still Assassins but at the easy end of the difficulty range. Each of them must have taken me at least two hours
Walkthrough by mhparker:
Hi folks,

Here's my walkthrough for the V1.5. It took a couple of strong pushes right at the start to get the proverbial ball rolling, but once it was in motion there was no stopping it...

After that, this puzzle proved to be very much routine, with no real surprises, and often a good selection of choices as to how to progress. This is in contrast to the most difficult Assassins, where one has to be thankful for small mercies in being able to find any way at all of shaving off the next candidate.

Maybe Ruud is just giving us a short break and this is the simply the calm before the storm?

Anyway, enough talk for now, here's my V1.5 walkthrough:

Walkthrough - Assassin 47 V1.5

1. 22/3 at R1C7 = {(58|67)9} -> no 9 elsewhere in R1

2. 13/4 at R1C5 = {1(237|246|345)} (no 8,9) -> no 1 elsewhere in N2

3. 11/3 at R2C3: no 9

4. 22/3 at R3C4 = {(58|67)9} -> no 9 elsewhere in C4

5. 9 in R1 now locked in R1C23 -> no 9 elsewhere in N1

6. 9 in R2 now locked in R2C789 -> no 9 elsewhere in N3

7. 22/3 at R3C6 = {(58|67)9} -> no 9 elsewhere in C6

8. 9 in N8 now locked in R789C5 -> no 9 elsewhere in C5

9. 12/2 at R7C1: no 1,2,6

10. 12/2 at R7C5: no 1,2,6

11. 11/3 at R7C6: no 9 in R7C7

12. 6/2 at R7C8 = {15|24}

13. 20/3 at R8C7: no 1,2

14. 5/2 at R9C1 = {14|23}

15. 12/2 at R9C8: no 1,2,6

16. Innies N7: R79C3 = 13/2 -> no 1,2,3

17. Innies N9: R79C7 = 7/2 -> no 7,8,9

18a. Innies R8: R8C456 = 10/3 = {127|136|145|235} (no 8,9)
18b. Cleanup: no 3,4 in R7C5

19a. 9 in R8 cannot be in 15/3 cage
({249} blocked due to 5/2 at R9C1, {159} blocked due to h10/3 at R8C456 (step 18))
-> 9 locked in 20/3 at R8C7 = {(38|47|56)9} -> no 9 elsewhere in R8 or N9
19b. 12/2 at R9C8 = {48|57} (no 3)

20. 20/3 at R8C7 must contain exactly two of {6789}
12/2 at R9C8 must contain exactly one of {6789}
6/2 ar R7C8 cannot contain any of {6789}
-> Innies N9 (see step 17) must contain exactly one of {6789}
-> R79C7 = {16} -> no 1,6 elsewhere in C7 or N9

21. 6/2 at R7C8 = {24} (1 no longer available) -> no 2,4 elsewhere in R7 or N9

22. 12/2 at R9C8 = {57} (4 no longer available) -> no 5,7 elsewhere in R9 or N9

23. 20/3 at R8C7 = {389} -> no 3,8 elsewhere in R8

24. Cleanup from previous steps:
24a. 12/2 at R7C1 = {39|57} (no 8)
24b. 12/2 at R7C5 = {48|57} (no 9)

25. Hidden Single (HS) in C5 at R9C5 = 9

26. 8 in N7 locked in h13/2 at R79C3 = [58]

27. 12/2 at R7C1 = {39} (5 unavailable) -> no 3 elsewhere in R7 or N7

28. 5/2 at R9C1 = {14} (3 unavailable) -> no 1,4 elsewhere in R9 or N7

29. Naked Single (NS) at R9C7 = 6

30. NS at R7C7 = 1

31. HS in R8 at R8C4 = 1 -> R7C4 = 8 (last digit in cage)

32. NS at R7C5 = 7 -> R8C5 = 5

33. NS at R7C6 = 6 -> R8C6 = 4 (last digit in cage)

34. Only two possible combinations for 22/3 ({(58|67)9}).
Clearly, 22/3 at R3C4 and 22/3 at R3C6 would clash with each other if both had same combination
-> One of these cages must be {589} and the other {679}
-> 22/3 at R3C6 = {589} (6 unavailable) -> no 5,8 elsewhere in C4
-> 22/3 at R3C4 = {679} -> no 6,7 elsewhere in C4

35. NS at R1C4 = 5 -> R1C23 = [89]

36. Outie N14: R6C4 = 3 -> R9C46 = [23]

37. NS at R2C4 = 4

38. 16/4 at R3C5 = {1348} (no 2,6) (5,7,9 unavailable) -> no 1,3 in R12C5

39a. HS in C5 at R3C5 = 3
39b. 1 in C5 now locked in N5 -> no 1 in R6C6

40. HS in N2 at R3C6 = 8

41a. HS in R3 at R3C4 = 9
41b. 7 in C4 now locked in N5 -> no 7 in R6C6 -> R6C6 = 2

42. NS at R1C6 = 7 -> R1C78 = {24} -> no 2,4 elsewhere in R1 or N3

43. NS at R2C6 = 1

44. NS at R1C5 = 6 -> R2C5 = 2

45. Naked Pair (NP) on {24} in C8 at R17C8 -> no 2,4 elsewhere in C8

46. 18/4 at R1C1 = {6(147|237|345)} (8,9 unavailable, must have one of {13} due to R1C1)
-> no 6 elsewhere in N1

47. 18/4 at R1C1 and R2C3 at R2C3 form killer pair on {37} -> no 7 in R3C13

48. 18/4 at R1C9 = {13(59|68)} (no 7) (2,4 unavailable) -> no 1,3 elsewhere in N3

49. 18/4 at R1C9 and R3C79 form killer triple on {567} -> no 5,7 in R2C7

50. 16/3 at R2C7 = [853|952] -> R3C7 = 5

51. HS in N3 at R3C9 = 7 -> R4C89 = 5/2 -> R4C8 = {13}, R4C9 = {24}

52. R9C89 = [75]

53. Naked Pair (NP) on {24} in C9 at R47C9 -> no 2,4 elsewhere in C9

54. Split 15/2 cage at R6C78 = [78|96]

55. HS in C8 at R5C8 = 5

56. R45C6 = [59]

57. Innies R1234: R4C45 = 14/2 = [68]

58. NS at R5C4 = 7

59. 8 in C1 locked in 15/4 at R5C1 = {1248} -> no 1,2,4 elsewhere in N4; no 2 in R5C7

60. HS in C1 at R2C1 = 5

61. HS in C1 at R8C1 = 6

62. HS in C1 at R4C1 = 7

--- and it's all naked singles from now on... ---
Walkthrough by Jean-Christophe:
Here is my walkthrough for V 1.5. The difficulty was to spot the starting place, then all the rest flows quite easilly. Very enjoyable 8-)

BTW what's the meaning of these "versions" V1.5, V2... ? I'm new on this forum (but not so new to killers ;-) )

Cage 6/2 in N9 = {15|24} = {(2|5)..} = {(1|4)..}
Innies of N9 -> R79C7 = 7 <> {25} = {(1|4)..}
Cage 6/2 & R79C7 froms a complex naked pair on {14} -> no {14} elsewhere in N9
-> Cage 12/2 in N9 <> {48}
8 of N9 locked in R8C789 -> not elsewhere in R8
Cage 20/3 in N9 = {8..} = {389|578}
6 of N9 locked in R79C7 = {16} (NP @ N9, C7)
Cage 6/2 in N9 = {24} (NP @ N9, R7)
-> Cage 12/2 in N7 <> {48}
8 of N7 locked in R79C3
Innies of N7 -> R79C3 = 13 = {58} (NP @ N7, C3)
Cage 12/2 in N7 = {39} (NP @ N7, R7)
Cage 5/2 in N7 = {14} (NP @ N7, R9)
R79C7 = [16]
Cage 15/3 in N7 = {267} (NT @ R8)
Cage 20/3 in N9 = {389} (NT @ N9, R8)
Cage 12/2 in N9 = {57} (NP @ R9)
R79C3 = [58]

3 Cages 22/3 in R1, C4 & C6 = {9..} -> no 9 elsewhere in R1, C4 & C6
BTW Not a necessity, but notice: the two Cages 22/3 in C4 & C6 forms a X-Wing on 9 -> not elsewhere in N25, not in elsewhere R3
R9C5 = 9 (HS @ N8)
R78C4 = 9 = [81]
Cage 12/2 in R78C5 = [75]
R78C6 = [64]

Innies of C1234 -> R29C4 = 6 = [42]
Innies of C4 -> R16C4 = 8 = [53]
R9C46 = [23]
R1C23 = 17 = [89]
Innies of C6789 -> R2C6 = 1
R12C5 = 8 = {26} (NP @ N2, C5)
Innies of C6 -> R16C6 = 9 = [72]
R1C78 = 6 = {24} (NP @ N3, R1)
R12C5 = [62]
R3C5 = 3, R3C46 = [98]

Outies of N3 -> R4C789 = 7 = {124} (NT @ R4, N6)
R4C5 = 8, R56C5 = {14} = 5
Innies of R56 -> R5C46 = 16 = [79]
R4C46 = [65]
R4C123 = {379} (NT @ N4)
Cage 14/3 in R6 -> R6C23 = [56]
Cage 17/3 in R6 -> R6C78= {78}
R6C9 = 9
...
The rest are naked & hidden singles
Walkthrough by Andrew:
I like the quotation at the end of JC's message. "When you have eliminated the impossible, whatever remains, however improbable, must be the truth." Sherlock Holmes. Very appropriate for sudokus!

I found that V1.5 went very smoothly right from the beginning. If anyone used a solver to do the preliminary eliminations, they probably found it harder to get started. Unusually the methodical start, that I've been using for the past two to three months, proved to be just right for this puzzle.

It was no harder than Assassin 47, in fact probably slightly easier because there wasn't a difficult key move.

As soon as I saw the diagram I spotted from the two vertical 22(3) cages that there must be 9 in R3C46. In the walkthrough that comes out in steps 8 to 11 without needing to see that X-wing.

Here is my walkthrough.

1. R7C12 = {39/48/57}, no 1,2,6

2. R78C5 = {39/48/57}, no 1,2,6

3. R7C89 = {15/24}

4. R9C12 = {14/23}

5. R9C89 = {39/48/57}, no 1,2,6

6. R1C234 = 9{58/67}, 9 locked for R1

7. R234C3 = {128/137/146/236/245}, no 9

8. R345C4 = 9{58/67}, 9 locked for C4
8a. 9 in R1 locked in R1C23, locked for N1

9. R345C6 = 9{58/67}, 9 locked for C6

10. 9 in N8 locked in R789C5, locked for C5

11. 9 in N2 locked in R3C46, locked for R3

12. R8C789 = {389/479/569/578}, no 1,2

13. 13(4) cage in N2 = {1237/1246/1345} = 1{237/246/345}, no 8, 1 locked for N2

14. 45 rule on N7 2 innies R79C3 = 13 = {49/58/67}, no 1,2,3

15. 45 rule on N9 2 innies R79C7 = 7 = {16/34} (cannot be {25} which clashes with R7C89), no 2,5,7,8,9
[I didn’t spot that this makes a killer pair {14} in R79C7 and R7C89, as in Jean-Christophe’s walkthrough. This didn’t matter because of my next move. Maybe I was already looking ahead to it.]

16. 2 in N9 locked in R7C89 = {24}, locked for R7 and N9, clean-up: no 8 in R7C12, no 8 in R8C5, no 9 in R9C3 (step 14), no 3 in R79C7 (step 15), no 8 in R9C89
16a. Naked pair {16} in R79C7, locked for C7 and N9

17. 8 in N9 locked in R8C789, locked for R8, R8C789 = 8{39/57}

18. 45 rule on R1 3 innies R1C159 = 10 = {127/136/145/235}, no 8

19. 45 rule on R8 3 innies R8C456 = 10 = {127/136/145/235}, no 9, clean-up: no 3 in R7C5
19a. 1,2 only in R8C46 -> no 7 in R8C46

20. 8 in N7 locked in R79C3 = {58} (step 14), locked for C3 and N7, clean-up: no 7 in R7C12
20a. R7C12 = {39}, locked for R7 and N7, clean-up: no 2 in R9C12
20b. R9C12 = {14}, locked for R9 and N7 -> R9C7 = 6, R7C7 = 1

21. R8C4 = 1 (hidden single in R8)
21a. R7C34 = 13 = {58}, locked for R7 -> R7C56 = [76] -> R8C5 = 5, R8C6 = 4 (hidden single in N8), R7C34 = [58], R9C3 = 8, R9C5 = 9 (hidden single in R9), clean-up: no 3 in R9C89
21b. R9C89 = {57}, locked for N9

22. R345C6 = {589}, no 7, locked for C6

23. R345C4 = {679}, no 5, locked for C4 -> R1C4 = 5 -> R1C23 = 17 = [89]

24. 45 rule on C5 2 remaining innies R12C5 = 8 = {26}, locked for C5 and N2
24a. R2C46 = 5 = [41]
24b. 8 in N2 locked in R3C56, locked for R3

25. 2 in N5 locked in R6C46, locked for R6

26. R1C159 (step 18) = {127/136} = 1{27/36}, no 4, R1C5 = {26} -> no 2,6 in R1C19, 1 locked in R1C19 for R1

27. R1C678 = 4{27/36}
27a. R1C6 = {37} -> no 3,7 in R1C78
27b. 4 in N3 locked in R1C78, locked for N3

28. 18(3) cage in N14, no 8,9 in R3C1 -> no 1 in R4C12

29. R6C234 = {239/257/347/356}, no 1, no 3,6 in R6C2

30. R6C678 = {179/269/278/359/368/467} (cannot be {458} because no 4,5,6 in R6C6), no 3 in R6C7, no 3,4 in R6C8

31. 45 rule on N1 3 remaining outies R4C123 = 19, no 1
31a. Max R4C3 = 7 -> min R4C12 = 12, no 2 in R4C12

32. 45 rule on N14 2 remaining innies R6C23 = 11 -> R6C4 = 3, R9C46 = [23], R1C6 = 7, R6C6 = 2, R3C456 = [938]
32a. R6C23 = {47}/[56], no 9

33. R1C6 = 7 -> R1C78 = 6 = {24}, locked for R1 and N3 -> R12C5 = [62]

34. R6C6 = 2 -> R6C78 = 15 = {78}/[96], no 1,4,5, no 9 in R6C8

35. R234C3 = {137/146/236}, 1 only in R3C3 -> no 4,7 in R3C3

36. 4 in N1 locked in R3C12 -> no 4 in R4C2 which can “see” both of R3C12

37. 4 in C3 locked in R456C3, locked for N4
[This would have eliminated 4 from R4C2 but I spotted step 36 first.]
37a. R6C23 = [56/74], no 7 in R6C3

38. 45 rule on N1 3 remaining innies R23C3 + R3C1 = 10 = {127/136/235} (cannot be {145} because no 1,4,5 in R2C3), no 6 in R2C3, no 4,7 in R3C1

39. R3C2 = 4 (hidden single in N1) -> R9C12 = [41]

40. R234C3 (step 35) = {137/236} (cannot be {146} because no 1,4,6 in R2C3) = 3{17/26}, no 4, 3 locked for C3

41. 45 rule on N3 3 remaining innies R23C7 + R3C9 = 21 = {579/678} = 7{59/68}, no 1,3, 7 locked for N3
41a. 9 only in R2C7 -> no 5 in R2C7
41b. 6 only in R3C9 and 8 only in R2C7 -> no 7 in R2C7
41c. R234C7 = [952] ([853] isn’t consistent with step 41) -> R3C9 = 7 (step 41)

42. 12(3) cage in N36 R3C9 = 7 -> R4C89 = 5 = {14}, locked for R4 and N6

43. 18(4) cage in N1 cannot have three odd numbers in R1C1 + R2C12 -> 6 locked in R2C12, locked for R2 and N1
[This also comes from the remaining combinations {1467/3456} = 46{17/35}]

44. Naked pair {12} in R3C13, locked for R3 and N1 -> R1C1 = 3, R2C3 = 7, R3C8 = 6

45. Naked pair {38} in R2C89, locked for N3 -> R1C9 = 1

46. A few naked singles to clear up R1C78 = [42], R4C89 = [14], R7C12 = [93], R7C89 = [42], R9C89 = [75], R6C78 = [78], R5C7 = 3, R6C2 = 5, R6C3 = 6 (cage sum)

and the rest is naked singles


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PostPosted: Thu Jun 19, 2008 12:48 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 48 by Ruud (Apr 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:7168:7168:3074:3074:772:2309:2309:7175:7175:1801:7168:7168:3596:772:1806:7175:7175:3857:1801:4115:7168:3596:8726:1806:7175:4121:3857:1801:4115:4115:4894:8726:5152:4121:4121:3857:2340:2340:4894:4894:8726:5152:5152:2347:2347:5165:4142:4142:4894:8726:5152:3379:3379:3125:5165:4142:4920:2617:8726:3387:9020:3379:3125:5165:4920:4920:2617:2115:3387:9020:9020:3125:4920:4920:2634:2634:2115:845:845:9020:9020:
Solution:
+-------+-------+-------+
| 6 4 5 | 7 2 8 | 1 3 9 |
| 2 7 8 | 9 1 3 | 5 4 6 |
| 1 9 3 | 5 6 4 | 7 2 8 |
+-------+-------+-------+
| 4 5 2 | 3 7 9 | 8 6 1 |
| 3 6 9 | 1 8 5 | 4 7 2 |
| 8 1 7 | 6 4 2 | 3 9 5 |
+-------+-------+-------+
| 5 8 4 | 2 9 7 | 6 1 3 |
| 7 2 1 | 8 3 6 | 9 5 4 |
| 9 3 6 | 4 5 1 | 2 8 7 |
+-------+-------+-------+
Quote:
CathyW: It's certainly harder than 47! Still working on it - currently up to step 49 and counting ..Finally managed no. 48. The Lite version more enjoyable for me.
Psymar: all fairly standard killer stuff until step 49, which involves Medusa Coloring of a sort, with a bit of a twist at the end
Andrew: 3 walkthroughs for the original Assassin 48 and all had different breakthroughs from the way that I finished it
Walkthrough by Para:
Hi all

Ok this is probably the ugliest walk-through i have made in a while. Really just jotted down everything i did and in which order. I think i might clean it up a little later, so everyone can follow it. But anyway, this is how i solved it.

Walkthrough Assassin 48

This first bit is just from left to right top to bottom. You could this much more practical of course.
1. R1C34 = {39/48/57}: no 1,2,6
2. R12C5 = {12} -->> locked for C5 and N2
3. R1C56 = {36/45}/[72/81]: R1C5: no 9, R1C6: no 7,8,9
4. R234C1 = {124} -->> locked for C1
5. R23C4 = {59/68}
6. R23C6 = {34} -->> locked for C6 and N2
6a. Clean up: R1C3: no 8,9; R1C7: no 5,6
7. 34(5) in R3C5 = {46789} -->> locked for C5
7a. Naked Pair {35} in R89C5 -->> locked for N8
8. 3 in C4 locked in 19(4) cage in R4C4 -->> R5C3: no 3
8a. 19(4) in R4C4 = {1369/1378/2359/2368/3457}
9. R5C12 = {36}/[54/72/81]: R5C1: no 9; R5C2: no 5,7,8,9
10. R5C89 = {18/27/36/45}: no 9
11. R678C1 = {389/569/578}
12. 19(5) in R7C3 = {12349/12358/12367/12457/13456}: 1 locked in 19(5) cage for N7
13. R78C4 = {19/28/46}: no 7
14. R78C6 = {67} -->> locked for C6 and N8
14a. Clean up: R78C4: no 4; R1C7: no 2, 3
14b. Killer Pair {58} in R1C6 + R23C4 -->> locked for N2
14c Clean up: R1C3: no 4,7
15. 35(5) in R7C7 = {56789} -->> locked for N9
16. R9C34 = {28}/[64/91]: R9C3: no 3,4,5,7; R9C4: no 9
17. R9C67 = {12} -->> locked for R9
17a. R9C34 = [64]
18. 9 locked in C6 for N5
18a. 9 in C6 locked in 20(4) cage in R4C6-->> R5C7: no 9
18b. 20(4) in R4C6 = {1289/2459}: 9 locked and only place for 3, 4, 6 and 7 is R5C7: 20(4) can only have one of these digits -->> R5C7: no 3, 5, 6, 7
18c. 20(4) can’t have both {12} in R456C6 -->> R5C7: no 8
18d. 2 locked in 20(4) cage: R5C4: no 2
19. Naked Triple {124} in R159C7 -->> locked for C7
20. 12(3) in R6C9 needs 2 of {1234} in R78C9: can’t have both {12} in R78C9 because of R9C7.
20a. 12(3) = {[8]{13}/[7]{14}/[7]{23}/[6]{24}/[5]{14}: R6C9 = {5678}
Forgetting those 45-tests again.
21. 45 on C123: 2 innies: R15C3 = 14 = [59]
21a. R1C4 = 7; R1C67 = [81]; R12C5 = [21]; R9C67 = [12]; R5C7 = 4
21b. R23C4 = {59} -->> locked for C4 and N2
21c. R78C4 = {28} -->> locked for C4 and N8
21d. R3C5 = 6; R7C5 = 9
21e. Clean up: R5C189: no 5; R6C9: no 6
21f. Hidden single: R5C6 = 5
22. 45 on R5: 2 innies: R5C45 = 9 = [18]
23. 19(5) in R7C3 = {12349/12358/12457} -->> 2 locked in 19(5) cage for N7
24. 28(5) cage in R1C1 can have only one digit of {124} because of R23C1
24a. 28(5) = {13789/23689/34678} = 38{179/269/467}-->> 3 and 8 locked for N1 in 28(5) cage
25. 45-test on N9: 1 outie – 1 innie: R6C9 – R7C8 = 4
25a. 13(3) in R6C7 needs one of {134} in R7C8 -->> 13(3) = {139/148/157/238/247/346}
25b. 13(3) can’t be {157}: R7C8 = 1 -->> R6C9 = 5 -->> 2 5’s in R6, so R6C78: no 5
25c. 13(3) can’t be {148}: R7C8 = 4 -->> R6C9 = 8 -->> 2 8’s in R6
OK something more productive again.
26. 45-test on R6789: 3 innies: R6C456 = 12 = {37/46}[2] -->> R6C6 = 2; R4C6 = 9
27. 13(3) = {139/346}-->> R6C78: no 7,8
28. 15(3) can’t contain both {13},{14},{15},{34},{37} or {48} because of 12(3) in R6C9
28a. 15(3) = {168/249/258/267/456}: no 3
29. 45 on N3: 1 innie – 1 outie: R3C8 – R4C9 = 1 -->> R3C8: no 4, 5; R4C9: no 5
This looks a bit chaotic I know, it’s late. This should have come before I know.
30. 45 on N1: 1 innie – 1 outie = R3C2 – R4C1 = 5: R4C1 = {24}, R3C2 = {79}
30a. R3C1 = 1(hidden)
31. 16(3) in R3C2 = {169/178/259/349/457}: needs 7 or 9 in R3C2; {367} would clash with R4C4; 7 or 9 needs to go in R3C2, so R4C23: no 7
31a. Only place for 5 is R4C2 -->> R4C2: no 2
32. 45 on N7: 1 innie = 1 outie : R6C1 = R7C2 -->> R6C1: no 6; R7C2: no 4
33. Hidden triple {124} in R7C3 + R8C23 for N7
33a. 19(5) in R7C3 = 124{39/57} : no 8; R9C12 = {39/57}
33b. 8 in R9 locked for N9
33c. Killer Pair {35} in R9C12 + R9C5 for R9
Missing moves everywhere.
34. 13(3) in R6C7 = {139}: {36}[4] clashes with R6C4 or 9 locked in R6C78 (pick one)
34a. Clean up: R6C9: no 8
34b. 4 locked in N9 for C9
34c. 1 locked in 13(3) cage in R67C8 for C8
35. 16(3) in R3C8 = {259/268/358/367} -->> R3C8: no 7: would mean R4C78 = {36} which clashes with R4C4
35a. Clean up: R4C9: no 6
36. 8 in R6 locked for N4
37. 4 in N3 locked in 28(5) in R1C8; 28(5) = 4{2589/2679/3579/3678}
38. R6C789 = [917]/{39}[5]
38a. Killer Pair {37} in R5C89 + R6C789 for N6
38b. Clean up: R3C8: no 8
38c. R4C5 = 7(hidden); R6C5 = 4
39. 8 in C23 locked in 28(5) in R1C1 and 16(3) in R6C2
39a. 16(3) in R6C2 = {178/358}: no 6
39b. R6C4 = 6(hidden); R4C4 = 3
39c. Naked triple {124} in R478C3 locked for C3
39d. 2 in N1 locked for R2
40. 28(5) in R1C8 = 347{59/68} -->> {37} locked for N1
40a. Clean up: R4C9: no 2
40b. 15(3) in R2C9 = [681/528] -->> R2C9: no 8,9; R3C9: no 5,9
40c. 8 locked in 15(3) cage for C9
40d. R9C8 = 8(hidden)
41. 16(3) in R3C8 = [952/286] -->> R4C7: no 6; R4C8: no 5
42. 45-test on C9: 3 innies : R159C9 = 18 = [927/{36}9]: R5C9: no 7
42a. Clean up: R5C8: no 2
43.Hidden Killer Pair {56} in R4C2 + R4C78; One of {56} in R4C78, only other place for 5 or 6 is R4C2, so R4C2 = {56}
44. 45 on C1: 3 innies: R159C1 = {369}/[675] -->> R9C1: no 7
44a. Clean up: R9C2: no 5
Ok this breaks the puzzle.
45. 45 on N13: 2 innies – 2 outies: R3C28 – R4C19 = 6 = [92] – [41]/ [72] – [21]: [79] – [28] clashes with R4C78 (step 41)
45a. R3C8 = 2; R4C9 = 1
45b. Naked Singles: R23C9 = [68]; R4C78 = [86]; R4C2 = 5
45c. Hidden Singles: R6C9 = 5; R9C9 = 7; R1C9 = 9; R5C89 = [72]; R7C8 = 1; R6C2 = 1; R8C3 = 1
Really didn’t wanna collapse to singles yet.
46.R9C12 = {39} (step 33a) -->> locked for R9 + N7
And now it is all singles and basic cage sums.

greetings

Para
Walkthrough by Jean-Christophe:
Here is my walkthrough for V1. Will try the Hevvie
1. Cage 3/2 in R12C5 = {12} (NP @ N2, C5)
2. Cage 7/2 in R23C6 = {34} (NP @ N2, C6)
3. Cage 8/2 in R89C5 = {35} (NP @ N8, C5)
4. Cage 13/2 in R78C6 = {67} (NP @ N8, C6)
5. Cage 3/2 in R9C67 = {12} (NP @ R9)
6. Cage 10/2 in R9C34 = [64]
7. Since R1C4 <> 3, Cage 12/2 in R1 -> R1C3 <> 9
8. Outies of C4 -> R15C3 = 14 = [59], R1C4 = 7
9. Cage 9/2 in R1C67 = [81], R12C5 = [21], R9C67 = [12]
10. Outies of C6 -> R5C7 = 4
11. Cage 14/2 in R23C4 = {59} (NP @ N2, C4)
12. Cage 10/2 in R78C4 = {28} (NP @ N8, C4)
13. R37C5 = [69]
14. Two Cages 9/2 in R5 <> {45}, R5C6 = 5
15. Innies of R5 -> R5C45 = 9 = [18]
16. Cage 7/3 in R234C1 = {124} (NT @ C1)
17. 45 on N1 -> R23C1+R3C2 = 12 = {1(29|47)} -> R3C1 = 1, R3C2 = {79}, R4C123 = 11
18. 45 on N9 -> R7C89+R8C9 = 8 = {134}, R6C789 = 17
19. 45 on N3 -> R23C9+R3C8 = 16, R4C789 = 15
20. 45 on N7 -> R7C12+R8C1 = 20 <> {12..}, R6C123 = 16
21. Innies of R4 -> R4C456 = 19 = [(37|64)9], R6C6 = 2
22. Innies of R6 -> R6C45 = 10 = [37|64]
23. Cage 12/3 in R678C9 = {138|147|345} -> R6C9 = {578}
24. 9 of R6 locked in R6C78 -> Cage 13/3 in N69 = {139}, NT -> R45C8 <> {13}
25. 17/3 in R6C789 -> R6C9 = {57}
26. 8 of R6 locked in R6C123 -> not elsewhere in N4, 16/3 in R6C123 = {8(17|35)} <> {46..}
27. R6C45 = [64] (HS), R4C45 = [37]
28. 4 of N9 locked in R78C9 -> not elsewhere in C9
29. Cage 16/3 in R34C8+R4C7 = {259|268|358} <> {47..}
30. 45 on N7 -> R7C2 = R6C1 = {3578}
31. R78C3, R8C2 = HT on {124} in N7 -> R9C12 = 12 = {39|57}
32. R478C3 = NT {124} @ C3
33. R3C6 = 4 (HS @ R3), R2C6 = 3
34. C37 & R36 = X-Wing on 3 -> not elsewhere in R36
35. 2 of R3 locked in R3C89 -> 16/3 in R23C9+R3C8 = {2(59|68)}
36. Cage 16/3 in R34C8+R4C7 = {259|268} = {2..} -> R5C8 <> 2 = {67} -> R5C9 {23}
37. Innies of C9 -> R159C9 = 18 = {9(27|36)} -> R1C9 = {69}, R9C9 = {79}
38. R9C8 = 8 (HS @ R9)
39. 9 locked in R19C9 -> not elsewhere in C9
40. 16/3 in R23C9+R3C8 = {259|268} = {(6|9)..} & R1C9 = complex naked pair on {69} -> not elsewhere in N3
41. Cage 28/5 in N1 = {368(29|47)}, 2 of N1 locked in R2C12 -> R2C2 <> 9
42. Cage 14/2 in R23C4 = [95] (HS @ R2)
43. 45 on R12 -> R3C37 = R2C19 + 2
43b. 3 of R3 locked in R3C37 = {3(7|8)} = 10|11
-> R2C19 = 8|9 = [26|45]
43c. {26} of R2 locked in R2C129
43d. Since R2C2 can't hold both {26} -> R2C19 = [26]
This unlocks the puzzle, almost
...

Edited to fix typos
Walkthrough by PsyMar:
Here's my solution for V1 -- all fairly standard killer stuff until step 49, which involves Medusa Coloring of a sort, with a bit of a twist at the end. It might be easier to write it as a double implication chain in Eureka notation, but I didn't.

I am, as always, quite prone to errors and quite slow to fix them even after they're pointed out. I don't mean to seem ungrateful, I'm just lazy.

Maybe I'll find a few days this summer to fix them all.

1. sole combination for 3/2 in n2 = {12} pair -> elim {12} from rest of n2/c5
2. sole combination for 8/2 in n8 = {35} pair -> elim {35} from rest of n8/c5
3. sole combination for 7/2 in n2 = {34} pair -> elim {34} from rest of n2/c6
4. sole combination for 13/2 in n8 = {67} pair -> elim {67} from rest of n8/c6
5. innies of c789 = r159c7 = 7/3 = {124} triple -> elim {124} from rest of c7
6. sole combination for 3/2 in r9 = {12} pair -> elim {12} from rest of r9
7. sole permutation for 10/2 in r9 = [64] -> make eliminations
8. permutations for 9/2 in r1 = [54|81]
9. 9 of r6 locked in n5 -> elim from rest of n5
10a. combinations for 14/2 in c4 = {59|68} -> no 7
10b. combinations for 10/2 in c4 = {19|28}
10c. 14/2 and 10/2 in c4 form killer pair {89} -> elim {89} from r1456c4
11. sole combination for 12/2 in r1 = {57} pair -> elim {57} from rest of r1 -> 9/2 in r1 = [81] -> 3/2 in r9 = [12] -> 3/2 in n2 = [21] && r5c7 = 4
12. sole combination for 10/2 in n8 = {28} -> elim {28} from rest of c4/n8 -> r7c5 = 9
13. sole combination for 14/2 in n2 = {59} -> elim {59} from rest of c4/n2 -> 12/2 in r1 = [57] -> r3c5 = 6
14. outies of n5 = r5c3 = 9
15. innies of r5 = r5c456 = 14/3; only permutation is [185]
16. innies of r1234 = 19/3 = {379|469} -> r4c6 = 9 -> r6c6 = 2
17. sole combination for 35/5 in n9 = {56789} -> elim from rest of n9
18. outies-innies of n9 = r6c9-r7c8 = 4 -> r6c9 = {578}
19. sole combination for 7/3 in c1 = {124} -> elim from rest of c1
20. outies-innies of n1 = r4c1-r3c2 = -5 -> r3c2 = {79}, r4c1 = {24}
21. r3c1 = 1 {hidden single in c1}
22. outies-innies of n7 = r6c1-r7c2 = 0 -> r6c1 = r7c2 = {3578}
23. r79c12+r8c1 = naked quintuple {35789} -> elim from rest of n7
24. r78c3+r8c2 = {124} = 7; rest of 19/5 in n7 = r9c12 = 12/2 = {39|57}
25. split cage 12/2 in r9c12 forms killer pair {35} with r9c5; elim 5 from r9c89
26. 8 of r9 locked in n7; elim from rest of n7
27. outies-innies of n3 = r4c9-r3c8 = -1 -> r3c8 != 5, r4c9 != 5
28a. combinations for 12/2 in c9 = {138|147|345}
28b. innies of c9 = r159c9 = 18/3 without 1 or 5 = {279|369} (378|468 conflict with 28a) -> no 4|8 in r159c9, no 9 in rest of c9
29. r9c8 = 8 (hidden single in r9)
30. combinations for 15/3 in c9 = {168|258|267|456} (348|357 conflict with 28a) -> no 3 in 15/3 in r9
31. outies-innies of n3 = r4c9-r3c8 = -1 -> r3c8 != 4, r4c9 != 7
32. outies of n3 = r4c789 = 15/3 = {168|258} (does not have 4 or 9; 267|357 conflicts with 9/2 in n6) -> no 3|7 in r4c789; elim 8 from rest of r4c789
33. outies of n1 = r4c123 = 11/3 = {146|245} (must have 2 or 4; 236 conflicts with r4c4) -> no 3|7 in r4c123 && no 4 in rest of r4 -> r4c5 = 7 -> r6c5 = 4
34. r4c4 = 3 (hidden single in r4) -> r6c4 = 6
35. 8 of r4 locked in n6 -> elim from rest of n6
36. combinations for 12/3 in c9 = {147|345} -> must have 4 in r78c9 -> elim from rest of c9/n9
37. combinations for 15/3 in c9 = {168|258} -> no 7, and r4c9 cannot be 6 as only permutation for {168} is [681]
38. outies-innies of n3 = r4c9-r3c8 = -1 -> r3c8 != 7
39. combinations for 28/5 in n3 = {24679|34579|34678} (does not have 1; must have 4 and 7 (4 and 7 of n3 locked in cage)
40. outies of n1 = r4c123 = 11/3 = {146|245} (by step 33) -> either 5 or 6 is in r4c2 (only spot for 5 or 6, and one of the two must be there)
41. r478c3 = naked triple {124} -> elim from rest of c3
42. r3c6 = 4 (hidden single) -> r2c6 = 3
43. 2 of n1 locked in r2 -> elim from rest of r2
44. combinations for 28/5 in n1 with no 1 or 2 = {34579|34678} -> no 3 in rest of n3
45. outies-innies of n3 = r4c9-r3c8 = -1 -> r4c9 != 2
46. permutations for 15/3 in c9 = [528|681] = [{56}{28}{18}]
47. innies of c9 = r159c9 = 18/3 = [369|639|927] = [{369}{236}{79}]
48. innies of c1 = r159c1 = 18/3 = [369|639|675|963] = [{369}{367}{359}]
49a. Medusa: (9)r3c8 blue <-> (2)r3c8 red <-> (2)r3c9 blue <-> (8) r3c9 red <-> (8) r4c9 blue <-> (1)r4c9 red
49b. Medusa continued: by cage sums in 15/3 in c9, (6)r2c9 red and (5)r2c9 blue <-> (5)r6c9 red <-> (7)r6c9 blue <-> (7)r9c9 red <-> (7)r9c2 blue
49c. (9)r3c8 and (7)r9c2 both blue -> blue elims all digits from r3c2 -> elim all blue digits mentioned, place all red digits mentioned
49d. naked singles: r4c8 = 6 -> r4c7 = 8 -> r4c2 = 5
50. r1c9 = hidden single (9) in c9
51. r5c9 = hidden single (2) in c9 -> r5c8 = 7
52. r7c8 = hidden single (1) in n9
53. 9 of n1 locked in c2 -> elim from rest of c2 -> naked singles and last-digit-in-cage moves solve it


Now I'll work on the hard version...
Walkthrough by Andrew:
We now have 3 walkthroughs for the original Assassin 48 and all had different breakthroughs from the way that I finished it so here is my walkthrough. Looks like my way took Ruud's introductory comment more literally than the others.

All the other 3 have neat breakthroughs. Mine is a bit more routine but I hope still worth looking at.

Thanks for the corrections Ed. I've also added an extra comment to step 25.

1. R1C34 = {39/48/57}, no 1,2,6

2. R12C5 = {12}, locked for C5 and N2

3. R1C67 = [36/45/54/63/72/81], no 9, no 7,8 in R1C7

4. R23C4 = {59/68}

5. R23C6 = {34}, locked for C6 and N2, clean-up: no 8,9 in R1C3, no 5,6 in R1C7

6. R5C12 = {18/27/36/45}, no 9

7. R5C89 = {18/27/36/45}, no 9

8. R78C4 = {19/28/37/46}, no 5

9. R78C6 = {58/67}, no 1,2,9

10. R89C5 = {35}, locked for C5 and N8, clean-up: no 7 in R78C4, no 8 in R78C6
10a. Naked pair {67}in R78C6, locked for C6 and N8, clean-up: no 2,3 in R1C7, no 4 in R78C4

11. R9C67 = {12}, locked for R9

12. R9C45 = [64]

13. R234C1 = {124}, locked for C1, clean-up: no 5,7,8 in R5C2

14. 19(5) cage in N7 = 1{2349/2358/2457}, 1,2 locked for N7 [Edit. Unnecessary combinations removed.]

15. 35(5) cage in N9 = {56789}, locked for N9

16. Killer pair 8/9 in R23C4 and R78C4 for C4, clean-up: no 3,4 in R1C3

17. Naked pair {57} in R1C34, locked for R1 -> R1C67 = [81], R12C5 = [21], R9C67 = [12], clean-up: no 6 in R23C4, no 9 in R78C4
17a. Naked pair {59} in R23C4, locked for C4 and N2 -> R1C34 = [57]
17b. Naked pair {28} in R78C4, locked for C4 and N8

18. R3C5 = 6, R7C5 = 9 (naked singles)

19. R456C4 = {136} -> R5C3 = 9

20. R456C6 = {259} -> R5C7 = 4, clean-up: no 5 in R5C1, no 5 in R5C89

21. R5C6 = 5 (hidden single in R5)

22. 45 rule on R5 2 remaining innies R5C45 = 9 = [18]
[If this hadn’t fixed these two cells, there was killer triple 6/7/8 in R5C12, R5C5 and R5C89 for R5 which eliminates 6 from R5C4]

23. 45 rule on N9 1 outie R6C9 – 4 = 1 innie R7C8 -> R6C9 = {578}

24. 45 rule on N7 1 innie R7C2 = 1 outie R6C1 -> no 6 in R6C1, no 4 in R7C2 [Edited because of correction to step 14.]

25. Hidden triple {124} in R7C3 +R8C23
[Alternatively naked quint {35789} in R7C12 + R8C1 + R9C12, locked for N7. I think I saw the hidden triple first.]
25a. 19(3) cage (step 14) = 124{39/57}, no 8 -> R9C12 = {39/57}

26. Killer pair 3/5 in R9C12 and R9C5 for R9

27. 8 in R9 locked in R9C89, locked for N9

28. 8 in C1 locked in R678C1
28a. R678C1 = 8{39/57}, 9 only in R8C1 -> no 3 in R8C1

29. 45 rule on N1 1 innie R3C2 – 5 = 1 outie R4C1 -> R3C2 = {79}, R4C1 = {24} -> R3C1 = 1 (hidden single in C1)

30. 45 rule on N3 1 innie R3C8 – 1 = 1 outie R4C9 -> no 5 in R3C8, no 5,9 in R4C9

31. 45 rule on R1234 3 innies R4C456 = 19 = [379/649] -> R4C6 = 9, R6C6 = 2

32. 9 in N6 locked in R6C78, 13(3) cage in N69 = {139}, no 1,3 in R45C8, R6C7 = {39}, R6C8 = {139}, R7C8 = {13}, clean-up: no 6 in R5C9, no 8 in R6C9 (step 23}

33. 4 in N9 locked in R78C9, locked for C9

34. 8 in R6 locked in R6C123, locked for N4
34a. 45 rule on N7 3 outies R6C123 = 16 = 8{17/35}, no 4,6

35. R6C5 = 4 (hidden single in R6) -> R4C5 = 7, clean-up: no 8 in R3C8 (step 30)

36. R6C4 = 6 (hidden single in R6) -> R4C4 = 3, clean-up: no 4 in R3C8 (step 30)

37. Naked triple {124} in R478C4, locked for C3 [Edit. My usual error of calling a naked ... a killer ...]

38. R3C6 = 4 (hidden single in R3) -> R2C6 = 3

39. 2 in R3 locked in R3C89, locked for N3

40. 4 in N4 locked in R4C123
40a. 45 rule on N1 3 outies R4C123 = 11 = [254/452/461] -> R4C2 = {56}

41. 8 in N6 locked in R4C789
41a. 45 rule on N3 3 outies R4C789 = 15 = 8{16/25}
41b. 1 only in R4C9 -> no 6 in R4C9, clean-up: no 7 in R3C8 (step 30)

42. 45 rule on C1 3 innies R159C1 = 18 = 6{39/57}
42a. 5 only in R9C1 -> no 7 in R9C1, clean-up: no 5 in R9C2 (step 25a)

43. 45 rule on N3 3 innies R2C9 + R3C89 = 16 = 2{59/68}, no 3,7, clean-up: no 2 in R4C9 (step 30)
43a. 6 only in R2C9 -> no 8 in R2C9

44. R678C9 = 4{17/35}
44a. R234C9 = {168/258} (cannot be {159} which clashes with R678C9) = 8{16/25}, no 9, 8 locked for C9
44b. 2 only in R3C9 -> no 5 in R3C9

45. 9 in C9 locked in R19C9
45a.45 rule on C9 3 innies R159C9 = 18 = 9{27/36}
45b. 2 only in R5C9 -> no 7 in R5C9, clean-up: no 2 in R5C8
45c. 6 only in R1C9 -> no 3 in R1C9 [Edit. Typo corrected]

46. R9C8 = 8 (hidden single in R9)

47. 6 in C9 locked in R12C9, locked for N3

48. R4C789 (step 41a) = 8{16/25}, 2 only in R4C8 -> no 5 in R4C8
48a. 1 only in R4C9 and no 8 in R4C8 -> no 6 in R4C7
[This also comes from the 16(3) cage in N36 with 2 locked in R34C8]

49. 6 in C7 locked in R78C7, locked for N9

50. 16(3) cage in N47 = 8{17/35} (step 34a with R7C2 = R6C1 from step 24), 1 only in R6C2 -> no 7 in R6C2

[While looking for how to make further progress, I spotted that R6C1 = R7C2 (step 24) produces a situation similar to step 34 in Mike’s Assassin 46 walkthrough; Ed told me that he did a similar move in Ruud’s second Special Killer-X. However it didn’t help here because at this stage, R6C1/R7C2 contain the same numbers as R123C3. I haven’t put this comment after step 24 because this process doesn’t work the other way round; it can’t eliminate candidates from R123C3.]

Ruud wrote "You really need wings to complete this journey."

These are presumably the four 5 cell cages, the wings of a butterfly. Until now I've made limited use of the two lower wings in N7 and N9. Now the upper left wing in N1 provides the final breakthrough, in the spirit of Ruud's introductory message.

51. 28(5) cage in N1 = 368{29/47}
51a. 28(5) cage cannot be {23689} because R2C2 = 2 => R1C12 = {69} clashes with R1C9 [Edit. Typo corrected.]
51b. 28(5) cage = {34678}, no 2,9, locked for N1
51c. 4 in N1 locked in R12C2, locked for C2
51d. 4 in N7 locked in R78C3, locked for C3
[Step 51a could have been used immediately after step 45c, although I didn't see it at that stage. By leaving it in the order that I made these steps, step 51 proves to much more powerful as can be seen below.]

and the rest is naked singles although it’s quicker if naked pairs and hidden singles are also used


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PostPosted: Thu Jun 19, 2008 12:50 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Assassin 48-Hevvie by Ruud (April 07)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6656:6656:3074:3074:2052:2821:2821:6151:6151:3337:6656:6656:3596:2052:1806:6151:6151:4881:3337:3603:6656:3596:7190:1806:6151:2841:4881:3337:3603:3603:5406:7190:3104:2841:2841:4881:3620:3620:5406:5406:7190:3104:3104:2091:2091:4141:3630:3630:5406:7190:3104:4147:4147:3637:4141:3630:5688:825:7190:3899:7996:4147:3637:4141:5688:5688:825:2371:3899:7996:7996:3637:5688:5688:2890:2890:2371:3149:3149:7996:7996:
Solution:
+-------+-------+-------+
| 1 6 5 | 7 2 8 | 3 4 9 |
| 2 3 7 | 9 6 4 | 8 1 5 |
| 4 8 9 | 5 1 3 | 2 7 6 |
+-------+-------+-------+
| 7 4 2 | 6 9 5 | 1 3 8 |
| 9 5 3 | 8 7 1 | 4 6 2 |
| 8 1 6 | 4 3 2 | 9 5 7 |
+-------+-------+-------+
| 5 7 4 | 1 8 9 | 6 2 3 |
| 3 9 1 | 2 5 6 | 7 8 4 |
| 6 2 8 | 3 4 7 | 5 9 1 |
+-------+-------+-------+
Quote:
sudokuEd: Very
mhparker: (Rating of) 2.5: Requires a team effort and several short to medium length hypotheticals. The TJK18 and A48-Hevvie would probably fall into this category.
rcbroughton: not an easy one by any means
Tag solution: sudokuEd, mhparker, rcbroughton & Glyn
sudokuEd: still in shock from Richard's 8 outtie's move to get a placement (step 102). Another world record for Richard. Too r-rated for a simplified walk-through though.
sudokuEd: Any puzzle raters out there who can give this puzzle a rating?
Andrew (in 2014): Another puzzle I've just tried for the first time. A tough one; it took me four days, but I was making progress each day. Unfortunately the original Tag solution isn't available, so I don't know how long that took.
In the condensed/simplified Tag walkthrough sudokuEd wrote:
Now that its easier to digest, can any one find a shortcut?
Yes, my step 35 can be used after step 53 of the condensed/simplified version.
edit: infamous step 102:
102. 45 rule on rows 1 to 2. Outies r3c3 r3c7 r3c1 r4c1 r3c4 r3c6 r3c9 r4c9 equal 44
102a. r4c19 potentially [18]/[19]/[28]/[29]/[78]/[79] - only possible placements are:
479315
7____8

165493
7____9

425396
7____8

569413
7____9

469325
7____8

175483
7____9

495326
7____8

265483
7____9

all the others blocked by no valid combos for 13(3)c1 or 19(3)c9

102b. only 7 allowed at r4c1 and no 1 at r3c3


103. 14(3)n14 can't be {347} - no 3 at r3c2


[For anyone intereseted in the combinations not allowed at step 102, here they are:
Code:
279485 179485 589426 269485 285496 295486_529486 589426
1____9 1____9 1____9 1____9 1____9 1____9_1____9 1____9

529486 179485 269485 285496 295486 589426
2____8 2____8 2____8 2____8 2____8 2____8

169485 185496 195486 519486 589416
2____9 2____9 2____9 2____9 2____9

569423 275483 175493 579413 265493
7____8 7____8 7____8 7____8 7____8

425386 485326 415396 469315 495316
7____9 7____9 7____9 7____9 7____9
Condensed/simplified walkthrough, though STALE, by sudokuEd:
Glyn wrote:
..(a move) to crack the problem of the Hevvie ...wonder how early it might have worked.
Step 57. Kind of.

Here is a condensed and simplified, though STALE (Still-Takes-A-Lotof-Effort) walk-through for Assassin 48 Hevvie.

Managed to get it down to using "just" 57 of the original 110 steps before Glyn's fine forcing chain could be used to crack it. Have included a bunch of new clean-up-at-the-end steps to muck it.

Many of the original steps have been changed/re-ordered to make them simpler and to take account of short-cuts found later. Have left the step numbers the same in case you want to check something. Let me know if its too confusing doing it this way.

Am still in shock from Richard's 8 outtie's move to get a placement (step 102). Another world record for Richard. Too r-rated for a simplified walk-through though.

Any puzzle raters out there who can give this puzzle a rating? Now that its easier to digest, can any one find a shortcut?


Assassin 48 Hevvie: Condensed/simplified walk-through.

1. 3(2)n8 = {12}:locked for c4, n8

2. 9(2)n8 = {36/45}(no 789) = [3/5..]

3. 8(2)n2 = {17/26}(no 3589) ({35} blocked by 9(2)n8 step 2)
3a. 8(2)n2 = [1/6..]

4. 7(2)n2 = {25/34}(no 16789} ({16} blocked by 8(2)n2 step 3a)
4a. 7(2)n2 = [2/3,4/5..]

5. 1 in c6 in 12(4)n5 = 12{36/45}(no 789)
5a. 1 locked for n5 and no 1 r5c7
5b. 12(4) must have 2 -> no 2 r5c5

6. complex hidden Killer pair 2/3 r23456c6. Here's how.
6a. 12(4)n5 = {1236} ->2/3 must be in r5c7 so r456c6 won't clash with 7(2)n2 (step 4b.) -> [2/3] both locked in 7(2) and r456c6 for c6
6b. 12(4)n5 = {1245} -> 4 must be in r5c7 so r456c6 won't clash with 7(2) (step 4b) -> [2/3] both locked in 7(2) and r456c6
6c. r5c7 = {234}
6d. no 4 r456c6
6d. 2 and 3 locked for c6 in r23456

7. 11(2)n2 = [47/56/65/74/83/92]
7a. r1c7 = 2..7

8. 12(2)n8 = [48/57/75/84/73](no 1,2,6)
8a. r9c7 no 9

9. "45"c789 -> 3 innies r159c7 = 12 = h12(3)c7 = {237/[624]/345}(no 8)
9a. no 4 r9c6

10. A neat little contradiction chain eliminates [624] from h12(3)c7
10a. from step 6a, 2 in r5c7 -> r456c6 = {136} -> 7(2)n2 = {25}
10b. 6 in r1c7 -> 5 in r1c6: but 2 5's c6
10c. h12(3) = {237/345} = 3{27/45}(no 6)
10d. no 5 r1c6
10e. 3 locked c7

11. 8 and 9 in c5 only in 28(5)
11a. 28(5) {14689} blocked by 8(2)n2
11b. 28(5) = {13789/23689/24589}
11c. 1 only in r3c5 -> no 7 r3c5

12. 4 in c6 only in n2: 4 locked for n2

13. 14(2)n2 = {59/68}

14. 12(2)n1 = {39/[48]/57}(no 1,2,6, no 8 r1c3)

15. "45"n2: 3 innies = 16 = h16(3)n2 = {178/259/268/349}
15a. {169/367} blocked by 8(2)
15b. {358/457} blocked by 7(2)
15c. 1 and 2 are only available in r3c5 for 3 innies -> no 5,6,8 r3c5
15d. r1c46 + r3c5 = {[78]1/[59]2/[86]2/[34]9/[94]3}. Note: blocked combo's
i.[87]1 :blocked 2 4's r1c37
ii.[39]4 :blocked 2 9's r1c37
15e. no 7 r1c6 -> no 4 r1c7

16. 7 in c6 only in n8: 7 locked for n8
16a. r9c3, no 1,4,9

17. "45"n8 -> 3 innies = 18 = h18(3)n8
17a. h18(3) = {369/378/459} ({468/567} blocked by 15(2))
17b. {369} combo: if r9c46 = [39] means r9c467 = [393](2 3's r9) -> {369} combo must have 3 in r7c5 -> no 6 r7c5 (since no other combo with 6)
17c. {378} combo: 7 only in r9c6 -> no 8 r9c6 (since only combo with 8)
17d. no 4 r9c7
17e. {459} combo:
i.if r9c46 = [45] -> r9c3467 = [7457] clash -> 9 must be in r9c46 -> no 9 r7c5
ii.if r7c5 + r9c46 = 4[59] -> 9(2)n8 = {36} and r9c37 = [63]: clash in r9!
iii. -> can only have r7c5 + r9c46 = 4[95]/5[49]
iv. no 5 r9c4

17f. In summary: h18(3)n8 = r5c5 + r9c46 = [3[69]/3[87]/8[37]/4[95]/5[49]
17g. no 6 r7c5,
17h. no 8 r9c6 -> no 4 r9c7
17i. no 9 r7c5
17j. no 5 r9c4 -> no 6 r9c3

20. 19(3)n3: no 1
20a.14(2)n4 = {59/68}
20b. 8(2)n6: no 4,8,9

The first of the clever "45"s with another chain move.
23. 45 on n147 & n5 (!!) - outies: r1c4 r3c5 r7c5 r5c7 r9c4 = 23
i. -> no 3 at r3c5. Here's how
23a. from step 15d, 3 in r3c5 -> 9 in r1c6 -> r3c5 + r1c4 = [39] = 12
23b. 3 in r3c5 -> 28(5) = {13789/13689}(no 4,5) -> r7c5 can only be 8
23c. from step 17f. the only combo in h18(3)n8 with 8 in r7c5 is
23d. 8[37]: -> r7c5 + r9c4 = [83] = 11
23e. These 4 outies of n147 & 5 with r3c5 = 3 sums to 12 + 11 = 23. But the 5 outies must = 23, not just the 4.
23g. -> no 3 at r3c5
23h. ->h16(3)n2 = r1c46 + r3c5 = {[78]1/[59]2/[86]2/[34]9}
23i. no 9 r1c4
23j. no 3 r1c3

A big contradiction chain - but essential to solve this puzzle.
26. no 9 r1c3. Here's how.
26a. r1c3 = 9 -> r1c46 = [34] (step 23h.)-> r1c7 = 7
26b. -> h12(3)c7: r159c7 = [723]
26c. -> r9c46 = [69/49](step 17f.) -> r9c3 = 5/7
26d. "45"c123: r159c3 = 16 = h16(3)c3
26e. but 7 is blocked from r9c3 because cannot have r19c3 = [97] = 16 in the h16(3)c3
26f. -> h16(3)c3 = [925]
26g. however,this last combo is also blocked since the 2 in r5c3 clashes with the 2 already in c7 (step 26c). Dizzy yet?
26h. conclusion: no 9 in r1c3 -> no 3 r1c4

27. 3 now locked in r23c6 for n2 - locked for c6
27a. 7(2)n2={34} - locked for n2 and c6
27b. 2 in c6 only in n5: 2 locked for n5 and no 2 r5c7
27d. cleanup 11(2)n23 - no 7 at r1c7

28. 5 now locked in r123c4 for n2 - 5 locked for c4
28a. cleanup: 21(4)n45 - no 7 possible at r5c3: since no 1,2,5 in r456c4

29. cleanup 28(5)c5=1{3789}/2{3689}/2{4589} = [1/2]
29a. 1 and 2 only available in r3c5 -> no 9 at r3c5

25. no 9 r5c3. Here's how - with a short chain.
25a. "45"c123: r19c4 - 7 = r5c3
25b. r5c3 = 9 -> r19c4 = 16 = [79]
25c. r19c4 = [79] -> 14(2)n2 = {68}
25d. but this leaves no 5 for c4
25e. -> no 9 r5c3

30. no 9 at r9c4. Here's how.
30a. "45"c123: r19c4 - 7 = r5c3
30b. max. r5c3 = 8 -> max. r19c4 = 15
30c. ->if r9c4 = 9 can only have r19c4 = [59] = 14:
30d. -> r5c3 must = 7: but no 7 in r5c3
30b. -> no 9 at r9c4
30c. cleanup 11(2)n78 - no 2 at r9c3

31. 11(2) & 12(2) r9 must use 3 (i.e. {38}{75}/[56][93]/[74][93]) -
31a. 3 locked for r9
31b. cleanup 9(2)n8 - no 6 at r8c5

32. Cleanup from step 30b.
32a. from step 17f. h18(3)n8 now = [3[69]/3[87]/8[37]/5[49]
32b. no 4 r7c5
32c. no 5 r9c6
32d. no 7 r9c7

35. 9 in n8 now only in c6: 9 locked for c6
35a. no 2 r1c7

41. naked pair {35} at r19c7: 3 and 5 locked for c7
41a. r5c7=4
41b. 12(4)n56 = {125}4 - no 6

38. 9 now locked in 14(2) for n2 = {59}: locked for c4
37a. 12(2)n12 no 7 at r1c3
38a. 21(4)n45 - no 5 or 9 r456c4 so cannot have 1 at r5c3

39. 8 locked in r1c46 for n2: 8 locked for r1

40. 5 locked in r789c6 for c6 - 5 locked for n5

45. "45" on c123: r159c3 = 16 = h16(3) = [457]/5{38} = 5{47/38}
45a. r5c3 = {358}
45b. r9c3 = {378}
45c. no 6 r9c4
45d. 5 locked for c3

42. 5 & 8 locked in 12(2)/11(2) for r1 - no 5 in rest of r1

43. 21(4) n45 = 3{468/567}
43a. must use 3 - eliminates 3 at r5c5

46. 6 locked in n5 for c4 - 6 nowhere else in n5

Now: a very clever "45"
47. 45 on r5: innies = h19(4) = {1279}/{1378}/{2359}/{2368}
i.combos with 89, 56 or 69 blocked by 14(2)n3
47a. {1279} blocked by 1,2 only in r5c6
47b. -> h19(4)r5 = 3{178/259/268} = [1/2..]
47c. must use 3 - locked for r5
47d. cleanup - no 5 in 8(2)n6
47e. 1 or 2 required in h19(4) only in r5c6 -> no 5 r5c6
47f. h19(4) = {378}[1]/[5392]/[3682] ([3691]blocked by 14(2)n4)

49. from 47c. 3 locked in r5 is actually locked in c34 - eliminate 3 from r46c4 for 21(4) cage
49a. 21(4) now = {3468}/53{67}(no other combo with 5 at r5c3 step 47f.)
49b. no 7 at r5c4
49c. h19(4)r5 now = [{38}71/5392/3682]

48. 5 only in n4 for r5 - 5 nowhere else in n4

The next steps require careful unpicking of combinations.

57. No 6 in r5c4 because of a hard to spot contradiction.
57a. from step 49c, 6 in r5c4 -> h19(4)r5 must have 8 in r5c5
57b. the only combo for the 21(4)n45 with r5c4 = 6 is {3468} with 3 in r5c3 -> 8 in r46c4
57c. but this means 2 8's n5
57d. -> no 6 r5c4
57e. h19(4) now = {38}[71]/[5392]
57f. no 8 r5c5


58. 21(4)n45 - {3468} combo is only combo with 8 with {38} only in r5c34
58a. no 8 at r46c4

Another very clever "45" which only works because of the combo work above.
63. 4 Innies r1234: r1c5+r2c456 = 21. r3c5 = 1/2 -> r4c456 = 19 or 20
63a. At least one of r4c456 must be >= 8 (else max. = {567} = 18) -> r4c5 = {89} only
63b. {89} in r4c5 -> no {12} possible in r4c456
63c. -> r4c6 = 5
63d. r4c5 = {89}
63e. 3 remaining innies r4c4 + r34c5 = 16 = 6[19]/7[18]/6[28] (no 4 r4c4)
63f. -> r34c5 =[19/18/28]
63g. -> 28(5) = [197{38}/18973/28945]

65a. combining parts of steps 63e and g: r4c4 = 7 -> r3456c5 = [1897]
65b. but this means 2 7's n5
65c. -> r4c4 != 7
65d. r4c4 = 6
65e. from 63g: r34567c5 = [197]{38} or [28945] -> no 7,9 in r6c5
65f. Cleanup: no 4 in r3c8 (due to {456} unavailable in r4c78}


Now: it's finally time to move the focus: to n469
.
64. 4 in r4 now locked in n4 -> not elsewhere in n4
64a. Cleanup: no 8 in r7c2 (due to {45} unavailable in r6c23)

54. 45 on n3 - outies total 20(4). r1c6 = {68} -> r4c789 = 12 or 14
54a. if r4c789 = 12(3) = {138} ({129/237} blocked by 8(2)n6
54b. if r4c789 = 14(3) = [239]
54c. In summary, r4c789 = {138}/[239] = [8/9] not both
54d. no 7 r4c789
54e. no 2 r4c89

55. cleanup 11(3) n36 = {128/137/236}
55a. no 1,3,5,8 at r3c8

66. 7 in r4 now locked in n4 -> not elsewhere in n4
66a. Cleanup: no 1,6 in r7c2 (due to {457} unavailable in r6c23)

68. Outies n1: r1c4+r4c123 = 20
68a. r1c4 = {78} -> r4c123 = 12 or 13
68b. 7 already locked in r4c123 (step 66) -> no 8,9 in r4c123
68c. max. of r4c23 = 4 + 7 = 11 -> no 1, 2 in r3c2

69. "45"n9: r6c789 - 16 = r9c7
69a. r9c7 = {35} -> r6c789 = 19/21
69a. r6c789 must contain 1 of {89} for n6 (only place in n6 outside of r4c789 step 54c),
69b. r6c789 must contain exactly one of {67} - only place in n6 outside of 8/2 cage
69c. r6c789 must have 5 for n6
69d. if r9c7 = 3 -> r6c789 = 19 = {568} only
i. -> r6c78 + r7c8 = 16(3) = [{68}2] ([{58}3] blocked by 3 in r9c7 step 69a:[655] not valid for 16(3) cage)
ii. -> r6c789 = {68}[5]
69e. if r9c7 = 5 -> r6c789 = 21 = {579}
i. -> r6c78 + r7c8 = 16(3) = [754/952]
ii. -> r6c789 = [759/957]
69f. r7c8 = 2,4
69g. r6c9 = 5,7,9
69h. no 123 r7c78


70. 3 in n6 now locked in r4 -> not elsewhere in r4

53. 4 innies on n9 = 14 = h14(4).
i. from step 69d and e, r9c7 + r7c8 = [32/52/54]
53a. if r9c7 + r7c8 = [32] = 5 -> r6c9 = 5 (step 69dii) and r78c9 = 9 = {18} only ({27/36} blocked by [23])
53b. if r9c7 + r7c8 = [54] = 9 and r6c9 = 9 (step 69e) -> r78c9 = 5 = {23} ({14} blocked by r7c8)
53c. if r9c7 + r7c8 = [52] = 7 and r6c9 = 7 (step 69e) ->r78c9 = 7 = {34} ({16} means 14(3) = {167}-> 19(3)n3 = {289} which clashes with r5c9)

53d. In summary: r78c9 = {18/23/34}(no 5,6,7,9)
53f. h14(4)n9 = {1238/2345} = 23{18/45}
53g. 2 and 3 locked for n9
53h. 14(3)n6 = {158/239/347}

73. 2 in r9 locked in n7 -> not elsewhere in n7

75. Common Peer Elimination (CPE): r3c8 can see all candidate positions with digit 2 in c7
-> no 2 in r3c8
75a. 11/3 at r3c8 = {137/236} = 3{17/26} ({128} blocked by r3c8)
75b. r4c8 = 3
75c. r4c7 = {12}

This step uses a clever overlap of innies and combinations.
87. no 5 r9c89. Here's how.
87a. "45" on r9: 5 innies = 22 = h22(5) and must have 1 and 2 for r9 (and no 3) = {12469}/{12568} ({12478} blocked by 11(2)r9)
87b. 31(5)n9 = {16789}/{45679}
87c. Only combo with 5 in h22(5)r9 is {12568} -
i. 2 locked at c12
ii. c89={16}/{18}/{68}(from combos in 31(5)n9) (note:{56} blocked by r9c5 which must have 5 or 6)
87d. no 5 in c89

105a. i/o difference(n6): r4c9 + r6c789 = r3c8 + 22
-> no 6 in r6c8
(reason: would force r3c8 to 7 -> r4c9 + r6c79 = 23 = {689} -> 2 6's in n6)
105b. Only combo with 6 in 16(3) = {268} -> No 8 in r6c7

A final forcing chain to crack open this puzzle then some new mucking steps
1(a)If R1C7=5 requires R1C6=6 to satisfy 11(2) cage. R1C6 forces R9C5=6.
1(b)If R1C7=5 forces R1C3=4 forces R5C3=5 (since must have h16(3)c3 = [457])
1(c) ->The 14(2) cage in N4 must be {68}
1(d) forces R6C7=6 (single n6).
1(a) and 1(d) together force R8C8=6. (Nonet 9 is blocked for 6 in R9 and C7).

2) R1C7=5 forces R9C7=3 forces R8C8=5.

3) R8C8 cannot equal both 5 and 6 therefore R1C7<>5.

4)r1c67 = [83], r1c34 = [57], r9c67 = [75], r6c4 = 4
4a. r5c34 = {38}: locked for r5

5) 14(2)n4 = {59}:locked for n4, r5

6) r5c5 = 7 -> rest of 28(5) = {1389}
6a. r34c5 = [19]

7) 15(2) n8 = {69}: locked for n8, c6

8) r89c5 = [54]

9) r67c5 = {38}

10. 8(2)n6 = {26}: locked for n6

11. r4c79 = [18], r3c8 = 7

12. r56c6 = [12]

13. 11(2) n7 = {38}:locked for r9

14. "45" n9: r6c9 - 5 = r7c8
14a. no 5 r6c9
14b. r6c8 = 5

15. 24(5)n3 = {12489}(no 6):locked for n3
15a. r23c9 = {56}:locked for c9

16. r5c89 = [62]

17. 14(3)n6 = [7]{34} ({149} blocked by r1c9)
17a. {34} locked for c9, n9

18. r6c7 = 9, r7c8 = 2, r78c4 = [12]

19. r23c7 = {28}:locked for c7
19a. r8c8 = 8 (hsingle n9)

20. r9c89 = {19}:locked for r9

21. r9c12 = {26}:locked for n7
21a. 22(5)n7 = 246{19/37}
21b. 4 also locked for n7

22. naked pair {38} r59c3:locked for c3

23. 14(3)n7: = {158/167/356} = [5/7..]
23a. 5 and 7 only in r7c2 = {57}

24. deleted

25. 14(3)n1 r4c23 = {247} -> 14(3) = {248/347} = [3/8..]
25a. 3 or 8 only in r3c2 = {38}

26. "45" n1: 3 innies = 14
26a. r3c2 + r23c1 = 14 = 3[74]/8{24}
26b. r23c1 = {247}
26b. 4 must be in r23c1: locked for c1, n1
26c. 13(3)n1 = [742]/{24}7 = {247}:locked for c1
26d. r4c1 = {27}

27. r9c12 = [62]

28. 16(3)n4 = {358} = [853]

29. r5c12 = [95], r5c34 = [38], r67c5 = [38], r9c34 = [83]

30. r78c9 = [34], r7c2 = 7, r4c2 = 4, r78c7 = [67], r78c6 = [96]

31. r7c3 = 4, r1c19 = [19], r12c8 = [41], r1c25 = [62], r2c5 = 6

32. r23c9 = [56], r23c4 = [95]

33. split 26(5)n1: {289} blocked by r2c7
33. = {379} only

Time for a beer.
Walkthrough by Andrew:
Prelims

a) R1C34 = {39/48/57}, no 1,2,6
b) R12C5 = {17/26/35}, no 4,8,9
c) R1C67 = {29/38/47/56}, no 1
d) R23C4 = {59/68}
e) R23C6 = {16/25/34}, no 7,8,9
f) R5C12 = {59/68}
g) R5C89 = {17/26/35}, no 4,8,9
h) R78C4 = {12}
i) R78C6 = {69/78}
j) R89C5 = {18/27/36/45}, no 9
j) R9C34 = {29/38/47/56}, no 1
k) R9C67 = {39/48/57}, no 1,2,6
l) 19(3) cage at R2C9 = {289/379/469/478/568}, no 1
m) 11(3) cage at R3C8 = {128/137/146/236/245}, no 9
n) 12(4) cage at R4C6 = {1236/1245}, no 7,8,9

1. Naked pair {12} in R78C4, locked for C4 and N8, clean-up: no 7,8 in R89C5, no 9 in R9C3
1a. R12C5 = {17/26} (cannot be {35} which clashes with R89C5), no 3,5 in R12C5
1b. R23C6 = {25/34} (cannot be {16} which clashes with R12C5), no 1,6 in R23C6

2. 1 in C6 only in R456C6, locked for N5 and 12(4) cage at R4C6, no 1 in R5C7
2a. 12(4) cage at R4C6 = {1236/1245}
2b. 6 of {1236} must be in R456C6 (R456C6 cannot be {123} which clashes with R23C6), no 6 in R5C7
2c. 5 of {1245} must be in R456C6 (R456C6 cannot be {124} which clashes with R23C6), no 5 in R5C7
2d. 4 of {1245} must be in R5C7 (R456C6 cannot be {145} which clashes with R23C6), no 4 in R4C456C6

3. 45 rule on C789 3 innies R159C7 = 12 = {237/246/345}, no 8,9, clean-up: no 2,3 in R1C6, no 3,4 in R9C6
3a. 4 in C6 only in R123C6, locked for N2, clean-up: no 8 in R1C3

4. 45 rule on N8 3 innies R7C5 + R9C46 = 18 = {369/378/459} (cannot be {468/567} which clash with R89C5)
4a. 45 rule on N8 2 innies R7C5 + R9C4 = 1 outie R9C7 + 6, IOU no 6 in R7C5
4b. 45 rule on N8 2 innies R7C5 + R9C6 = 1 outie R9C3 + 7, IOU no 7 in R7C5
4c. R9C34 = 11, R9C67 = 12, R9C3 cannot equal R9C7 -> R9C6 cannot be 1 more than R9C4
4d. 9 of {369} must be in R9C6, 9 of {459} must be in R9C46 (R9C46 cannot be [45] because R9C6 cannot be 1 more than R9C4) -> no 9 in R7C5
4e. {378} can only be [387/837] (cannot be [378] because R9C6 cannot be more than R9C4) -> no 7 in R9C4, no 8 in R9C6, clean-up: no 4 in R9C4, no 4 in R9C7
4f. R7C5 + R9C46 = {459} cannot be [459] (because R7C5 + R9C3467 = [46593] clashes with R9C5) -> no 5 in R9C4, clean-up: no 6 in R9C3
4g. R7C5 + R9C46 = {369} => R9C34 = [56] or R7C5 + R9C46 = {378} => R9C67 = [75] or R7C5 + R9C46 = {459}, 5 locked for N8 -> no 5 in R9C5, clean-up: no 4 in R8C5
4h. 7 in N8 only in R789C6, locked for C6, clean-up: no 4 in R1C7

5. 45 rule on N2 3 innies R1C46 + R3C5 = 16 = {178/259/268/349} (cannot be {169/358} which clash with R23C4, cannot be {367} which clashes with R12C5, cannot be {457} which clashes with R23C6)
5a. 1,2 of {178/259/268} must be in R3C5 -> no 5,6,7,8 in R3C5

6. R159C7 (step 3) = {237/345} (cannot be {246} because no 2,4,6 in R9C7), no 6, 3 locked for C7, clean-up: no 5 in R1C6

7. 45 rule on N9 4 innies R7C89 + R8C9 + R9C7 = 14 = {1238/1247/1256/1346/2345}, no 9
7a. 45 rule on N9 2 innies R7C8 + R9C7 = 1 outie R6C9
7b. Min R7C8 + R9C7 = 4 -> min R6C9 = 4
7c. Max R7C8 + R9C7 = 9, min R9C7 = 3 -> max R7C8 = 6

8. 45 rule on C123 2 outies R19C4 = 1 innie R5C3 + 7
8a. 7 in C4 either in R1C4 => R5C3 = R9C4 = {34689} or R456C4, locked for 21(4) cage at R4C4, no 7 in R5C3 -> no 7 in R5C3

9. 45 rule on C123 3 innies R159C3 = 16 = {178/259/349/358/367/457} (cannot be {169} because no 1,6,9 in R9C3, cannot be {268} because no 2,6,8 in R1C3)
9a. Consider combinations for R23C4 = {59/68}
R23C4 = {59}, locked for N2 => R23C6 = {34}, locked for N2 => no 3 in R1C4 => no 9 in R1C3
or R23C4 = {68}, locked for C4 => no 6 in R9C4 => no 5 in R9C3
-> R159C3 cannot be [925] -> no 2 in R5C3
[Note that this doesn’t eliminate the {259} combination, there’s still 2 in R9C3.]
9b. R7C5 + R9C46 (step 4) = {369/378/459}
9c. Again consider combinations for R23C4 = {59/68}
R23C4 = {59}, locked for N2 => no 5 in R1C4 => no 7 in R1C3
or R23C4 = {68}, locked for N2 => 8 in C6 only in R78C6 = {78}, locked for N8 => R7C5 + R9C46 = {369/459}, 6 of {369} must be in R9C4 => no 3 in R9C4 => no 8 in R9C3
-> R159C3 cannot be [718] -> no 1 in R5C3

10. 21(4) cage at R4C4 = {3459/3468/3567}, CPE no 3 in R5C56

11. R159C3 (step 9) = {259/349/358/367/457}, 21(4) cage at R4C4 (step 10) = {3459/3468/3567}
11a. R1C34 = 12, R9C34 = 11, R1C4 cannot equal R9C4 -> R1C3 cannot be 1 more than R9C3
11b. R159C3 cannot be [493] because R1C3 cannot be 1 more than R9C3, cannot be [943] because R1C34 + R5C3 = [934] clashes with 21(4) cage at R4C4 -> R159C3 cannot be {349} -> R159C3 = {259/358/367/457}
11c. Consider placement for 3 in 21(4) cage
3 in R5C3 => R159C3 = {358/367}
or 3 in 21(4) cage in C4, locked for C4, no 3 in R1C4 => no 9 in R1C3
-> no 9 in R1C3, clean-up: no 3 in R1C4

12. R1C46 + R3C5 (step 5) = {178/259/268/349}
12a. 1,2,3 only in R3C5 -> R3C5 = {123}
12b. 9 in C5 only in R456C5, locked for N5

13. R159C3 (step 11b) = {259/358/367/457}, 21(4) cage at R4C4 (step 10) = {3459/3468/3567}
13a. Consider combinations for R1C46 + R3C5 (step 5) = {178/259/268/349}
R1C46 + R3C5 = {178}, locked for N2 => R23C4 = {59}, locked for C4 => 21(4) cage at R4C4 = {3468/3567} (cannot be {3459} because 5,9 only in R5C3)
or R1C46 + R3C5 = {259/268/349}, no 7 in R1C4 => no 5 in R1C3
-> R159C3 = {358/367/457} (cannot be {259} = [592]), no 9 in R5C3, no 2 in R9C3, clean-up: no 9 in R9C4
13b. 9 in N8 only in R789C9, locked for C9, clean-up: no 2 in R1C7
13c. Hidden killer pair 7,9 in R78C6 and R7C9 for C9, R78C6 contains one of 7,9 -> R9C6 = {79}, clean-up: no 7 in R9C7
13d. R159C7 (step 6) = {237/345}
13e. 2,4 only in R5C7 -> R5C7 = {24}

14. R7C5 + R9C46 (step 4) = {369/378/459}
14a. 5 of {459} must be in R7C5 -> no 4 in R7C5
14b. 4 in N8 only in R9C45, locked for R9

15. R7C89 + R8C9 + R9C7 (step 7) = {1238/1256/1346/2345} (cannot be {1247} because R9C7 only contains 3,5), no 7 in R78C9

16. 1,2 in R9 only in R9C1289
16a. 45 rule on R9 5 innies R9C12589 = 22 = {12469/12478/12568} (cannot be {12379} which clashes with R9C6), no 3 in R9C12589, clean-up: no 6 in R8C5

17. R1C46 + R3C5 (step 5) = {178/268/349} (cannot be {259} because R1C6 only contains 4,6,8), no 5, clean-up: no 7 in R1C3

18. 21(4) cage at R4C4 (step 10) = {3468/3567}
18a. 4 of {3468} must be in R456C4 (R456C4 + R5C3 cannot be {368}4 which clashes with 12(4) cage at R4C6) -> no 4 in R5C3

19. R159C3 (step 13a) = {358/367/457} -> R159C3 contains one of 5,6
19a. Consider combinations for 21(4) cage at R4C4 (step 10) = {3468/3567}
6 of {3468} must be in R456C4 (R5C3 cannot be 6 => no 5 in R9C3, no 6 in R9C4 when R456C4 = {348} clashes with R9C4)
or 6 of {3567} must be in R456C4 (R456C4 cannot be {357} which clashes with 12(4) cage at R4C6)
-> no 6 in R5C3
19b. 21(4) cage at R4C4 = {3468/3567}, 6 locked for C4 and N5, clean-up: no 8 in R23C4, no 5 in R9C3
19c. R9C12589 (step 16a) = {12469/12568} (cannot be {12478} which clashes with R9C34), no 7

20. 12(4) cage at R4C6 = {1245} (only remaining combination) -> R5C7 = 4, R456C6 = {125}, locked for C6 and N5
20a. Naked pair {34} in R23C6, locked for N2, clean-up: no 7 in R1C7
20b. Naked pair {35} in R19C7, locked for C7

21. R1C46 + R3C5 (step 17) = {178/268}, no 9, 8 locked for R1, clean-up: no 3 in R1C3

22. 1,2 in R1 only in R1C12589
22a. 45 rule on R1 5 innies R1C12589 = 22 = {12379/12469}, no 5

23. Hidden killer pair 1,2 in R5C6 and R5C89 for R5 -> R5C6 = {12}, R5C89 = {17/26}, no 3,5
23a. 3 in R5 only in R5C34, locked for 21(4) cage at R4C4, no 3 in R46C4
23b. 5 in R5 only in R5C123, locked for N4

24. R159C3 (step 13a) = {358/457}, 5 locked for C3

25. Consider combinations for 21(4) cage at R4C4 (step 19a) = {3468/3567}
21(4) cage = {3468}, CPE no 8 in R5C5
or 21(4) cage = {3567} => R5C3 = 5, R5C12 = {68}=> R5C5 = 9 (hidden single in R5)
-> R5C5 = {79}
25a. Killer triple 6,7,9 in R5C12, R5C5 and R5C89, locked for R5
25b. 3,8 of {3468} must be in R5C34, locked for 21(4) cage -> no 8 in R46C4

26. 45 rule on R1234 4(1+3) innies R3C5 + R4C456 = 21
26a. Max R4C45 = 16 -> min R3C5 + R4C6 = 5 -> R4C6 = 5
26b. R3C5 + R4C45 = 16 = [169/178/268] -> R4C4 = {67}, R4C5 = {89}
26c. 4 in N5 only in R6C45, locked for R6
26d. 8,9 in C5 only in 28(5) cage at R3C5 = {13789/24589}
26e. R3C5 + R4C45 = [169/268] (cannot be [178] which clashes with 28(5) cage) -> R4C4 = 6
26f. R34C5 = [19/28] -> 28(5) cage = [19738/19783/28945], no 7,9 in R6C5

27. 11(3) cage at R3C8 = {128/137/236} (cannot be {146/245} because 4,5,6 only in R3C8), no 4,5 in R3C8
27a. 8 of {128} must be in R4C78 (R4C78 cannot be {12} which clashes with R5C89) -> no 8 in R3C8
[I didn’t think it was worth using interactions between the 11(3) cage and R5C89 at this stage.]

28. 45 rule on C1 3 innies R159C1 = 16 = {169/178/259/268/358/367} (cannot be {349/457} because 3,4,7 only in R1C1), no 4 in R1C1

29. 14(3) cage at R6C9 = {149/158/167/239/248/257/347/356}
29a. 7 of {167} must be in R6C9, 6 of {356} must be in R78C9 (R78C9 cannot be {35} which clashes with R9C7) -> no 6 in R6C9
29b. R7C8 + R9C7 = R6C9 (step 7a)
29c. R9C7 = {35}, no 4,6 in R6C9 -> no 1 in R7C8

30. 4 in R4 only in R4C123
30a. 45 rule on N1 3 outies R4C123 = 1 innie R1C3 + 8
30b. R1C3 = {45} -> R4C123 = 12,13 = {147/148/247}, no 3,9
30c. R4C23 = {14/24/17/18/27} = 5,6,8,9 -> R3C2 = {5689} (cage sum)
30d. 45 rule on N1 2 innies R1C3 + R3C2 = 1 outie R4C1 + 6
30e. Min R1C3 + R3C2 = 9 -> no 1,2 in R4C1
30f. R4C1 = {478} -> R1C3 + R3C2 = 10,13,14 = [46/49/58/59], no 5 in R3C2

31. 3 in R4 only in R4C89, locked for N6
31a. 45 rule on N3 3 outies R4C789 = 1 innie R1C7 + 9
31b. R1C7 = {35} -> R4C789 = 12,14 = {138/239} (cannot be {237} which clashes with R5C89), no 7 in R4C789, 3 locked for N6
31c. 2 of {239} must be in R4C7 -> no 2 in R4C89
31d. R4C9 = {389} -> R4C78 = {18/13/23} -> R3C8 = {267} (cage sum)
31e. Killer pair 1,2 in R4C789 and R5C89, locked for N6
31f. Killer pair 8,9 in R4C5 and R4C789, locked for R4
31g. 7 in R4 only in R4C123, locked for N4

32. 16(3) cage at R6C7 = {259/268/358/457} (cannot be {349} because 3,4 only in R7C8, cannot be {367} which clashes with R5C89)
32a. 2,3,4 only in R7C8 -> R7C8 = {234}
32b. 5 of {259/457} must be in R6C8 -> no 7,9 in R6C8

33. 45 rule on N7 3 outies R6C123 = 1 innie R9C3 + 7
33a. R9C3 = {378} -> R6C123 = 10,14,15 = {136/239/168} -> R6C23 = {16/36/23/29/39/18} -> R7C2 = {23579} (cage sum)
33b. 8 of {18} must be in R6C2 (R6C23 cannot be [18] which clashes with R9C3, IOD clash) -> no 8 in R6C3

34. R7C89 + R8C9 + R9C7 (step 7) = {1238/1256/1346/2345}
34a. 45 rule on C9 3 innies R159C9 = 12 = {129/138/147/156/237/246} (cannot be {345} because R5C9 only contains 1,2,6,7)
34b. 14(3) cage at R6C9 = {158/239/248/257/347} (cannot be {149/356} because R7C8 + R9C7 cannot be {36/24}, cannot be {167} which clashes with R159C9), no 6 in R78C9
34c. R7C89 + R8C9 + R9C7 = {1238/2345}, 2,3 locked for N9
34d. 2 in R9 only in R9C12, locked for N7
[I missed 2 in C7 only in R234C7, CPE no 2 in R3C8 -> R3C8 = {67} gives R4C8 = 3 a bit sooner than in my solving path.]

[I’ve simplified the remaining steps by moving the ‘obvious’ continuation after step 34c to here.]

35. R7C89 + R8C9 + R9C7 (step 34c) = {1238/2345} -> 31(5) cage at R7C7 = {16789/45679}
35a. Consider placements for R9C7 = {35}
R9C7 = 3 => R9C6 = 9, R9C34 = [74], R9C5 = 6 => R9C89 = {158} => 31(5) cage = {16789}
or R9C7 = 5 => 31(5) cage = {16789}
-> 31(5) cage = {16789}, locked for N9

36. 45 rule on N3 2 innies R1C7 + R3C8 = 1 outie R4C9 + 2
[I had looked at this earlier but replaced it by step 31a, which was more powerful at the time.]
36a. R4C9 = {389} -> R1C7 + R3C8 = 5,10,11 = [32/37/56] -> R1C7 + R3C8 + R4C9 = [323/378/569]
36b. 19(3) cage at R2C9 = {289/379/568} (cannot be {469} = {46}9 which clashes with R1C7 + R3C8 + R4C9 = [569], cannot be {478} = {47}8 which clashes with R1C7 + R3C8 + R4C9 = [378]), no 4 in R23C9
36c. 14(3) cage at R6C9 (step 34b) = {239/248/347} (cannot be {257} which clashes with 19(3) cage), no 5
36d. R159C9 (step 34a) = {129/147/156/237/246} (cannot be {138} which clashes with 19(3) cage), no 8 in R9C9
36e. 4 in C9 only in R159C9 = {147/246} or in 14(3) cage = {248/347} -> R159C9 = {129/147/156/246} (cannot be {237} which clashes with 14(3) cage = {248/347}, locking-out cages), no 3 in R1C9
36f. 4 of {147} must be in R1C9 -> no 7 in R1C9
[Cracked at last, the rest is fairly straightforward.]

37. R9C7 = 5 (hidden single in R9) -> R9C6 = 7, R1C7 = 3 -> R1C6 = 8, R1C4 = 7 -> R1C3 = 5, R6C4 = 4, R9C5 = 4 (hidden single in R9) -> R8C5 = 5, R45C5 = [97] (hidden pair in C5), R3C5 = 1 (step 26f), clean-up: no 1 in R5C89
37a. Naked pair {26} in R5C89, locked for R5 and N6 -> R56C6 = [12], clean-up: no 8 in R5C12
37b. Naked pair {38} in R59C3, locked for C3
37c. Naked pair {59} in R5C12, locked for N4
37d. Naked pair {38} in R9C34, locked for R9

38. Naked triple {138} in R4C789, locked for R4 and N6 -> R6C8 = 5
38a. 19(3) cage at R2C9 (step 36b) = {289/568} (cannot be {379} which clashes with R6C9), no 3 -> R4C9 = 8, R4C78 = [13], R3C8 = 7 (cage sum)

39. R6C9 = 7 (hidden single in C9) -> R78C9 = 7 = {34}, locked for C9 and N9 -> R7C8 = 2

40. R1C3 + R3C2 = R4C1 + 6 (step 30d)
40a. R1C3 = 5 -> R3C2 = R4C1 + 1 -> R3C2 = 8, R4C1 = 7 -> R23C1 = 6 = {24}, locked for C1 and N1

41. 2 in C7 only in R23C7, locked for N3
41a. R4C9 = 8 -> R23C9 = 11 = {56}, locked for C9 and N3 -> R5C89 = [62]
41b. Naked pair {19} in R9C89, locked for R9 and N9 -> R8C8 = 8, R9C12 = [62], R4C23 = [42]
41c. R2C23 = [37] (hidden pair in N1)
41d. R6C23 = {16}, locked for R6, R7C2 = 7 (cage sum)

and the rest is naked singles.
Rating Comment:
It's interesting that Mike defined typical examples for 2.5 as A48-Hevvie and TJK 18; since I've now solved A48-Hevvie, but have so far made little progress on TJK 18, I'll assume these two are at the lower and upper limits of 2.5.

Since Mike quoted A48-Hevvie as a typical 2.5, which I assume applies to the original (no longer available) Tag solution, I haven't changed the entry in the rating table. However I wouldn't rate my walkthrough any higher than Hard 1.75.


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