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PostPosted: Sat Jun 07, 2008 11:27 am 
Offline
Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
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.16 1.25 0.95|SKX1 1.00|A.22 0.80|
|UTA H1.50 1.90|A.19 1.00 0.80|SKX2 1.40|
|UTAv2 H1.50 1.50|A.20 1.00 0.75|A.23 0.70|
|A.17 1.00 0.70|A.20v2 1.25 1.20|A.24 1.25|
|A.17v2 E1.50 1.10|A.21 0.70| |
|A.18 1.25 0.95|A.21v2 1.50| |
|====================================================================|
page #2
Old scores from SSv3.3.0:
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.16 1.25 1.00|SKX1 !1.20|A.22 0.80|
|UTA H1.50 !2.25|A.19 1.00 0.80|SKX2 1.75 !2.95|
|UTAv2 H1.50 !1.60|A.20 1.00 0.80|A.23 0.75|
|A.17 1.00 0.70|A.20v2 1.25 1.15|A.24 H1.25 1.25|
|A.17v2 E1.50 !1.30|A.21 0.70| |
|A.18 1.25 !1.15|A.21v2 H1.50 1.55| |
|====================================================================|
page #2
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.16 Ruud 1.25 1.05|SKX1 Ruud 1.05|A.22 Ruud 0.95|
|UTA Ed H1.50 1.85|A.19 Ruud 1.00 0.95|SKX2 Ruud 1.75 2.35|
|UTAv2 Ed H1.50 1.45|A.20 Ruud 1.00 0.90|A.23 Ruud 0.90|
|A.17 Ruud 1.00 0.85|A.20v2 Ed 1.25 1.00|A.24 Ruud H1.25 1.35|
|A.17v2 Ed E1.50 1.25|A.21 Ruud 0.85| |
|A.18 Ruud 1.25 1.05|A.21v2 Ed H1.50 1.60| |
|=========================================================================================|
page #2


Assassin 16 by Ruud (Sept 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6144:6144:3586:3586:3588:3589:3589:3079:2312:6144:6144:3586:3596:3588:3588:4111:3079:2312:3346:3596:3596:3596:4118:4118:4111:3865:3865:3346:3612:3613:3613:7455:4118:4111:4111:3865:3612:3612:3613:7455:7455:7455:4394:4907:4907:2349:4654:4654:4912:7455:4394:4394:4907:1589:2349:2349:4654:4912:4912:3643:3643:3643:1589:3391:2624:4654:2626:2626:3643:2885:6726:6726:3391:2624:3914:3914:2626:2885:2885:6726:6726:
Solution:
+-------+-------+-------+
| 2 8 7 | 3 1 5 | 9 4 6 |
| 9 5 4 | 2 6 7 | 1 8 3 |
| 6 1 3 | 8 4 9 | 7 5 2 |
+-------+-------+-------+
| 7 4 5 | 1 9 3 | 6 2 8 |
| 1 9 8 | 6 5 2 | 4 3 7 |
| 3 6 2 | 4 7 8 | 5 9 1 |
+-------+-------+-------+
| 4 2 9 | 7 8 6 | 3 1 5 |
| 8 7 1 | 5 3 4 | 2 6 9 |
| 5 3 6 | 9 2 1 | 8 7 4 |
+-------+-------+-------+
Quote:
Ruud, lead-in: a tough nut to crack
sudokuEd: Indeed it was. Took ages to find the right nut-cracker to use - lots of nooks and crannies to search through
Andrew: I found it the hardest Weekly Assassin so far..I.. avoid putting in candidates until necessary and never more than four candidates in a cell unless I'm stuck and forced to put in more
Oscar: After 4 days, I just couldn't make it without the "hint on the nook" from sudokuEd :oops: . It is narrow gorge that allows to pass to the other side of the mountain. I would rate this one 1.25
Walkthrough by sudokuEd:
Quote:
a tough nut to crack.
Indeed it was. Took ages to find the right nut-cracker to use - lots of nooks and crannies to search through. Very enjoyable. A walk-through follows - but if you want just a hint on the nook to start with - in tiny text here 4 innies of N89

Assassin 16 –

Step 1
“45” on N89 -> 4 innies = 29
Max r7c45 + r9c4 = 7+8+9 = 24 -> min r7c9 = 5
->r67c9 = [15] and r7c45 & r9c4 = {789} only -> no 7,8 or 9 elsewhere in N8
“45” on N89 -> 2 outies = 10 -> r9c3 = {678} r6c4 = {234}

Step 2
“45” on c89 -> 2 innies = 3 -> r47c8 = [21]
“45” on r12 -> 2 innies = 3 -> r2c47 = [21]
r3c9 = 2 (hidden single r3)
9(2) cage in N3 {36} only -> no 3 or 6 elsewhere in N3 or c9
->14(2) cage in N23 = {59} or [68]

Step 3
“45” on r6789 -> 4 innies = 29 -> r6c5678 = {5789} only -> no 5,7,8 or 9 in r6c1234
->17(3) cage in N56 = {359/458} (not 368/467 since 3,4 & 6 only in r5c7)
-> 17(3) cage = 5{39/48} with r5c7 = {34} only, r6c67 = {589} with 5 locked in r6-> no 5 elsewhere in r6
“45” on N36 -> 2 outies = 13 -> r16c6 = [58] ([85] not possible since 8 has been eliminated from r1c6 in step 2), r16c7 = [95], r5c7 = 4, r34c7 = 13 = [76], r5c8 = 3 (hidden single N6), rest of 19(3) cage {79}
12(2) cage in N3 = {48} only -> no 4 or 8 elsewhere in N3 or c8 -> r3c8 = 5, r4c9 = 8

Step 4
“45” on r89 -> 2 innies = 5 -> r8c36 each max 4
-> 5 in N7 locked in 13(2) cage = {58} -> no 5 or 8 elsewhere in N7 or c1
-> r9c34 = [69/78] -> 7 in N8 only in r 7 (no 7 elsewhere in N8 - see step 1)-> no 7 elsewhere in r7
9(3) cage in N47 now {234} only
-> 6 in r6 now locked in 18(4) cage in N47 ->18(4) cage {1269} only
-> r78c3 = [91], r7c6 = 6 (hidden single r7), -> rest of 14(4) cage in N89 = 7 -> r7c7 = 3, r8c6 = 4, r89c7 = {28} = 10 -> r9c6 = 1, r9c9 = 4 (hidden single N9)

Step 5
r7c12 = {24} = 6 -> r6c1 = 3, r6c4 = 4
10(2) cage in N7 = {37} only -> no 3 or 7 elsewhere in N7 or c2 -> r9c34 = [69]
2 in N5 only in 29(5) cage -> no 1 in 29(5) cage (since the last 3 cells in that cage could not total 26) -> r4c4 = 1 (hidden single N5),
“45” on N5 -> r4c6 = 3
the rest is pretty straightforward

Edited again to make some points clearer (thanks again Andrew) and fix more typos
Walkthrough by Andrew:
Ruud wrote:
A slightly altered version of my recent Ruudiculous Killer. It is easier, but still a tough nut to crack.
I agree completely with sudokuEd, a tough nut to crack which is hardly surprising since it's stated to have come from a Ruudiculous Killer. I found it the hardest Weekly Assassin so far.

I'm also posting my walkthrough for this puzzle. While it does, of course, follow roughly the same method as used by sudokuEd there are differences which I think come from our different solving approaches. sudokuEd puts in all possible candidates very early while I take work from the opposite approach and avoid putting in candidates until necessary and never more than four candidates in a cell unless I'm stuck and forced to put in more. That explains why my walkthroughs sometimes state the obvious in the final cage of a nonet before I move on to another part of the puzzle.

Step 1
45 rule on R12 2 innies = 3 -> R2C47 = {12}, 45 rule on C89 2 innies = 3 -> R47C8 = {12}, 45 rule on R6789 4 innies = 29 -> R6C5678 = {5789}, 9 in R7 locked in R7C345 because 14(4) cage in N89 cannot contain 9 -> 9 in N9 locked in 26(4) cage

Step 2
6(2) cage in N69 = {15/24}, 45 rule on N89 4 innies = 29 = {5789}* -> R67C9 = [15] -> R7C45 + R9C4 = {789}, R4C8 = 2, R2C47 = [21], R7C8 = 1, R6C4 = {34}

* This was “loose thinking” by me. Thanks to sudokuEd for pointing this out to me in a private message. Fortunately it happens to be correct in this case because the 5 is in N9 and the {789} are in N8. However there are cases when applying the 45 rule to multiple nonets where this would not be true. For example, with two nonets, alternative candidates for 4 innies totalling 29 could be {4799}, {4889} or {3899} where the repeated digits are not in the same row/column/nonet while it is possible with three/four nonets for the combination to be {2999}!

Step 3
6 in R6 locked in N4, it cannot be in R6C1 because {12} in R7C12 would clash with the 1 in R7C8 so the 6 is in R6C23, 9(3) cage in N47 = {234}, 6 in R7 must be in R7C67 because there is already a 6 in the 18(4) cage in N47, 45 rule on R89 2 innies = 5 = {14/23} -> R8C6 = {234}, R8C3 = {123}, 14(4) cage in N89 must be {1346} (it cannot be {1256} because 5 cannot go in any cell of this cage), 2 in R7 is locked in N7 -> R8C36 = [14], R7C67 = {36}, R7C12 = {24}, R6C1 = 3, R6C4 = 4, R6C23 = {26}, R7C3 = 9, R7C45 = {78}, R9C4 = 9, R9C3 = 6, R6C23 = [62], R89C1 = {58}, R89C2 = {37}

Step 4
45 rule on C12 2 innies = 7 -> R3C2 = 1, R5C1 = 1, 45 rule on N8 2 innies = 7 -> R79C6 = [61], R7C7 = 3, 10(3) cage in N8 = {235} with the 2 in C5, R5C6 = 2, R5C7 = {1234} from the valid combinations in R6C67 = 4 because {123} are all blocked, R6C67 = {58}, 45 rule on N5 4 innies = 16, because R6C6 is {58} the remaining 3 innies must total 8 or 11 and cannot contain 9 -> the 9 in N5 must be in the 29(5) cage in C5, R34C7 = {67} because {58} would clash with the {58} in R6C7, R89C7 = {28}, R6C67 = [85], R1C67 = [59] (hidden single in C7), 26(4) cage in N9 = {4679} with R8C89 = {69} and R9C89 = {47}, R89C2 = [73]

Step 5
45 rule on N3 2 outies = 14 -> R4C79 = [68], R3C7 = 7, neither the 7(2) sub-cage nor the 12(2) cage in N3 can contain 6 -> R12C9 = {36}, R3C89 = [52], R12C8 = {48}, R9C89 = [74], R6C58 = [79], R8C89 = [69], R5C89 = [37], the remaining 3 cells in the 29(5) cage in N5 are {569} with the 9 in C5 and 6 in R5, R4C46 = [13], 45 rule on N4 -> R4C1 = 7, R3C1 = 6, R3C56 = [49], R45C2 = [49] because 9 in N4 blocked from C3, R45C3 = [58], R3C34 = [38] and carry on, the rest is filling in the remaining candidates and simple elimination

This was written more as how I solved this puzzle, with minor rearrangement of sub-steps, rather than deliberately as a walkthrough.

As an example of the different approaches, sudokuEd was able to fix a hidden single early in his step 2 but I didn't fill that cell until my step 5.


Last edited by Ed on Sun Apr 11, 2010 8:17 am, edited 8 times in total.

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PostPosted: Sat Jun 07, 2008 11:42 am 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
UTA - Ruudiculous Tag Killer by sudokuEd (Sept 06)

Puzzle pic: 1-9 cannot repeat on the diagonals:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:6913:6913:5122:5122:5122:5134:5134:5134:5134:5138:6913:6913:6913:5122:1792:1792:2320:2320:5138:4373:4373:4373:5654:5654:5654:1305:1305:5138:4373:4390:4390:6439:5941:5941:5941:5941:5915:4390:4390:6439:6439:6439:3882:5931:5941:5915:5915:5915:5915:6439:3882:3882:5931:3911:3894:3894:2362:2362:4355:4355:5931:5931:3911:1343:1343:2370:2370:5966:4355:4355:4432:3911:5704:5704:5704:5704:5966:5966:5966:4432:4432:
Solution:
+-------+-------+-------+
| 4 5 6 | 3 7 1 | 2 8 9 |
| 7 1 9 | 8 4 2 | 5 6 3 |
| 8 3 2 | 5 6 9 | 7 4 1 |
+-------+-------+-------+
| 5 7 1 | 6 9 8 | 3 2 4 |
| 9 2 8 | 7 3 4 | 1 5 6 |
| 3 6 4 | 1 2 5 | 9 7 8 |
+-------+-------+-------+
| 6 9 5 | 4 1 7 | 8 3 2 |
| 1 4 7 | 2 8 3 | 6 9 5 |
| 2 8 3 | 9 5 6 | 4 1 7 |
+-------+-------+-------+
Quote:
sudokuEd, lead-in: This puzzle takes SumoCue 17 seconds to find a solution, compared to Uluru's 10. So, a definite UTA
Tag solution for 30 steps (not in archive) then swapped to UTA2: by Nasenbaer, sudokuEd, Andrew
Para: worked through this killer in the last few days. Now done both killers from before i came here...Really tough work to finish it
Andrew in 2011: Para and I have both shown that the original UTA is also solvable by logic....rate..at Hard 1.5
2020: Revisit forum thread here
Condensed Walkthrough of SudokuSolver log by sudokuEd:
Finally worked out how rcbrougton's solver solved UTA original! Very interesting path taken too. Lots of innies and hidden cages that we never thought of - r7, r9, n9. Oh yeah - and did I mention combinations? Lots of combination conflicts from many different directions at once.

This walk-through is the sweetened-condensed version of rcbroughton's - lots of steps left out, changed the order of many - and lots of my own explanations included. The step numbers should correspond.

Oh yeah - and did I mention combination charts? Make sure you've got one that works :wink: Good luck - a very rough road ahead :twisted: [Many thanks to Andrew for navigating through it!!]

Just a quick reminder, UTA is a diagonals puzzle: 1-9 cannot repeat on diagonals. [edit: colours changed to get better match with burgundy]

1. Must use 9 in cage 22(3) at r3c5
-> Removed 9 from r3c1234

19. 45 rule on r12 -> r2c1 = 7

19a. r34c1 = 13 = [49]/{58}
-> Removed candidate 369 from r3c1
-> Removed candidates 346 from r4c1

21. Only combinations {18/36/45} allowed in cage 9(2) at r2c8
-> Removed candidate 2 from r2c89

22. Only combinations {69/78} allowed in cage 15(2) at r7c1
-> Removed candidate 8 from r7c2

13. 45 rule on N3. Included cells r23c7 minus excluded cell r1c6 equals 11
-> Min of included cells is 12. Set min candidate in r2c7 (3)
-> min r3c7 = 7 - (can't have r23c7 = {66})

-> Max of excluded cells is 4. Set max candidate in r1c6

14. Only combinations {16/25/34} allowed in cage 7(2) at r2c6
-> Removed candidates 56 from r2c6

27. 3 locked in n3 in 9(2) or 5(2)

27a.if 9(2) = [1/4]->5(2) = {23};

27b.if 5(2) = {14} ->9(2) = {36})
-> Removed candidate 3 from r2c7, r1c789

28. Only combinations {16/25/34} allowed in cage 7(2) at r2c6
-> Removed candidate 4 from r2c6

7. 45 rule on row 9. Excluded cells r8c5 r8c8 equal 17
-> Only combination {89} allowed

8. Only combinations with 8 or 9 (not both) allowed in cage 17(3) at r8c8.
-> Removed candidates 89 from r9c89

9. Naked pair 89 found in row 8 at r8c58

9a.-> Removed value 8 from r5c5 (because both cells connected through D\)

9b.-> Removed value 8 from r5c5 r9c7 (because r9c7 is in the same cage as r8c5 and same n as r8c8)

9c.-> Removed value 8 from r5c5 r8c34 r8c679

9d.-> Removed value 9 from r5c5(as above) r9c7(as above) r8c6 r8c7 r8c9

9e.-> Removed combination {18} from cage 9(2) at r8c3

9f.-> Removed candidate 1 from r8c34

10. Value 4 locked in row 8 of combined cages 9(2) at r8c3 & 5(2) at r8c1

10a.ie 5(2) = {23} -> 9(2) = {45};
9(2) = [2/3] -> 5(2) = {14}
-> Removed 4 from r8c679

34. 45 rule on N7, N8. Excluded cells r89c7 equal 10
-> Only combinations {37/46} allowed
-> Removed candidates 125 from r8c7
-> Removed candidates 1256 from r9c7
-> Found a hidden cage 10(2) at r89c7

This is a tricky step - but absolutely critical. Get your combination tables ready!
35. 45 rule on row 9. Included cells r9c56789 equal 23 = h23(5) cage

35a.-> Cage h10(2) at r8c7 doesn't allow permutations in r9c89 with {36}
->17(3)n9 {368} combo eliminated

35b. h23(5)r9 must include {347} (r9c7)
-> {12569} excluded

35c. Possible combinations for h23(5)r9 must have a triple overlap of combination from r9c56:23(4) and then double overlap from r9c89:17(3), AND must agree with the {89} pair in r8 AND must agree with the h10(2)n9

ie. combinations h23(5)r9 = {r9c56}[r9c7]{r9c89}

-{12389} Blocked (must have 3 in r9c7 but no triple overlap of combination with r9c567:23(4))

-{12479} = {29}[4]{17} (7 in r9c7 blocked - no triple overlap of combination with r9c567:23(4))

-{12578} Blocked (only be {25}[7]{18} but no 8 in r9c89)

-{13469} Blocked (r9c7 = [3/4] but no triple overlap with r9c567:23(4))

-{13478} Blocked (only be {347} in r9c567: but no 8 left in r9c89)

-{13568} Blocked (only be {56}[3] in r9c567: but no 8 left in r9c89)

-{14567} = {56}[4]{17}/{16}[7]{45}

-{23459} = {29}[4]{35} (3 in r9c7 blocked: no triple overlap with r9c567:23(4))

-{23468} Blocked ({28}[4]{36} blocked - see step 35a; 3 in r9c6: no triple overlap with r9c567:23(4))

-{23567} = {57}[3]{26}/{35}[7]{26}
-({56}[3]{27} blocked since 7 must be in r8c7 in h10(2) -> 2 7's n9)
-({25}[7]{36} blocked since no {36} possible r9c89 - step 35a)
-({26}[7]{35} blocked since 3 must be in r8c7 in h10(2) -> 2 3's n9)

35d.In summary h23(5)r9 =
{12479} = [{29}4{17}]
{14567} = [{56}4{17}/{16}7{45}]
{23459} = [{29}4{35}]
{23567} = [{57}3{26}/{35}7{26}]

-> Removed candidates 48 from r9c56

35e. from step 35d. r8c5 + r9c56 (part of 23(4) that's in n8) = {289/568/169/578/358}

35f. from step 35d. the only combinations in 23(4)n89 = {2489/4568/1679/3578} = {1679/2489/3578/4568}

35g. from step 35d. 17(3)n9 = {179/458/359/269} = {179/269/359/458} = [8/9..]

39. r8c8 = [8/9]. Only other place for [8/9] in nonet 9 is in row 7

39a. 15(2) at r7c1 = {69/78} = [8/9..]

39b.->Killer pair {89}: locked for r7
-> Removed candidate 8 from r7c3456
-> Removed candidate 9 from r7c56

39c. ->17(4)n89 = {1367/1457/2357/2456}

40. Only combinations {27/36/45} allowed in cage 9(2) at r7c3
-> Removed candidate 1 from r7c34

18. 45 rule on N7
-> Found a hidden cage 15(3) at r789c4

41. Only combinations {249/258/267/348/357/456} allowed in cage h15(3) at r7c4
-> Removed candidate 1 from r9c4

43. from step 41. h15(3)n8 = {249/258/267/348/357/456}

43a. {267} blocked
-from step 39c.the only combination in 17(4)n89 possible is {1457} with {145} in n8. All other combinations are blocked since they have 2 candidates in common with {267}.

-But{267-145} in h15(3) + 3 cells from 17(4) in n8 is blocked by r8c5 + r9c56 (see step 35e.)

43b.{357} blocked
-the only available combination in 17(4)n89 (step 39c.) without a 5 is {1367},
-but they have both {37} in common -> blocked

43c.
-> Only combinations allowed in h15(3) = {249/258/348/456} (no 7)
-> Removed candidate 7 from r78c4
-> Removed candidates 237 from r9c4 (since is the only cell with an 8 or 9)

44. Only combinations {27/36/45} allowed in cage 9(2) at r7c3
-> Removed candidate 2 from r7c3

45. Only combinations {27/36/45} allowed in cage 9(2) at r8c3
-> Removed candidate 2 from r8c3

47. 45 rule on N1, N2, N3, N4. Excluded cells r46c4 equal 7
-> Only combinations {16/25/34} allowed
-> Removed candidates 789 from r46c4
-> Found a hidden cage 7(2) at r46c4

48. h15(3) in n8
-> Cage h7(2) at r4c4 doesn't allow permutations with {456}
-> Only combinations {249/258/348} allowed (no 6)
-> Removed candidate 6 from r78c4
-> Removed candidates 456 from r9c4 (only cell with 8/9)

49. Only combinations {45/63/27} allowed in cage 9(2) at r7c3
-> Removed candidate 3 from r7c3

50. Only combinations {27/45/63} allowed in cage 9(2) at r8c3
-> Removed candidate 3 from r8c3

51. Naked pair 89 found in N8 at r8c5 r9c4
-> Removed value 9 from r9c56
-> Removed combination {2489} from cage 23(4) at r8c5
-> removed 2 from r9c56

51a.from step 35d. h23(5)r9
-> Only combinations =
{14567} = [{56}4{17}/{16}7{45}]
{23567} = [{57}3{26}/{35}7{26}]
=567{14/23}

51b.-> no 3 r9c89

51c.->17(3) = {179/269/458}

51d.->567 locked for r9

51e.->22(4)r9 = 89{14/23}

51f. combining the information from step 51a with the h10(2)n9
->r8c78 + r9c789 = [69417/38745/79326/39726] = 7{1469/2369/3458}

51g.-> 7 locked for n9 in h10(2) and 17(3)

51h. "45"n9 -> 4 innies = 18(4) = {1269/1458/2358} ({1359/1368/2349/3456} blocked by combined h10(2)-17(3) step 51f)

51i. 22(4)n78 = {1489/2389} = 89{14/23}
-since {14/23} must be in n7 -> Killer quad with 5(2)n7
-> no 4 r78c3

53. Only combinations {27/36/45} allowed in cage 9(2) at r7c3
-> Removed candidate 5 from r7c4

54. Only combinations {27/36/45} allowed in cage 9(2) at r8c3
-> Removed candidate 5 from r8c4

55. Value 5 locked in column 3 of N7
-> Removed 5 from r123456c3

56. Since 5 is locked in r78c3 -> Value 4 locked in r78c4 (since connected by 2 9(2)cages)
-> 4 locked for c4
-> Removed 4 from r123456c4

57. Only combinations {16/25} allowed in cage h7(2) at r4c4
-> Removed candidate 3 from r46c4

58. Value 4 locked for column 4 in N8
-> Removed 4 from r7c56

58a. 17(4)n89 = {1367/2357} = 37{16/25}

59. Must use 37 in cage 17(4) at r7c5
-> Removed candidate 3 from r8c4

60. Only combinations {27/45} allowed in cage 9(2) at r8c3
-> Removed candidate 6 from r8c3

61. Value 6 locked in row 7 of N7
-> Removed 6 from r7c56789

62. Value 2 locked in row 8 of combined cages 9(2) at r8c3 & 5(2) at r8c1
-> Removed 2 from r8c69

65. 45 rule on row 89. Included cells r8c679 = h14(3)

65a. = [176/671/761/356/365/536] ([563] not compatable with 17(4) combo's)
-> Cage h15(3) at r7c4 doesn't allow permutations in r8c56 with {39}
-> Only combinations {167/356}
-> Removed candidate 3 from r8c9

68a.From step 51h. we know that innies n9 = 18(4) = {1269/1458/2358}

68b. -> with r8c9 = {156} -> {r7c789} = {269/129/458/148/238}

68c. "45" r7 -> 5 innies = h21(5)

68d. combining steps 68b and c and keeping an eye on the combinations in the 17(4)n89: {1367/2357}
-> h21(5) = {r7c56}{r7c789}
= {13-269}/{36-129}/{13-458}/{35-148}/{17-238}

68e. -> h21(5) = {12369/12378/13458} = 13{269/278/458}

68f. -> no 3 r7c4

68g. and no 3 in r8c7 (since 3 locked in h21(5) r7 is in the same cage or nonet as r8c7)

68h. -> no 7 r9c7

68i.removed 6 from r7c3

69. from step 68h and 51a.
h23(5)r9 = {14567} = [{56}4{17}]
= {23567} = [{57}3{26}]

69a. -> Only combinations {3578/4568} allowed in cage 23(4) at r8c5

69b. = 58{37/46} (no 19)

69c. -> r8c5 = 8

69d. 5 is locked for n8 and r9 in r9c56 = 5[6/7](no 3)

75. Only combinations {249} with 9 locked in r9c4 allowed in cage h15(3)n8 at r7c4
-> Set candidate 9 in r9c4

75a. r78c4 = {24}:locked for n8, c4

75b. r8c8 = 9 placed for D\

75c. r9c89 = {17/26}(no 3,4)

86. Only combination {16} allowed in cage h7(2) at r4c4
-> Removed candidate 5 from r4c4
-> Removed candidate 5 from r6c4

86a. {16}locked for c4,n5

87. "45"n5 -> r46c6 = 13 = [94]/{58}

88. Naked pair {57} found in r78c3
-> Removed value 7 from r456c3 r7c2
-> Removed combination {1457} from cage 17(4) at r4c3
-> Removed combination {23459} from cage 23(5) at r5c1
-> Removed combination {78} from cage 15(2) at r7c1 = {69} only

89. Naked pair {16} found in cage h7(2)c4
-> Removed combination {1469} from cage 20(4) at r1c3
-> cage 25(5) at r4c5 = 237{49/58}

91. Must use {1367} in cage 17(4) at r7c5

91a. 6 is locked for r8
-> Removed candidate 6 from r8c9

99 from step 68d. "45" r7 -> 5 innies = h21(5)

99a. -> {r7c56}{r7c789} = {13-458}/{17-238}

99b. -> 1 locked in r7c56 for r7, n8

99c. -> h21(5) = {12378/13458} = 138{27/45}


Now moving into n6 - and the key moves that finally unlock this puzzle
37. 45 rule on N7, N8, N9. Excluded cells r56c8 r6c9 = 20

37a.-> Only combinations {389/479/569/578} allowed

37b.-> Removed candidates 12 from r5c8

37c.-> Removed candidates 12 from r6c8

37d.-> Found a hidden cage 20(3) at r56c8 r6c9

80. 23(4)n69 = {2678/3578/4568}.

80a.From step 37, r56c8 + r6c9 = 20.
-r6c9 = 3..9 -> r56c8 = 11..17 -> r7c78 = 6..12

80b.from step 99a. ->{r7c789} = {458}/238}
from 80a. r7c78 = 6..12 -> = {45/48/28/38}

80c. from step 80 ->r56c8 = {68/56/67/57}

80d. But since r56c8 are part of a h20(3) (step 37a) -> {67/68} are blocked

80e. ->r56c8 = {56/57} = 5{6/7} = 11/12

80f.->5 locked for c8,n6 and no 5 in r7c78

80g. -> r6c9 = 8/9 only (from h20(3)n6)

80h. from 80e. the rest of 23(4)n69 = {38/48} = 8{3/4} (no 2)

80i. 15(3)n69 = [825/951] with r7c9 = {25}

80j. -> 15(3) = 5{28/19}: 5 locked for c9, n9

80k. -> r6c9 = {89}, r7c9 = {25}, r56c8 = {567} (no 8), r7c78 = {348} = 8{3/4}

80l. 23(4) n69 = {3578/4568} = 58{37/46} with 5 locked for c8 -> no 4 r2c9

80m. 9(2)n3 = {18/36}(no 4) = [1/3...]

80n. 5(2)n3 = {14/23} = [1/3..] -> Killer pair {13} locked for n3


109/114. 45 rule on N3. Included cells r23c7 minus excluded cell r1c6 equals 11

-1 in r1c6 -> r23c7 = 12 = {57}
({48} blocked since requires 9(2)n3 = {36} and 5(2) = {23} = 2 3's)

-2 in r1c6 -> r23c7 = 13 = {49}
({58} blocked since requires 9(2)n3 = {36} and 5(2) = {14} = but this leaves {279} for r1 -> 2 2's in r1; {67} blocked by r8c7)

-3 in r1c6 -> r23c7 = 14 = {59}
({68} blocked by 9(2)n3)

-4 in r1c6 -> r23c7 = 15 = blocked
({69} requires 9(2) = {18}, 5(2) = {23} -> 2 4's in r1)

109a. r23c7 = [57/49/59]
-> Removed candidate 6 from r2c7
-> Removed candidate 8 from r3c7
-> removed 4 from r1c6

110. Only combinations {25/34} allowed in cage 7(2) at r2c6
-> Removed candidate 1 from r2c6

111. Only combinations {589/679} allowed in cage 22(3) at r3c5
-> Removed candidate 5 from r3c6

112. Value 1 locked for column 7 in N6
-> Removed 1 from r4c89 r5c9

115. Value 4 locked in column 5 of N2
-> Removed 4 from r456c5

90. Must use 9 in cage 27(5) at r1c1
-9 for 27(5)n12 only in n1
-> no 9 in r1c3

117. Value 4 locked for column 5 in cage 20(4) at r1c3
-> Removed 4 from r1c3

117a. 20(4) must have 4 = {1478/2459/2468/3458/3467} ({1469} blocked since no candidates in r1c4)

117b. 45 on r123 -> r4c12 = 12 = [57/84/93]: r4c2 = {347}

117c. 45 on n1 -> r23c4 - 7 = r1c3.

-r1c3 = 1 -> r23c4 = 8 = {35} -> rest of 20(4) = {478}

-r1c3 = 2 -> r23c4 = 9 = no options -> no 2 in r1c3

-r1c3 = 3 -> r23c4 = 10 = [37] -> rest of 20(4) = {458} ({467} blocked by r23c4)

-r1c3 = 6 -> r23c4 = 13 = {58} -> rest of 20(3) = {347} ({248} blocked by r23c4)

-r1c3 = 8 -> r23c4 = 15 = [87] -> rest of 20(3) = {345} ({147} blocked by r23c4;{246} blocked since no candidates in r1c4)

117d. In summary, r23c4 = {35/58}[37/87]

117e. In summary, rest of 20(3) in n2 = 4{78/58/37/35} (no 1269)

117f. -> r2c3 = 9 (hsingle r2)

117g. -> 9 for n2 locked in 22(3) -> no 9 in r3c7

117h. r3c7 = 7

126a. naked quint on {34578} in n2 -> r12c6 = [12], r3c56 = {69} -> r2c7 = 5, r89c7 = [64], r7c78 = {38} locked for r7, r7c34 = [54], r78c6 = [73], r56c8 = {57}, 15(3)n56 = 9{28/46} (note: may still have {258} combo so this next bit may be wrong) with 9 locked for n6, c7 -> r1c9 = 9, r1c78 = {28}, 9(2)n3 = {36}, r3c89 = [41] etc
Walkthrough by Para:
Hi all

Also worked to through this killer in the last few days. Now done both killers from before i came here. Really nice puzzle. Really tough work to finish it.

Walk-through Assassin UTA-1

1. R234C1 = {389/479/569/578}: no 1,2

2. R2C67 = {16/25/34}: no 7,8,9

3. R2C89, R7C34 and R8C34 = {18/27/36/45}: no 9

4. R3C567 = {589/679}: no 1,2,3,4; 9 locked for R3

5. R3C89 = {14/23}: no 5,6,7,8

6. R7C12 = {69/78}: no 1,2,3,4,5

7. R8C12 = {14/23}: no 5,6,7,8,9

8. 45 on R12: 1 innie: R2C1 = 7
8a. In 20(3) at R2C1: R34C1 = 13 = [49/58/85]: no 3,6; R4C1: no 4
8b. Clean up: R7C2: no 8; R2C89: no 2

Let’s first attack the bottom
9. 45 on R9: 2 outies: R8C58 = 17 = {89} -->> locked for R8; pointing: R5C5 + R9C7: no 8,9
9a. Clean up: R8C34: no 1

10. 45 on R89: 3 innies: R8C679 = 14 = {167/257/356}: {347 blocked by R8C12): no 4

11. 45 on N78: 2 outies: R89C7 = 10 = [37/64/73]: no 1,2,5 ; R9C7: no 6

12. 17(3) at R8C8 = [9]{17/26/35}/[8]{27/36/45} -->> R9C89 = {17/26/35/27/36/45}: no 8,9

13. Hidden Killer Pair {89} in N9: R8C8 = {89} and R7C789 needs one of {89}
13a. Killer Pair {89} in R7C12 + R7C789 -->> locked for R9
13b. Clean up: R7C34: no 1

14. Hidden Killer Pair {89} in N8: R8C5 = {89} and R9C456 needs one of {89}

15. Hidden Killer Pair {89} in R9: R9C456 needs one of {89} and R9C123 needs one of {89}

16. Either 22(4) at R9C1(when R9C4 ={89}) or 23(4) at R8C5(when one of R9C56 = {89}) needs 2 of {89}(from step 14)
16a. 22(4) = {1489/2389} or 23(4) = {2489}({1589} blocked by R9C7)
16b. Hidden Killer Quad {1234} in R9: R9C89 = {1|2|3|4..}, 22(4) or 23(4) with {89} had 2 of {1234}, so 22(4) or 23(4) with one of {89} needs one of {1234} -->> 22(4) at R9C1: {2479/3469/3478} blocked; 23(4): {3479 blocked}
16c. 23(4) at R8C5 = {1679/2489/2579/2678/3569/3578/4568}: When 4, it has to go in R9C7 -->> R9C56: no 4

17. 45 on N1234: R46C4 = 7 = {16/25/34}: no 7,8,9

18. 45 on N7: 3 outies: R789C4 = {249/258/267/348/357}: ({159} blocked because 1,9 only in R9C4; {456} blocked by R46C4) -->> R9C4: no 1,4

19. 1 in N8 locked within 17(4) cage at R7C5 or 23(4) cage at R8C5
19a. Either 17(4) = {1367/1457} or 23(4) = {1679} -->> R9C7 = 7 -->> R8C7 = 3
19b. So either 17(4) = {1367/1457} or R8C7 = 3 -->> 17(4) at R7C5 = {1367/1457/2357}: 7 locked within cage -->> R8C4: no 7
19c. Clean up: R8C3: no 2

20. 17(4) at R7C5: {1457} blocked: When {1457}; R7C56 + R8C6 = {145} and R8C7 = 7 -->> R9C7 = 3 -->> R8C5 + R9C56 (within 23(4) cage at R8C5) = {569/578}: 2 5’s in N8
20a. 17(4) = {1367/2357}: no 4; 3 locked within cage -->> R8C4: no 3
20b. 4 locked in N8 locked within R78C4 -->> locked for N4; h15(3) at R789C4 = {24}[9]/[348] -->> R7C4 = {234}; R8C4 = {24}; R9C4 = {89}
20c. Naked Pair {89} within R8C5 + R9C4 -->> locked for N8
20d. Clean up: R46C4: no 3; R78C3 = {57}/[65]; 5 locked for C3 and N7; R7C3 = {567}; R8C3 = {57}

21. 21(4) at R9C1 needs 2 of {89}(R9C123 and R9C4) -->> 21(4) = {1489/2389}: no 6,7
21a. 6 in N7 locked for R7

22. Killer Pair {24} in R8C12 + R8C4 -->> locked for R8

23. R8C12 and R8C4679 need either both {23} or neither.
23a. 17(4) in R7C5 = {1367/2357}: 3 locked
23b. When 3 in R7C56 + R8C6 -->> R78C4 = {24}
23c. When 3 in R8C7 -->> R8C4 = 2(step 23)
23d. Conclusion: 2 locked in R78C4 -->> locked for C4 and N8
23e. Hidden Pair: R78C4 = {24} -->> R9C4 = 9(step 20b)
23f. R8C58 = [89]
23g. Clean up: R46C4 = {16} -->> locked for C4 and N5
23h. Clean up: R78C3 = {57} -->> locked for C3 and N7
23i. Clean up: R7C1: no 8

24. 17(4) at R7C5 = {1367}: no 5

25. 23(4) at R8C5 = 8{357/456}: no 1; 5 locked for R9

26. 17(3) at R8C8 = 9{17/26}: no 3,4
26a. Killer Pair {67} in R89C7 + R9C89 -->> locked for N9

27. 45 on R789: R56C8 + R6C9 = 20 = {389/479/569/578}: no 1,2
27a. 9 only in R6C9 -->> R6C9: no 3,4,6

28. 15(3) at R6C9 = [9]{15}/[825]/[7]{35}: ([843] blocked by R89C7): R6C9 = {789}; R7C9 = {1235}; 5 in R78C9 for C9 and N9

Slowly moving up.
29. 23(4) at R5C8 = {2678/3578/4568}: no 1; R56C8 = {567}(only place in cage for these numbers and needs at least 2 of these)

30. h20(3) at R56C8 + R6C9 = {56}[9]/{57}[8] -->> R6C9 = {89}; R56C8 = {56/57} -->> 5 locked for C8 and N6
30a. 23(4) at R5C8 = {3578/4568}(needs 5 in R56C8): no 2
30b. 15(3) at R6C9 = [9]{15}/[825]: no 3
30c. Clean up: R2C89: no 4

31. Killer Pair {13} in R2C89 + R3C89 -->> locked for N3
31a. R2C67 = {25}/[34]: {16} blocked by R2C89; R2C6: no 1,4,6; R2C7: no 1,6

Now for some work on top.
32. 45 on N3: 2 outies and 1 innie: R3C7 = R12C6 + 4: Min R12C6 = 3 -->> Min R3C7 = 7; R3C7 = {789} -->> R12C6 = [12/13/23/32]: no 4,5,6,7,8,9
32a. Clean up: R2C7: no 2
32b. 4 in N2 locked for C5 and 20(4) cage at R1C3

33. 45 on N23: R1C3 + 7 = R23C4 : R1C3 = {123689} -->> R23C4 can be 8/9/10/13/15/16: R23C4 = {35/58}/[37/87] = 8/10/13/15 -->> R1C3 = {1368}
33a. R23C4 = {3|8..},{5|7..}

34. 20(4) at R1C3 = {1478/2459/2468/3458/3467}(needs 4 in R12C5): {1469} blocked by R1C4
34a. 20(4) = {2468} blocked -->> R12C6 = {2|3..}; R23C4 = {3|8..}; When {2468}, R12C5 = {24} and R1C4 = 8 -->> clash with R12C6 + R23C4
34b. 20(4) = {1478/2459/3458/3467} = {5|7..}: only {57} in R1C45 + R2C5

And now the easier work.
35. Killer Pair {57} in R1C45 + R2C5 and R23C4 -->> locked for N2
35a. 22(3) at R3C5 = {679} -->> R3C7 = 7; R3C56 = {69} -->> locked for R3 and N2
35b. R2C3 = 9(hidden); R12C6 = [12](step 32); R2C7 = 5; R78C3 = [57]; R78C4 = [42]
35c. R89C7 = [64](step 11); R8C6 = 3; R7C56 = [17]; R7C9 = 2; R8C9 = 5(hidden); R6C9 = 8

36. Clean up: R8C12 = {14} locked for N7; R56C8 = {57}(step 30) -->> locked for N6 and C8
36a. R9C89 = [17]

37. 45 on N23: R1C3 + 7 = R23C4 -->> R1C3 = 6; R23C4 = [85]
37a. Clean up: R2C9: no 1
37b. R2C2 = 1(hidden); R2C5 = 4(hidden); R46C4 = [61]; R8C12 = [14]; R1C9 = 9; R4C6 = 8

38. 45 on N5: 1 innie: R6C6 = 5
38a. R9C56 = [56]; R56C8 = [57]; R3C56 = [69]; R5C6 = 4
38b. R3C9 = (hidden); R3C8 = 4; R3C1 = 8; R4C1 = 5; R1C12 = [45](hidden)
38c. R4C9 = 4(hidden) R6C3 = 4(hidden); R7C7 = 8(hidden); R7C8 = 3
38d. R1C78 = [28]; R2C89 = [63]; R4C8 = 2; R5C9 = 6; R456C7 = [319]; R4C3 = 1
38e. R4C5 = 9(hidden) ; R4C2 = 7; R5C4 = 7(hidden); R1C45 = [37]

39. 17(4) at R4C3 = 16{28} (last possible combo): R5C23 = {28} -->> locked for R5 and N4
39a. R56C5 = [32]; R3C23 = [32]; R9C123 = [283]; R5C123 = [928]
39b. R6C12 = [36]; R7C12 = [69]

greetings

Para
2011 walkthrough by Andrew:
While checking my files I found that I had played a small part in the original UTA "tag" but my files was incomplete because the "tag" was switched to UTA2. I therefore had another go at UTA, starting from the beginning.

SudokuEd wrote:
Now the good news. I've changed N457 and now it is solvable by logic. Hence UTA2.
Para and I have both shown that the original UTA is also solvable by logic. ;-)

We both made our key breakthrough steps in the same area. Para used interactions between the 22(4) cage at R9C1 and the 23(4) cage at R8C5; I used interactions between the hidden 15(3) cage R789C4 and the 23(4) cage at R8C5 which I think was a bit easier.

Here is my walkthrough for the original UTA.

Prelims

a) R2C67 = {16/25/34}, no 7,8,9
b) R2C89 = {18/27/36/45}, no 9
c) R3C89 = {14/23}
d) R7C12 = {69/78}
e) R7C34 = {18/27/36/45}, no 9
f) R8C12 = {14/23}
g) R8C34 = {18/27/36/45}, no 9
h) 20(3) cage at R3C1 = {389/479/569/578}, no 1,2
i) 22(3) cage at R3C5 = {589/679}

1. 22(3) cage at R3C5 = {589/679}, 9 locked for R3

2. 45 rule on R12 1 innie R2C1 = 7, clean-up: no 2 in R2C89, no 8 in R7C2
2a. 20(3) cage at R3C1 = {479/578}, no 3,6
2b. 9 of {479} must be in R4C1 -> no 4 in R4C1

3. 45 rule on R123 2 outies R4C12 = 12 = [57/84/93], R4C2 = {347}

4. 45 rule on R9 2 outies R8C58 = 17 = {89}, locked for R8, clean-up: no 1 in R8C34
4a. Naked pair {89} in R8C58, CPE no 8,9 in R9C7
4b. Naked pair {89} in R8C58, CPE no 8,9 in R5C5 using D\
4c. 17(3) cage in N9 can only contain one of 8,9 -> no 8,9 in R9C89

5. 45 rule on N3 1 innie R3C7 = 2 outies R12C6 + 4
5a. Min R12C6 = 3 -> min R3C7 = 7
5b. Max R12C6 = 5, no 5,6,7,8,9 in R12C6, clean-up: no 1,2 in R2C7

6. 45 rule on N1234 2 outies R46C4 = 7 = {16/25/34}, no 7,8,9

7. 45 rule on N6789 2 outies R46C6 = 13 = {49/58/67}, no 1,2,3

8. 45 rule on R89 3 innies R8C679 = 14 = {167/257/356} (cannot be {347} which clashes with R8C12), no 4

9. 45 rule on R789 3 outies R5C8 + R6C89 = 20 = {389/479/569/578}, no 1,2
9a. 3 of {389} must be in R56C8 (R56C8 cannot be {89} which clashes with R8C8) -> no 3 in R6C9

10. 45 rule on N78 2 outies R89C7 = 10 = [37/64/73], no 1,2,5, no 6 in R9C7

11. Hidden killer pair 8,9 in R7C789 and R8C8 for N9, R8C8 = {89} -> R7C789 must contain one of 8,9
11a. Killer pair 8,9 in R7C12 and R7C789, locked for R7, clean-up: no 1 in R7C34

12. 45 rule on R3 4 innies R3C1234 = 18 = {1458/1467/2358} (cannot be {1278/1368/2457/3456} which clash with R3C89, cannot be {2367} because R3C1 only contains 4,5,8)
12a. 7 of {1467} must be in R3C4 -> no 6 in R3C4

13. 45 rule in N7 2 innies R78C3 = 1 outie R9C4 + 3
13a. Min R78C3 = 5 -> min R9C4 = 2

14. R2C89 = {18/36/45}, R3C89 = {14/23} -> combined cage R23C89 = {18}{23}/{36}{14}/{45}{23}, 3 locked for N3, clean-up: no 4 in R2C6

15. 23(4) cage at R8C5 cannot be {1589} because R9C7 only contains 3,4,7
15a. 45 rule on N7 3 outies R789C4 = 15 = {249/258/267/348/357} (cannot be {456} which clashes with R46C4)
15b. Variable killer pair 8,9 in R789C4 and 23(4) cage for N8, either R789C4 must contain one of 8,9 in R9C4 or 23(4) cage must contain both of 8,9
15c. R789C4 = {249/258/348/357} (cannot be {267} which clashes with 23(4) cage = {2489}, only combination for 23(4) cage which contains both of 8,9), no 6, clean-up: no 3 in R7C3, no 3 in R8C3
15d. Min R78C3 = 7 (cannot be {24} = 6 which clashes with R8C12) -> min R9C4 = 4 (step 13)
15e. 8,9 of {249/348} must be in R9C4 -> no 4 in R9C4

16. 17(4) cage at R7C5 = {1367/1457/2357/2456}
16a. R789C4 (step 15c) = {249/258/348} (cannot be {357} which clashes with 17(4) cage, ALS block for {1367}, 5 in the other combinations of 17(4) cage must be in N8), no 7, clean-up: no 2 in R7C3, no 2 in R8C3
16b. 8,9 of {249/258/348} must be in R9C4 -> R9C4 = {89}
16c. Naked pair {89} in R8C5 + R9C4, locked for N8

17. 8,9 in R9 only in 22(4) cage at R9C1 = {1489/2389}, no 5,6,7
17a. Killer quad 1,2,3,4 in R8C12 and 22(4) cage at R9C1, locked for N7, clean-up: no 5 in R7C4, no 5 in R8C4
17b. 5 in N7 only in R78C3, locked for C3

18. R789C4 (step 16a) = {249/348}, 4 locked for C4 and N8, clean-up: no 3 in R46C4 (step 6)

19. 17(4) cage at R7C5 (step 16) = {1367/2357}, CPE no 3 in R8C4, clean-up: no 6 in R8C3
19a. 6 in N7 only in R7C123, locked for R7

20. Killer pair 2,4 in R8C12 and R8C4, locked for R8
20a. R8C12 = {14/23}, R8C34 = [54/72] -> combined cage R8C1234 = {14}[72]/{23}[54]

21. 17(4) cage at R7C5 (step 19) = {1367} (only remaining combination, cannot be {2357} because {23}[57] clashes with R8C3, {25}{37} clashes with combined cage R8C1234 and {27}[53] clashes with R8C34 which must be [54] when 2 in R7C56), no 2,5, 1 locked for N8, 6 also locked for R8
21a. 5 in N8 only in R9C56, locked for R9

22. 45 rule on R89 2 outies R7C56 = 1 innie R8C9 + 3
22a. R7C56 = {13/17/37} = 4,8,10 -> R8C9 = {157}, no 3

23. 17(3) cage in N9 = {179/269/278} (cannot be {368} which clashes with R89C7, cannot be {467} because R8C8 only contains 8,9), no 3,4
23a. Killer pair 6,7 in R89C7 and 17(3) cage, locked for N9
23b. Killer pair 1,2 in 22(4) cage at R9C1 and 17(3) cage, locked for R9

24. R8C9 = {15} -> R7C56 (step 22a) = 4,8 = {13/17}, 1 locked for R7 and N8

25. R78C4 = {24} (hidden pair in N8), locked for C4, R9C4 = 9 (step 18), R8C5 = 8, R8C8 = 9, placed for D\, clean-up: no 5 in R46C4 (step 6), no 4 in R4C6 (step 7), no 6 in R7C3
25a. Naked pair {57} in R78C3, locked for C3 and N7, clean-up: no 8 in R7C1
25b. Naked pair {16} in R46C4, locked for C4 and N5, clean-up: no 7 in R46C6 (step 7)

26. R5C8 + R6C89 (step 9) = {389/479/569/578}
26a. 9 of {479/569} must be in R6C9 -> no 4,6 in R6C9

27. 15(3) cage at R6C9 = {159/258/357} (cannot be {249/348} because R8C9 only contains 1,5), no 4, 5 locked for C9, clean-up: no 4 in R2C8
27a. 2 of {258} must be in R7C9 -> no 8 in R7C9
27b. 7,8,9 of {159/258/357} must be in R6C9 -> R6C9 = {789}
27c. 15(3) cage = {159/258/357}, 5 locked for N9

28. 8 in N9 only in R7C78, locked for 23(4) cage at R5C8, no 8 in R56C8
28a. R5C8 + R6C89 (step 9) = {479/569/578} (cannot be {389} because 8,9 only in R6C9), no 3
28b. 8,9 only in R6C9 -> R6C9 = {89}, no 7, clean-up: no 3 in R7C9 (step 27)
28c. Killer pair 1,2 in 15(3) cage at R6C9 and 17(3) cage, locked for N9

29. 23(4) cage at R5C8 must contain 8 = {3578/4568}, 5 locked for C8 and N6, clean-up: no 4 in R2C9
29a. 5,6 of {4568} must be in R56C8 -> no 4 in R56C8

30. Killer pair 1,3 in R2C89 and R3C89, locked for N3
30a. 1 in C7 only in R456C7, locked for N6

31. R2C67 = [25/34] (cannot be [16] which clashes with R2C89), no 1,6

32. Hidden killer pair 7,9 in 20(4) cage at R1C6 and R3C7 for N3, 20(4) cage at R1C6 can only contain one of 7,9 (cannot be {1379} because 1,3 only in R1C6) -> R3C7 = {79}

33. R3C7 = R12C6 + 4 (step 5)
33a. R3C7 = {79} -> R12C6 = 3,5 = [12/23/32], 2 locked for C6 and N2

34. 22(3) cage at R3C5 = {679} (cannot be {589} which clashes with R123C4, ALS block), locked for R3, 6 also locked for N2
34a. 8 in N2 only in R123C4, locked for C4

35. 27(5) cage at R1C1 = {13689/14589/23589/34569}, 9 locked for N1

36. 4 in N2 only in R12C5, locked for C5 and 20(4) cage at R1C3, no 4 in R1C3

37. 45 rule on N23 2 innies R23C4 = 1 outie R1C3 + 7
37a. R23C4 = {35/58} = 8,13 (cannot be {38} = 11 because no 4 in R1C3) -> R1C3 = {16}
37b. R23C4 = {35/58}, 5 locked for C4 and N2

38. 20(4) cage at R1C3 must contain 4 (step 36) = {1478/3467} (cannot be {1469} because R1C4 only contains 3,7,8), no 9
38a. R1C3 = {16} -> no 1 in R12C5

39. R1C6 = 1 (hidden single in N2), R2C6 = 2 (hidden single in C6), R2C7 = 5, R1C3 = 6
39a. 20(4) cage at R1C3 (step 38) = {3467} (only remaining combination), 3,7 locked for N2, 7 also locked for R1 -> R2C4 = 8, R3C4 = 5, clean-up: no 1 in R2C89, no 8 in R4C1 (step 2a), no 4 in R4C2 (step 3)

40. Naked pair {36} in R2C89, locked for R2 and N3 -> R2C5 = 4, R2C2 = 1, placed for D\, R2C3 = 9, R4C4 = 6, placed for D\, R6C4 = 1, placed for D/, clean-up: no 2 in R3C89, no 4 in R8C1
40a. Naked pair {14} in R3C89, locked for R3 and N3 -> R3C1 = 8, R4C1 = 5 (step 2a), R4C2 = 7 (step 3), clean-up: no 8 in R6C6 (step 7)

41. R1C12 = [45] (hidden pair in N1), 4 placed for D\ -> R6C6 = 5, R4C6 = 8 (step 7), placed for D/

42. R3C7 = 7 (hidden single in R3), clean-up: no 3 in R89C7 (step 10)
42a. R89C7 = [64]
42b. Naked pair {38} in R7C78, locked for R7 -> R7C6 = 7, R7C5 = 1, R8C6 = 3, R7C3 = 5, R7C9 = 2, R7C4 = 4, R8C34 = [72], R8C12 = [14], R8C9 = 5, R1C9 = 9, R6C9 = 8, R9C9 = 7, placed for D\, R9C8 = 1, R3C89 = [41]

43. R2C8 = 6 (hidden single on D/), R2C9 = 3, R45C9 = [46], R56C8 = [57]

44. R7C7 = 8 (hidden single on D\), R7C8 = 3, R1C78 = [28], R4C8 = 2, R4C7 = 3 (cage sum)

45. R4C34 = [16] = 7 -> R5C23 = 10 = {28}, locked for R5 and N4 -> R5C5 = 3, placed for both diagonals

and the rest is naked singles.

I'll rate my walkthrough for UTA at Hard 1.5 based on steps 15 and 16. The only other difficult step was the analysis in step 21.
UTA v2 by sudokuEd (Sept 06)
Puzzle pic: 1-9 cannot repeat on the diagonals:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:6913:6913:5122:5122:5122:5134:5134:5134:5134:5138:6913:6913:6913:5122:1792:1792:2320:2320:5138:4373:4373:4373:5654:5654:5654:1305:1305:5138:4373:4390:4390:3623:3844:3844:3125:3125:4635:4390:4390:3077:3623:3844:3882:5931:3125:4635:4635:3077:3077:3623:3882:3882:5931:3911:3894:3894:2362:2362:4355:4355:5931:5931:3911:1343:1343:2370:2370:5966:4355:4355:4432:3911:2632:2632:3078:3078:5966:5966:5966:4432:4432:
Solution:
+-------+-------+-------+
| 4 5 6 | 3 7 1 | 2 8 9 |
| 7 1 9 | 8 4 2 | 5 6 3 |
| 8 3 2 | 5 6 9 | 7 4 1 |
+-------+-------+-------+
| 5 7 1 | 6 9 8 | 3 2 4 |
| 9 2 8 | 7 3 4 | 1 5 6 |
| 3 6 4 | 1 2 5 | 9 7 8 |
+-------+-------+-------+
| 6 9 5 | 4 1 7 | 8 3 2 |
| 1 4 7 | 2 8 3 | 6 9 5 |
| 2 8 3 | 9 5 6 | 4 1 7 |
+-------+-------+-------+
Quote:
sudokuEd: I've changed some cages around and now it is solvable by logic. Hence UTA2
tag solution: by sudokuEd & Nasenbaer (support from Andrew; and more support from Andrew! :) )
Andrew in 2011: I used the same breakthrough steps that I used for my UTA walkthrough. I'll therefore also rate UTA2 at Hard 1.5
Condensed Walkthrough by sudokuEd:
Here is the condensed walk-through V2 for UTA2

Preliminary eliminations for 2 cell cages
R2C67 = 7 -> no 7, 8 or 9
R2C89 = 9 -> no 9
R3C89 = {14/23}
R7C12 = {69/87}
R7C34 = 9 -> no 9
R8C12 = {14/23}
R8C34 = 9 -> no 9
R9C12 = 10 -> no 5
R9C34 = 12 -> no 1,2 or 6

1. "45" on R12 -> R2C1 = 7 -> no 2 in R2C89, no 3 in R9C2, R7C12 = {69/[87]}

2. 22(3) in R3 = {589/679} -> 9 locked in R3, nowhere else

3. R34C1 = 13 = [49]/{58}

4. "45" on R123 -> 2 outies = 12 -> R4C12 = [57/84/93]

5. "45" on R9 -> 2 outies = 17 -> R8C58 = {89} -> 8 and 9 locked in R8 -> no 1 in R8C34

6. since R8C58 = {89}-> 8 and 9 eliminated from R9C7 (same cage as R8C5) and R5C5 (linked through D\)

7. 17(3) cage in N9 must have 8 or 9 but cannot have both since 8 + 9 = 17 ->no 8 or 9 in R9C89

8. this step moved to 10a

9. this step moved to 10b

10. "45" on R89 -> 3 innies = 14 -> R8C679 = {167}/{257}/{356} (cannot be {347} because that would clash with the 5(2) cage in R8C12) -> no 4

10a. "45" on N78 -> R89C7 = 10 ->{37/[64]}

10b. R9C7 = {347} -> 10(2) cage in R9C12 cannot be [37] because{347} in R9C127 would clash with the 12(2) cage in R9 -> no 3 or 7 in R9C12
[Alternatively: R9C12 cannot be [37] which clashes with R9C34 and R9C7, Killer ALS block]

11. In N9, the only other place (apart from R8C8) for [8/9] must be in R7C789. 15(2) cage in R7C12 also must have [8/9] -> Killer pair on 8/9 in r7 -> no 8/9 elsewhere R7 -> no 1 in R7C34

12. 5 in N7 locked in C3 -> no 5 elsewhere in C3

13. "45" on N7 -> 3 innies = 15 [hidden (h)15(3) cage] and must have 5 -> other 2 cells = 10
-Cannot be {19} since no 1 in R789C3
-Cannot be {28} as that requires 15(2) cage = {69} but this leaves no 2, 6 or 9 for 10(2) cage in N7
-> h15(3) cage in N7 = 5{37/46} -> no 2, 8 or 9 -> no 7 in R78C4, no 3 or 4 in R9C4

14. h15(3) in N7 = [6/7...], 15(2) cage in N7 = {69/[87]} = [6/7...] -> Killer pair on 6 and 7 -> not elsewhere in N7 -> 10(2) cage now {19/28}

15."45" on N3 -> 1 outie + 11 = 2 innies -> min R23C7 = 1 + 11 = 12
-Max. R3C7 = 9 -> min. R2C7 = 3 -> no 1 or 2 in R2C7, no 5 or 6 in R2C6
-Max R23C7 = {69} = 15 -> Max R1C6 = 4

16. 7(2) cage in N23 cannot have [43] since that would require 9(2) cage in N3 = {18} but [3{18}] in R2C789 would leave no 3 or 1 for 5(2) cage in N3 -> no 3 in R2C7 -> no 4 in R2C6
[Alternatively: Combined cage R23C89 gives eliminations for R1C789]

17. "45" on N3 -> 2 outies + 4 = 1 innie. R12C6min = {12} = 3 -> min R3C7 = 7

18. "45" on N7 -> R789C4 = 15

19. "45" on N1234 -> R456C4 = 14

20. "45" on C4 -> R123C4 = 16 = h16(3) cage

21. "45" on C1234 -> 2 innies = 9 -> no 9 in R1C34 and no 2 or 4 in R1C4

22. 9 in N1 now locked in 27(5) cage -> no 9 in R2C4

23. "45" on C1234 -> 2 outies = 11 -> no 1 in R12C5 and no 4 in R1C5

24. 22(3) cage in R3 = {589/679} = [5/6...] with the 5 or 6 locked in N2 -> no {56} combination in R12C5 since = 11 (step 23) -> no 5 or 6

25. The only other place for [5/6] in N2 is in R123C4 -> h16(3) cage = {268/358/367/457} -> no 1 in R123C4, no 8 in R1C3

26. 1 locked in R12C6 for N2 -> 1 locked for C6 -> 1 in N8 locked in C5 and 1 locked in C4 for N5

27. we know that 5 is locked in R789C3 -> [4/7] must be in R789C4 (since R789C34 are linked through 2 9(2) cages or the 12(2) cage -> h16(3) cage in N2 cannot be {457} (since 4/7 must be in R789C4)-> h16(3) cage in N2 = {268/358/367} = [3/8,5/6...] -> no 4 in R23C4

28. R789C4 = 15 and must contain a 4 or 7 = {249/267/357} ({348/456} are blocked because a 3/8 or 5/6 are needed for R123C4- step 27) -> no 8 in R789C4. Each combination needs a 7 or 9 which are only available in R9C4 -> R9C4 = {79} -> R9C34 = [39/57]

29. 8 in N8 now locked in 23(4) cage in N89
-Cannot be {1589} as R9C7 has none of these
-Cannot be {2678} as the {267} must be in R9 which means both the 12(2) and 10(2) cages in R9 need 9's!
-Cannot be {3578} as the {357} must be in R9. But that clashes with 12(2) cage in R9

30. Only combinations left are 8{249/456} -> only candidate available in R9C7 is 4 -> R9C7 = 4, R89C7 = 10 -> R8C7 = 6, R2C67 = [25], R1C6 = 1(hidden single N2), R3C7 = 7 (step 15), no other 7s on D/ , no 1, 3 or 7 in R9C56

31. R3C56 = 15 = {69} -> 5 in N2 locked in C4 in hidden 16(3) cage = {358} -> r12c5 = [74] -> R1C34 = 9 = [63], R23C4 = [85], no 1 in R2C89, h15(3) cage in N8 = {249/267} = 2{49/67}, 2 cannot be in R7C4 since no 7 available in R7C3 -> R8C4 = 2 (hidden single N8), R8C3 = 7, R7C12 = {69}, R9C12 = {28}, R8C12 = {14}, R7C4 = 4, R7C3 = 5 placed for D/, R9C34 = [39], 17(4) cage in N89 now {1367} only ->R78C6 = [73], R7C5 = 1, R8C58 = [89], no other 9 on D\, R8C9 = 5

32. 2 in N5 locked in 14(3) cage = {239}, R456C6 = {458}, R3C56 = [69], R9C56 = [56]

33."45" on N9 -> 3 outies = 20 -> no 1 or 2 in R56C8 or R6C9, max. R56C8 = {78} = 15 -> min. R7C78 = 8 so can only be {28/38} = 12/13 -> 8 locked in R7C78 -> no 8 in R7C9 or in R56C8

34. R67C9 = 10 = [73/82]

35. 9(2) cage in N3 = {36}, R3C89 = {14}, R1C789 = {289}
the rest are singles, (remembering to use the diagonals for elimination)

[edited 24Oct: condensed walkthrough V2. Since there have been 606 views for this thread, probably only you and me will read this Andrew - so great job with this final walkthrough. Your suggestions were all spot on. We're slowly getting better at this! I also like the double spacing - works great in notepad, don't get lost nearly as much. So UTA is finally (really) finished ;)
2011 Walkthrough by Andrew:
And now for UTA2. This was originally solved as a "tag" at Ed’s suggestion when he posted it. I’m now trying it by myself several years later. This puzzle is a simplified variant of UTA; the "tag" started with steps from the "tag" for UTA so I’ve done the same thing using as many of the steps from my walkthrough for UTA as possible or until I spotted something simpler.

Prelims

a) R2C67 = {16/25/34}, no 7,8,9
b) R2C89 = {18/27/36/45}, no 9
c) R3C89 = {14/23}
d) R7C12 = {69/78}
e) R7C34 = {18/27/36/45}, no 9
f) R8C12 = {14/23}
g) R8C34 = {18/27/36/45}, no 9
h) R9C12 = {19/28/37/46}, no 5
i) R9C34 = {39/48/57}, no 1,2,6
j) 20(3) cage at R3C1 = {389/479/569/578}, no 1,2
k) 22(3) cage at R3C5 = {589/679}

1. 22(3) cage at R3C5 = {589/679}, 9 locked for R3

2. 45 rule on R12 1 innie R2C1 = 7, clean-up: no 2 in R2C89, no 8 in R7C2, no 3 in R9C2
2a. 20(3) cage at R3C1 = {479/578}, no 3,6
2b. 9 of {479} must be in R4C1 -> no 4 in R4C1

3. 45 rule on R123 2 outies R4C12 = 12 = [57/84/93], R4C2 = {347}

4. 45 rule on R9 2 outies R8C58 = 17 = {89}, locked for R8, clean-up: no 1 in R8C34
4a. Naked pair {89} in R8C58, CPE no 8,9 in R9C7
4b. Naked pair {89} in R8C58, CPE no 8,9 in R5C5 using D\
4c. 17(3) cage in N9 can only contain one of 8,9 -> no 8,9 in R9C89

5. 45 rule on N3 1 innie R3C7 = 2 outies R12C6 + 4
5a. Min R12C6 = 3 -> min R3C7 = 7
5b. Max R12C6 = 5, no 5,6,7,8,9 in R12C6, clean-up: no 1,2 in R2C7

6. 45 rule on R89 3 innies R8C679 = 14 = {167/257/356} (cannot be {347} which clashes with R8C12), no 4

7. 45 rule on R789 3 outies R5C8 + R6C89 = 20 = {389/479/569/578}, no 1,2
7a. 3 of {389} must be in R56C8 (R56C8 cannot be {89} which clashes with R8C8) -> no 3 in R6C9

8. 45 rule on N78 2 outies R89C7 = 10 = [37/64/73], no 1,2,5, no 6 in R9C7

9. Hidden killer pair 8,9 in R7C789 and R8C8 for N9, R8C8 = {89} -> R7C789 must contain one of 8,9
9a. Killer pair 8,9 in R7C12 and R7C789, locked for R7, clean-up: no 1 in R7C34

10. 45 rule on R3 4 innies R3C1234 = 18 = {1458/1467/2358} (cannot be {1278/1368/2457/3456} which clash with R3C89, cannot be {2367} because R3C1 only contains 4,5,8)
10a. 7 of {1467} must be in R3C4 -> no 6 in R3C4

11. R2C89 = {18/36/45}, R3C89 = {14/23} -> combined cage R23C89 = {18}{23}/{36}{14}/{45}{23}, 3 locked for N3, clean-up: no 4 in R2C6

[The main breakthrough step I used for solving UTA doesn’t work immediately for UTA2. I’ve therefore added a few steps which weren’t available in the UTA cage pattern.]

12. 45 rule on N1234 1 outies R4C4 = 1 innie R6C3 + 2, no 1,2 in R4C4, no 8,9 in R6C3

13. 45 rule on N6789 1 outie R6C6 = 1 innie R4C7 + 2, no 8,9 in R4C7, no 1,2 in R6C6

14. 45 rule on C1234 2 outies R12C5 = 11 = {29/38}/[74] (cannot be {56} which clashes with 22(3) cage at R3C5), no 1,5,6, no 4 in R1C5
14a. 45 rule on C1234 2 innies R1C34 = 9 = {18/36/45}/[27], no 9, no 2 in R1C4
14b. 9 in N1 only in R1C12 + R2C23, locked for 27(5) cage at R1C1, no 9 in R2C4

15. 5 in N7 only in R789C3, locked for C3, clean-up: no 4 in R1C4 (step 14a), no 7 in R4C4 (step 12)
15a. 45 rule on N7 3 innies R789C3 = 15 = {258/357/456} (cannot be {249/348} which clash with R8C12, cannot be {267} which clashes with R7C12), no 9, clean-up: no 3 in R9C4

[With hindsight there’s also 1 in N7 only in R8C12 = {14} or R9C12 = {19} -> R9C12 cannot be {46} (locking-out cages). However that’s not needed after step 18.]

16. Hidden killer pair 5,6 in R123C4 and R3C56 for N2, R3C56 contains one of 5,6 -> R123C4 must contain one of 5,6

[Now I can return to my UTA steps with a slight change to step 17a.]

17. 23(4) cage at R8C5 cannot be {1589} because R9C7 only contains 3,4,7
17a. 45 rule on N7 3 outies R789C4 = 15 = {249/258/267/348/357} (cannot be {456} which clashes with R123C4)
17b. Variable killer pair 8,9 in R789C4 and 23(4) cage for N8, either R789C4 must contain one of 8,9 in R9C4 or 23(4) cage must contain both of 8,9
17c. R789C4 = {249/258/348/357} (cannot be {267} which clashes with 23(4) cage = {2489}, only combination for 23(4) cage which contains both of 8,9), no 6, clean-up: no 3 in R7C3, no 3 in R8C3
17d. 8,9 of {249/348} must be in R9C4 -> no 4 in R9C4, clean-up: no 8 in R9C3

18. Hidden killer pair 8,9 in R7C12 and R9C12 for N7, R7C12 contains one of 8,9 -> R9C12 must contain one of 8,9 -> R9C12 = {19/28}, no 3,4,6,7
18a. Killer pair 1,2 in R8C12 and R9C12, locked for N7, clean-up: no 7 in R7C4, no 7 in R8C4

19. 17(4) cage at R7C5 = {1367/1457/2357/2456}
19a. R789C4 (step 17c) = {249/258/348} (cannot be {357} which clashes with 17(4) cage, ALS block for {1367}, 5 in the other combinations of 17(4) cage must be in N8), no 7, clean-up: no 5 in R9C3
19b. 8,9 of {249/258/348} must be in R9C4 -> R9C4 = {89}, clean-up: no 7 in R9C3
19c. Naked pair {89} in R8C5 + R9C4, locked for N8
19d. Killer pair 3,4 in R8C12 and R9C3, locked for N7, clean-up: no 5 in R7C4, no 5 in R8C4

20. R789C4 (step 19a) = {249/348}, 4 locked for C4 and N8, clean-up: no 2 in R6C3 (step 12)

21. 17(4) cage at R7C5 (step 19) = {1367/2357}, CPE no 3 in R8C4, clean-up: no 6 in R8C3
21a. 6 in N7 only in R7C123, locked for R7

22. Killer pair 2,4 in R8C12 and R8C4, locked for R8
22a. R8C12 = {14/23}, R8C34 = [54/72] -> combined cage R8C1234 = {14}[72]/{23}[54]

23. 17(4) cage at R7C5 (step 19) = {1367} (only remaining combination, cannot be {2357} because {23}[57] clashes with R8C3, {25}{37} clashes with combined cage R8C1234 and {27}[53] clashes with R8C34 which must be [54] when 2 in R7C56), no 2,5, 1 locked for N8, 6 also locked for R8
23a. 5 in N8 only in R9C56, locked for R9

24. 45 rule on R89 2 outies R7C56 = 1 innie R8C9 + 3
24a. R7C56 = {13/17/37} = 4,8,10 -> R8C9 = {157}, no 3

25. 17(3) cage in N9 = {179/269/278} (cannot be {368} which clashes with R89C7, cannot be {467} because R8C8 only contains 8,9), no 3,4
25a. Killer pair 6,7 in R89C7 and 17(3) cage, locked for N9
25b. Killer pair 1,2 in 22(4) cage at R9C1 and 17(3) cage, locked for R9

26. R8C9 = {15} -> R7C56 (step 24a) = 4,8 = {13/17}, 1 locked for R7 and N8

27. R78C4 = {24} (hidden pair in N8), locked for C4, R9C4 = 9 (step 20), R8C5 = 8, R8C8 = 9, placed for D\, R9C3 = 3, clean-up: no 6 in R1C4 (step 14a), no 3 in R12C5 (step 14), no 5 in R4C4 (step 12), no 7 in R4C7 (step 13), 7 in R6C3 (step 12), no 6 in R7C3, no 2 in R8C12, no 7 in R8C7 (step 8), no 1 in R9C12
27a. Naked pair {14} in R8C12, locked for R8 -> R8C9 = 5, R8C34 = [72], R7C3 = 5, placed for D/, R7C4 = 4, clean-up: no 4 in R2C89, no 8 in R7C1
27b. Naked triple {238} in R7C789, locked for R7 and N9 -> R8C7 = 6, R9C7 = 4 (step 8), R8C6 = 3, R2C7 = 5, R2C6 = 2, clean-up: no 9 in R12C5 (step 14), no 1 in R4C7 (step 13), no 6,7,8 in R6C6 (step 13)

28. R12C5 = [74], R1C6 = 1, R7C56 = [17], clean-up: no 2,8 in R1C3 (step 14a), no 8 in R1C4 (step 14a)

29. R5C8 + R6C89 (step 7) = {389/479/569/578}
29a. 9 of {479/569} must be in R6C9 -> no 4,6 in R6C9

30. R8C9 = 5 -> 15(3) cage at R6C9 = {258/357}, no 9
30a. 2 of {258} must be in R7C9 -> no 8 in R7C9

31. 8 in N9 only in R7C78, locked for 23(4) cage at R5C8, no 8 in R56C8
31a. R5C8 + R6C89 (step 7) = {578} (only remaining combination) -> R6C9 = 8, R7C9 = 2 (step 30), R56C8 = {57}, locked for C8 and N6, R9C89 = [17], clean-up: no 3 in R3C8, no 4 in R3C9
31b. 1 in C7 only in R56C7, locked for N6

32. 12(3) cage in N6 = {246} (only remaining combination) -> R4C8 = 2, R45C9 = {46}, locked for C9, R1C9 = 9, placed for D/, R4C7 = 3, R6C6 = 5 (step 13), placed for D\, R56C8 = [57], R9C56 = [56], R7C7 = 8, placed for D\, R7C8 = 3, R3C8 = 4, R3C9 = 1, R2C9 = 3, R2C8 = 6, placed for D/, R1C78 = [28], R2C234 = [198], 22(3) cage at R3C5 = [697], R4C4 = 6, placed for D\, R4C5 = 9, R45C9 = [46], R4C6 = 8, placed for D/, R5C6 = 4, R34C1 = [85], R4C2 = 7, R9C1 = 2, placed for D/, R9C2 = 8, R5C5 = 3, placed for D\, R1C1 = 4, R1C3 = 6, R1C4 = 3 (step 14a)

and the rest is naked singles.

Rating Comment. While the later steps for UTA2 were simpler than for the original UTA, I used the same breakthrough steps that I used for my UTA walkthrough. I'll therefore also rate my walkthrough for UTA2 at Hard 1.5.


Last edited by Ed on Sat Jun 14, 2008 12:54 am, edited 2 times in total.

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PostPosted: Sat Jun 07, 2008 11:48 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
Assassin 17 by Ruud (Sept 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6400:6400:6400:5123:5123:5123:4614:4614:4614:2313:4362:6400:5123:5645:2830:2830:4368:4614:2313:4362:5645:5645:5645:3863:2830:4368:4368:2313:4362:4362:4638:4638:3863:3863:4898:2339:4388:4388:4134:4134:4638:4638:4638:4898:2339:3885:3885:4134:3376:7729:7729:7729:4898:4898:3885:3885:4664:3376:3376:7729:6204:6204:6204:3903:2624:4664:2370:2370:2372:2372:3910:6204:3903:2624:4664:4664:2370:3661:3661:3910:3910:
Solution:
+-------+-------+-------+
| 5 4 7 | 6 8 2 | 3 1 9 |
| 3 8 9 | 4 7 1 | 6 2 5 |
| 2 6 1 | 9 5 3 | 4 8 7 |
+-------+-------+-------+
| 4 1 2 | 8 3 5 | 7 9 6 |
| 8 9 6 | 7 1 4 | 2 5 3 |
| 7 5 3 | 2 9 6 | 8 4 1 |
+-------+-------+-------+
| 1 2 4 | 5 6 7 | 9 3 8 |
| 9 7 5 | 3 2 8 | 1 6 4 |
| 6 3 8 | 1 4 9 | 5 7 2 |
+-------+-------+-------+
Quote:
Ruud, lead-in: This is a strange killer. It keeps resisting almost till the very end
Andrew: True! Apart from an easy start it does resist for a long time. The Assassins continue to get harder each time
Ruud: I create all my Killers (Assassins and Jigsaws) manually, and use SumoCue to validate and rate it. I keep rearranging the cages until the puzzle has the required difficulty, while it can still be solved by the program....I only needed to glue a few cages together to avoid an easy solution
Oscar: really keep working until the end. With less than two hours solving time I would rate it 1.0
Walk-through by Andrew:
Ruud wrote:
This is a strange killer. It keeps resisting almost till the very end.

True! Apart from an easy start it does resist for a long time. The Assassins continue to get harder each time.

Here is my walkthrough. Any comments or suggested improvements to the solving method are welcome. As with all my walkthroughs, this is essentially in the order that I solved the puzzle.

Step 1
30(4) cage in N568 = {6789}, 45 rule on N9 2 outies = 17 -> R89C6 = [89], R89C7 = [15], R67C6 = {67}, R6C57 = {89}, 45 rule on N3 1 outie = 1 -> R2C6 = 1, R5C12 = {89}, R89C1 = {69/78}, killer pair {89} in R589C1 so no 8 or 9 in rest of C1

Step 2
45 rule on C89 2 outies = 12 -> R17C7 = {39/48}, killer pair {89} in R167C7 -> no 8 or 9 in rest of C7, R23C7 = {37/46}, killer pair {34} in R1237C7 -> no 3 or 4 in rest of C7, R45C7 = {267} with 2 in N6 locked in C7

Step 3
45 rule on R89 innie/outie -> R7C3 = R8C9, 18(5) cage in N56 must contain {12} with the 1 in C45, R1345C6 = {2345} so R34C6 is 9 max -> R4C7 = {67}, R5C7 = 2 (hidden single in C7) -> no other 2 in 18(5) cage in N56, R34C6 = {3/4 5}, R5C6 = {34}, R1C6 = 2, R6C4 = 2 (hidden single in N5), 2 in N8 locked in R89C5 -> 9(3) cage in N8 = {126/234}, 18(5) cage in N56 cannot contain 9 and this is blocked from R5 -> R6C57 = [98] (hidden single 9 in N5), 8 in N5 locked in R4C45 so 18(5) cage in N56 must contain {12348}, R5C4 = {67}, R4C6 = 5, R17C7 = {39}, R23C7 = {46}, R4C7 = 7, R35C6 = [34], R45C9 = {36/45}, 19(4) cage in N6 contains 1 and 9 -> R4C8 = 9

Step 4
R7C45 = {47/56}, killer pair {67} in R7C456 -> no 6 or 7 in rest of R7 or N8 -> 9(3) cage in N8 = {234} with the 2 in C5, R9C4 = 1 (hidden single in N8), R7C45 = {56}, R67C6 = [67], R5C4 = 7, 24(4) cage in N9 = {3489} because the other candidate groups {2679} and {3678} contain both 6 and 7 which are blocked from R7C89, 15(3) cage in N9 = {267}

Step 5
45 rule on R6789 4 outies = 27, already have R4C8 = 9 and R5C4 = 7, 2 remaining outies = 11 -> R5C38 = {56} (7, 8 and 9 already blocked in R5) -> R5C5 = 1 (hidden single in R5) -> R45C9 = [63] (hidden single 3 in R5) -> R5C38 = [65], R6C3 = 3, R6C89 = {14}, R6C12 = {57}, R7C12 = {12}, 45 rule on R89 1 innie/1 outie R7C3 = R8C9 = {49}, R89C2 = {37/46}, R8C3 = 5 (hidden single in N7), R79C3 = [48], R8C9 = 4, R9C5 = 4, R8C45 = [32], R4C45 = [83], R89C2 = [73], R89C1 = [96], R5C12 = [89], R6C12 = [75], R8C8 = 6, R6C89 = [41]

Step 6
R4C123 = {124}, 45 rule on N1 outies – innies = 6 -> R3C3 = 1, R1C8 = 1 (hidden single in N3), R4C3 = 2, R12C3 = {79}, R1C12 = {36/45}, 8 in N1 locked in R23C2 -> 17(4) cage in N14 must contain {1268} -> R4C12 = 1, R23C2 = {68}, R4C1 = 4, R1C12 = [54], R23C1 = [32]

Step 7
9 in N2 cannot be in 20(4) cage (all possible candidates for the remaining 2 cells are blocked) -> R3C4 = 9, R23C5 = {57}, R1C5 = 8, R12C4 = [64], R23C7 = [64] , R23C2 = [86], R7C45 = [56]

Step 8
8 in N3 must be in R3C89 -> 17(3) cage = {278} -> R2C8 = 2, R3C89 = {78} and carry on, the rest is filling in the remaining candidates and simple elimination

Thanks to sudokuEd for some useful suggestions off-forum. They have helped to make this more logical and readable.

I don't think I've ever seen so many Killer Pairs in a puzzle. Ruud, was that your intention when you composed this Assassin?

Andrew
Assassin 17 V2 by sudokuEd (Sept 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6400:6400:6400:5123:5123:5123:4614:4614:4614:2313:4362:6400:5123:5645:2830:2830:4368:4614:2313:4362:5645:5645:5645:3863:2830:4368:4368:2313:4362:4362:4638:4638:3863:3863:4898:2339:4388:4388:4134:4134:4638:4638:4638:4898:2339:3885:3885:4134:3376:7729:7729:7729:4898:4898:3885:3885:4664:3376:3376:7729:6204:6204:6204:3903:2624:4664:4685:4685:3586:3586:3910:6204:3903:2624:4664:4664:4685:4685:3586:3910:3910:
Solution:
+-------+-------+-------+
| 5 4 7 | 6 8 2 | 3 1 9 |
| 3 8 9 | 4 7 1 | 6 2 5 |
| 2 6 1 | 9 5 3 | 4 8 7 |
+-------+-------+-------+
| 4 1 2 | 8 3 5 | 7 9 6 |
| 8 9 6 | 7 1 4 | 2 5 3 |
| 7 5 3 | 2 9 6 | 8 4 1 |
+-------+-------+-------+
| 1 2 4 | 5 6 7 | 9 3 8 |
| 9 7 5 | 3 2 8 | 1 6 4 |
| 6 3 8 | 1 4 9 | 5 7 2 |
+-------+-------+-------+
Quote:
sudokuEd: Look the liberty of making a small change to Assassin 17 to make it a bit harder. It's still solvable by logic
Andrew: When A17V2 first appeared I didn't get very far with it. I was still resisting using elimination solving..I found the other key move, step 26, difficult to spot so for that reason I'll rate A17V2 as an easier 1.5 rather than a harder 1.25
Walkthrough by sudokuEd:
Walk-through for Assassin 17 Version 2
1."45" on N3 -> r2c6 = 1
2. "45" N9 -> r8c6 = 8
3. 30(4) cage in N568 = {6789} with 8 locked in r6 -> no 8 elsewhere in r6
4. 17(2) cage in N4 = {89} -> no 8 or 9 elsewhere in r5 or N4
5. "45" on N12 -> 5 outties = 19 -> must have 1 -> no 1 elsewhere in r4 -> 1 required in 18(5) cage in N56 locked in r5 -> no 1 elsewhere in r5
6. 16(3) cage in N45 = {367/457} = 7{36/45}
7. "45" on r6789 -> 3 innies = 8 = {125/134} = 1{25/34} with 1 locked in r6c89 in N6 -> no 1 elsewhere in N6 or r6 -> r5c5 = 1 (hidden single N5)
8. 7 locked in 16(3) cage in N45 now only in r5 -> no 7 elsewhere in r5
9. from step 7, r6c389 = [5{12}/{134}] but r6c89 cannot be {12} = 3 since that requires r45c8 = 16 = {79} but 7 and 9 only in r4c8 -> r6c389 <> [5{12}] -> r6c389 = {134} only -> no 3 or 4 elsewhere in r6.
10. from step 9, r6c89 = {13/14} = 4 or 5 -> r45c8 = 14 or 15 = [95/86/96] -> r4c8 = {89}, r5c8 = {56}
11. from step 9, r6c3 = {34} -> r5c34 = {67/57} = {5/6} -> Killer pair with {56} in r5c8 -> no 5 or 6 elsewhere in r5
12. "45" on c89 -> 2 outies = 12 -> no 1,2 or 6 -> 1 in c7 locked in r89c7 = 6 = {15} only -> no 1 or 5 elsewhere in N9 or c7
13. "45" on N6 -> 3 innies = 17. Max. r56c7 = {49} = 13 -> min. r4c7 = 4 but this means 2 4's in c7 -> no 4 possible in r4c7
14. from step 12,r17c7 = 12 = {39/48}, r23c7 = {28/37/46} -> complex hidden single on 3 in r1237c7 -> no 3 in r5c7.
15.19(4) cage in N6 = 1{369/459/468} = {4/6, 6/9...} -> r456c7 = 17 <> {269/467} (since would leave no combinations for 19(4) cage. -> the only combination left for r456c7 = {278} with r5c7 = 2, {78} pair in r46c7 -> no 7 or 8 elsewhere in c7 or N6, r23c7 = {46} pair in c7 and N3, r4c8 = 9, r23c7 = {46} pair in N3
16."45" on N8 -> 4 outties - 24 = 1 innie -> r6c567 cannot = 24 since that would require r6c4 = r9c4 which is impossible since both are in c4 -> r6c567 <> {789} -> no 6 in r7c6 -> 6 required in 30(4) cage locked in r6 and in N5 -> no 6 elsewhere in r6 or N5.
17. rest of 18(5) cage in N56 now can only = {348/357} but cannot be {357} since r5c4 = {57} -> r4c45 + r5c6 = {348} only with 8 locked in r4 -> r46c7 = [78], r34c6 = [62/35]
18. "45" on N2 -> 1 outie + 2 = 1 innie. r3c6 = {36} -> r3c3 = {14} -> r3c36 [13/46] but cannot be [46] since r3c7 = {46} -> r3c36 = [13], r45c6 = [54], r45c9 = [63], r5c48 = [75], r56c3 = [63]
19. {69} pair in r6c56 -> r6c4 = 2, r7c6 = 7,
20. "45" on N8 -> r9c4 = 1, r89c7 = [15]
21. {57} pair in r6c12 = 12 -> rest of 15(4) cage = 3 = {12}, 18(3) cage in N7 = {458} only, r89c1 = [69} only, r89c2 = {37} only
22. "45" on r89 -> r7c3 = r8c9 = 4
the rest is pretty straight forward
Walkthrough by Andrew:
When A17V2 first appeared I didn't get very far with it. I was still resisting using elimination solving; my A17 walkthrough finished "and carry on, the rest is filling in the remaining candidates and simple elimination".

Recently Ed encouraged me to have another go at A17V2, having improved and learned some new techniques in the year or so since it first appeared.

Here is my walkthrough for A17V2. Step 7a, one of the key moves, is probably the most useful IOU I've come across. The hidden killer pair in step 16 was probably a slightly easier way to sort out R456C7 than Ed's hidden complex single in C7, which was itself a neat move.

I found the other key move, step 26, difficult to spot so for that reason I'll rate A17V2 as an easier 1.5 rather than a harder 1.25.

This puzzle differs from A17 because of changes to the cages at R8C4 and R8C6. There were still easy starters at R2C6 and R8C6 but the cage changes took away the other easy starter at R9C6 which now doesn't get placed until "the rest is naked singles".

Here is my walkthrough for A17V2.

Prelims

a) R45C9 = {18/27/36/45}, no 9
b) R5C12 = {89}, locked for R5 and N4, clean-up: no 1 in R4C9
c) R89C1 = {69/78}
d) R89C2 = {19/28/37/46}, no 5
e) R234C1 = {126/135/234}, no 7,8,9
f) 11(3) cage at R2C6 = {128/137/146/236/245}, no 9
g) 30(4) cage at R6C5 = {6789}
h) 18(5) cage at R4C4 = 12{348/357/456}, no 9

1. 45 rule on N3 1 outie R2C6 = 1
1a. R23C7 = 10 = {28/37/46}, no 5

2. 45 rule on N9 1 outie R8C6 = 8, clean-up: no 7 in R9C1, no 2 in R9C2
2a. R89C7 = 6 = {15/24}

3. 8 in 30(4) cage at R6C5 locked in R6C57, locked for R6

4. Killer pair 8,9 in R5C1 and R89C1, locked for C1

5. 45 rule on N2 1 remaining innie R3C6 = 1 outie R3C3 + 2, no 2 in R3C6, no 6,8,9 in R3C3

6. 45 rule on R89 1 outie R7C3 = 1 innie R8C9, no 8 in R7C3

7. 45 rule on N8 2 remaining innies R7C6 + R9C4 = 1 outie R6C4 + 6
7a. IOU no 6 in R7C6
7b. 6 in 30(4) cage at R6C5 locked in R6C567, locked for R6

8. 45 rule on R1234 4 innies R4C4589 = 26 = {2789/3689/4589/4679/5678}, no 1

9. 1 in 18(5) cage at R4C4 locked at R5C57, locked for R5, clean-up: no 8 in R4C9

10. 45 rule on R6789 3 innies R6C389 = 8 = 1{25/34}, 1 locked in R6

11. 16(3) cage at R5C3 = {367/457}, no 1,2
11a. 7 locked in R5C34, locked for R5, clean-up: no 2 in R4C9
11b. 3 of {367} must be in R6C3 -> no 3 in R5C34

12. R6C389 (step 10) = 1{25/34}, 1 locked in R6C89 for N6 and 19(4) cage
12a. 5 of {125} must be in R6C3 -> no 5 in R6C89
12b. Max R6C89 = 5 -> min R45C8 = 14, no 2,3,4, no 5,6,7 in R4C8
12c. Max R45C8 = 15 -> min R6C89 = 4, no 2, clean-up: no 5 in R6C3 (step 10), no 4 in R5C34 (step 11)

13. Naked triple {134} in R6C389, locked for R6

14. Naked triple {567} in R5C348, locked for R5, clean-up: no 3,4 in R4C9

15. R5C5 = 1 (hidden single in R5)

16. Hidden killer pair 8,9 in R456C7 and R4C8 for N6 -> R456C7 must contain 8/9
16a. 45 rule on N6 3 innies R456C7 = 17 = {269/278/359/368} (cannot be {458} which clashes with R89C7, cannot be {467} which doesn’t contain 8,9), no 4
16b. R5C7 = {23} -> no 2,3 in R4C7
16c. 2 in N6 locked in R5C79, locked for R5

17. 45 rule on C89, 2 outies R17C7 = 12 = {39/48/57}, no 1,2,6
17a. 1 in C7 locked in R89C7 = {15}, locked for C7 and N9, clean-up: no 7 in R17C7 (step 17)

18. Killer pair 8,9 in R17C7 and R46C7, locked for C7, clean-up: no 2 in R23C7
18a. Killer pair 3,4 in R17C7 and R23C7, locked for C7 -> R5C7 = 2, clean-up: no 7 in R4C9
[Alternatively R5C7 = 2 (hidden single in C7.]

19. 7 in N6 locked in R46C7, locked for C7, clean-up: no 3 in R23C7
19a. Naked pair {46} in R23C7, locked for C7 and N3, clean-up: no 8 in R17C7 (step 17)
19b. Naked pair {39} in R17C7, locked for C7
19c. Naked pair {78} in R46C7, locked for N6 -> R4C8 = 9
19d. Max R23C8 = 15 -> min R3C9 = 2
19e. Min R4C7 = 7 -> max R34C6 = 8, no 7,9, no 6 in R4C6, clean-up: no 5,7 in R3C3 (step 5)

20. 6 in 30(4) cage at R6C5 locked in R6C56, locked for N5

21. 18(5) cage at R4C4 = 123{48/57}, 3 locked for N5

22. Killer pair 7,8 in R4C45 and R4C7, locked for R4

23. 15(3) cage at R3C6 = {258/267/357} (cannot be {348} which clashes with R5C6), no 4, clean-up: no 2 in R3C3 (step 5)

24. 4 in N5 locked in 18(5) cage at R4C4 = {12348}, 8 locked for R4 and N5 -> R4C7 = 7, R6C7 = 8

25. 15(3) cage at R3C6 (step 23) = {267/357}
25a. 3 of {357} must be in R3C6 -> no 5 in R3C6, clean-up: no 3 in R3C3 (step 5)

26. R3C36 = [13] (cannot be [46] which clashes with R3C7), R4C6 = 5 (step 25), R5C6 = 4, R45C9 = [63], R5C8 = 5, R5C4 = 7, R5C3 = 6, R6C3 = 3
26a. R2C5 + R3C45 = 21 = {489/579/678}, no 2
26b. R7C6 = 7 (only remaining position for 7 in 30(4) cage)

27. R6C4 = 2 (hidden single in N5)
27a. R7C45 = 11 = {56} (only remaining combination), locked for R7 and N8
27b. 2 in N2 locked in R1C56, locked for R1

28. R6C12 = {57} -> R7C12 = 3 = {12}, locked for R7 and N7, clean-up: no 8,9 in R89C2
28a. 5 in N7 locked in R89C3, locked for C3

29. Killer pair 6,7 in R89C12, locked for N7
29a. 7 in C3 locked in R12C3, locked for N1 and 25(4) cage

30. 45 rule on N7 3 outies R6C12 + R9C4 = 13, R6C12 = 12 -> R9C4 = 1, R89C7 = [15]

31. 18(4) cage at R7C3 = {1458} (only remaining combination) = [458], R4C3 = 2, R8C9 = 4 (step 6) , R6C89 = [41], clean-up: no 7 in R8C1, no 6 in R89C2

32. Naked pair {69} in R89C1 -> R5C12 = [89]
32a. Naked pair {37} in R89C2 -> R6C12 = [75]
32b. Naked triple {468} in R123C2, locked for C2 and N1 -> R4C12 = [41], R7C12 = [12], clean-up: no 5 in R23C1 = [32], R1C1 = 5, R1C2 = 4 (cage sum)

33. Naked triple {389} in R7C789, locked for N9

34. R1C8 = 1 (hidden single in R1)

35. R9C5 = 4 (hidden single in R9)

36. 3 in N8 locked in R8C45, locked for R8 -> R89C2 = [73]

37. R7C8 = 3 (hidden single in C8), R7C79 = [98], R1C7 = 3

38. R1C78 = 4 -> R12C9 = 14 = [95]

and the rest is naked singles


Last edited by Ed on Sat Jun 14, 2008 12:57 am, edited 1 time in total.

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PostPosted: Sat Jun 07, 2008 11:57 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
Assassin 18 by Ruud (Sept 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6144:6144:4098:4355:4355:4355:4358:5127:5127:6144:4098:4098:7180:4355:7180:4358:4358:5127:6144:3859:3859:7180:7180:7180:5144:5144:5127:4123:3859:3859:2334:2334:2334:5144:5144:5411:4123:4123:4134:4134:2856:1833:1833:5411:5411:4123:6446:2351:2351:2856:2866:2866:7988:5411:5174:6446:6446:4665:4665:4665:7988:7988:4670:5174:5174:6446:6446:4665:7988:7988:4670:4670:5174:1865:1865:5195:5195:5195:2382:2382:4670:
Solution:
+-------+-------+-------+
| 6 8 2 | 3 1 7 | 9 5 4 |
| 3 9 5 | 4 6 2 | 7 1 8 |
| 7 4 1 | 9 5 8 | 6 2 3 |
+-------+-------+-------+
| 5 7 3 | 6 2 1 | 4 8 9 |
| 1 2 9 | 7 8 4 | 3 6 5 |
| 8 6 4 | 5 3 9 | 2 7 1 |
+-------+-------+-------+
| 4 3 8 | 2 7 5 | 1 9 6 |
| 9 5 7 | 1 4 6 | 8 3 2 |
| 2 1 6 | 8 9 3 | 5 4 7 |
+-------+-------+-------+
Quote:
Ruud, lead-in: It takes a lot of nibbling before you are able to place the first digit. Halfway there is another obstacle to remove
annette: toook me almost 2 hours
sudokuEd: I was greedy and got straight into the Mains - then got indigestion halfway through!
Andrew, in A19, A55 & 71 threads: A18 was the first one I found hard... That was the first one that I needed a hint
Oscar: SudokuEd Walk Through is really very good....I did turn my back to this one and had to restart a couple of times :oops: Arround 3 hours solving time I would rate it 1.25
Walkthrough by sudokuEd:
annette wrote:
Loved it 8-)

Me too.
Ruud wrote:
It takes a lot of nibbling before you are able to place the first digit. Halfway there is another obstacle to remove. This is a true Assassin. Enjoy!

I was greedy and got straight into the Mains - then got indigestion halfway through!

1. "45" on N9 -> 2 outies = 13 -> no 1, 2 or 3 in r6c8 or r8c6

2. "45" on N7 -> 2 outies = 7 -> no 7, 8 or 9 in r6c2 or r8c4

3. "45" on r789 -> 2 outies = 13 -> r6c28 = [67/58/49]

4. "45" on N8 -> 2 innies = 7 -> r8c46 = [16/25/34]

5. from step 3. r6c8 = {789}. "45" on N36 -> 3 innies = 12 -> r6c8 + r56c7 (with no 1 in r6c7) = [912/813/714/7{23}] -> r5c7 = {123}, r5c6 = {456}, r6c7 = {234}, r6c6 = {789}

6. Another way of expressing the information from step 5 is that r6c678 = [929/838/747/927/837]. Clearly, several of these have a problem! -> r6c8 = 7, r6c67 = [92/83] -> r6c6 = {89}, 2 innies of N36 now = 5 = {23} -> naked pair on {23} in c7 and N6 -> no 2 or 3 elsewhere in these. r5c6 = {45}

7.r8c46 = [16] (step 1, 4), r6c2 = 6 (step 3)

8. r5c34 = 16 = {79} -> no 7 or 9 elsewhere in r5 -> r5c34 = [97] (hidden single 7 in N5)

9. 21 (4) cage in N6 now {1569} only since 2,3 and 7 are blocked, with 9 only in c9 -> no 1,5,6 or 9 elsewhere in N6 and no 9 elsewhere in c9

10. r4c78 = {48} -> no 4 or 8 elsewhere in r4 or 20(4) cage in N36

11. "45" on N5 -> 2 outies = 1 innie. Since r56c7 = {23} = 5 -> r6c4 = 5, r6c3 = 4, r5c67 = [43], r6c67 = [92], r56c5 = 11 = [83], r6c19 = [81], r5c89 = {56} pair = 11 -> r4c9 = 9, r5c12 = {12} pair in N4 = 3 ->r4c1 = 5

12. r4c23 = {37} pair = 10 -> r3c23 = 5 = [41] (not {32} since {37} pair in same cage)

13. r4c78 = {48} = 12 -> r3c78 = 8 = [53/62]

14. 31(5) cage in N698 now = 18(3) = {189/459} = 9{18/45} = [5/8..] -> no 9 elsewhere in N9

15. 18(4) cage in N9 = {2367/2457/3456} (not {2358} blocked see step 14) -> no 8 in 18(4) cage

16. 8 in c9 now locked in N3 -> no 8 elsewhere in N3 (including r1c8)

17. "45" on r 9 -> 2 innies = 9 -> no 1 or 9 in r9c1, no 4 in r9c9

18. 25(5) cage in N478 now 18(3) cage = {279/378} = 7{29/38} -> no 5. No 7 elsewhere in N7 -> 7 in c1 locked in N1 in 24(4) cage -> no 7 elsewhere in N1 (including r1c2)

19. "45" on c9 -> 3 outies = 14. r5c8 = {56} -> r19c8 = 8/9. Min r8c8 = 2 -> max. r1c8 = 7 -> no 9 in r1c8

20. 9 in N3 locked in 17(3) cage = 9{17/26/35} -> no 4

21. 4 and 8(step 16) in N3 now locked in 20(4) cage = 4+8=12 -> other 2 cells= 8 = {17/26/35}

22. "45" on c1 -> 3 outies = 15. r5c2 = {12} -> r18c2 = 13/14= {58/59} = 5{8/9} -> no 5 elsewhere in c2 and no 2 or 3 in r18c2

23. r9c23 = 7 = [16/25] -> r9c2 = {12} -> naked pair on {12} with r5c2 -> no 1 or 2 elsewhere in c2

24. 4 in N7 locked in 20(4) cage = 4{169/259} (not{268} since [2/6] needed in 7(2) cage in N7 - (not {3458} since 5 and 8 in 20(4) cage both only in r8c2) -> 20(4) cage = 49{16/25} -> no 9 elsewhere in N7 and no 3 or 8 in 20(4) cage

25. 3 and 8 in N7 now locked in 18(3) cage = {378} only

26. 3 in c1 now only in N1 in 24(4) cage -> no 3 elsewhere in N1

27. 16(3) cage in N1 = {259/268} = 2{59/68} -> no 2 elsewhere in N1

28. naked triple on {589} in r128c2 -> no 8 in r7c2 -> 8 locked in r78c3 for c3 & r2c12 for N1

29. from step 24, the {1469} combination in the 20(4) cage in N7 is the only combination (in the 20(4) cage) that uses 1 or 6 with the 1 only available in r7c1 -> 6 must be in r9c1 in this combination -> no 6 in r7c1 ->r7c9 = 6 (hidden single r7), r5c89 = [65]

30. rest of 18(4) cage in N9 now 12(3) cage = {237/345} = 3{27/45} -> no 3 elsewhere in N9

31. "45" on c9 -> 2 outies = 8 -> r18c8 = {35} pair -> no 3 or 5 elsewhere in c8 -> r3c78 = 8 =[62]

32. 20(4) cage in N3 now {3458} with r1c8 = 5 and no 7.
the rest is basically hidden and naked singles

1Nov-edited - especially inaccuracies in step 24 - thanks Andrew
Special Killer X 5 Oct 06 (aka SKX1) by Ruud (Oct 06)
Puzzle pic: 1-9 cannot repeat on the diagonals:
Image
Code: Select, Copy & Paste into solver:
3x3:d:k:4352:4352:3586:3586:7684:1285:1285:3335:3335:4352:2570:2570:7684:7684:7684:5647:5647:3335:1554:2570:5396:5396:7684:3351:3351:5647:3610:1554:6684:5396:5396:7967:3351:3351:6178:3610:6684:6684:6684:7967:7967:7967:6178:6178:6178:3885:6684:4655:4655:7967:5938:5938:6178:821:3885:2871:4655:4655:4154:5938:5938:4669:821:3903:2871:2871:4154:4154:4154:4669:4669:3655:3903:3903:2378:2378:4154:4429:4429:3655:3655:
Solution:
+-------+-------+-------+
| 8 6 5 | 9 7 1 | 4 2 3 |
| 3 2 1 | 4 6 5 | 7 9 8 |
| 4 7 9 | 3 8 2 | 1 6 5 |
+-------+-------+-------+
| 2 5 8 | 1 3 4 | 6 7 9 |
| 1 9 7 | 8 5 6 | 2 3 4 |
| 6 4 3 | 2 9 7 | 5 8 1 |
+-------+-------+-------+
| 9 1 6 | 7 4 8 | 3 5 2 |
| 5 8 2 | 6 1 3 | 9 4 7 |
| 7 3 4 | 5 2 9 | 8 1 6 |
+-------+-------+-------+
Quote:
Nasenbaer: Took me really long to get to the first number, but from then on it unfolded rapidly
sudokuEd: If that is your first ever walk-through Nasenbaer, it is very impressive
Andrew, on SKX4 thread: Assassin level
Walkthrough by Nasenbaer:
That was a good one. Took me really long to get to the first number, but from then on it unfolded rapidly.

I used the diagonals in the "endgame", so maybe they aren't really necessary, I don't know. Has anyone tried it without them?

Walkthrough KillerX5oct06

1. 14(2) in r1 : {59/68} -> no 1,2,3,4,7
2. 5(2) in r1: {14/23} -> no 5,6,7,8,9
3. 6(2) in c1: {15/24} -> no 3,6,7,8,9
4. 15(2) in c1: {69/78} -> no 1,2,3,4,5
5. 14(2) in c9: {59/68} -> no 1,2,3,4,7
6. 3(2) in c9: {12} -> 1,2 locked in c9 -> nowhere else
7. 17(2) in r9: {89} -> 8,9 locked in r9 -> nowhere else
8. 16(5) in n8: {12346} -> 1,2,3,4,6 locked in n8 -> nowhere else
8a. -> r7c4 = {5789}
8b. -> r7c6 = {5789}
9. 9(2) in r9: using steps 8 and 7: r9c3={24}, r9c4={57}
10. "45" on n7 -> r7c13 + r9c3 = 19 -> {289/469/478} -> r7c3 = {6789}
11. 11(3) in n7 : no 9
12. 18(3) in n9 : at most one of 8 or 9 (one used for step 7)
-> {189} not possible -> no 1
13. "45" on n9 -> r7c79 + r9c7 = 13 -> {148/139/238} -> r7c7 = {34}
14. 18(4) in n4578 : min. r7c34 = 11 = {56}
-> max. r6c34 = 7 = {1234} (5 and 6 not possible)
15. "45" on r89 : r7c258 = 10 = {127/136/145/235} -> no 8,9
16. "45" on n3 : r3c79 + r1c7 = 10 = {127/136/145/235} -> no 8,9
16a. -> {127} not possible because r3c9 = {56} -> no 7
16b. -> r13c7 = {1234} (step 16a) -> no 5,6
17. r4c9 = {89} (because r3c9 = {56})
18. 22(3) in n3 : {589/679} -> 9 locked in 22(3), nowhere else
19. 22(3) in n3 has either 5 or 6, r3c9 = {56} -> 5,6 nowhere else in n3
20. 13(3) in n3 : {148/238/247} -> r1c8 = {12}
21. "45" on c89 -> r258c7 = 18 -> at most one of 8 or 9 (one used for step 7)
-> no 1
22. "45" on n6 : r46c79 = 21 -> because of {12} in r6c9 and {89} in r4c9 it
can have only one of {12} -> no 1,2 in r46c7
23. 1 locked in r13c7 for c7 -> r1c8 = 2
24. 5(2) in r1 = {14} -> 1,4 locked in r1
25. 2 locked in 18(3) in c7 (step 21) -> only {279} possible, 2,7,9 locked
in c7, nowhere else -> r9c7 = 8, r9c6 = 9
26. naked triple {134} in c7 -> 3,4 removed from r46c7 -> r46c7 = {56}
-> 5,6 locked in n6
27. 14(3) in n9 has to have 5 or 6 (from c9) -> no 9 possible
-> 9 locked in 18(3) in n9, locked in r8 -> 9 locked in r7c13 for r7
28. 19(3) in n7 (step 10) has 9 -> {478} not possible -> no 7
28a. 8 removed from r6c1
29. "45" on r6789 : r6c258 = 21 = {489/579/678} -> no 1,2,3

EDIT:
Old 30. 14(3) in n9 (continued from step 27) : {167/257/356} -> no 4
-> no 7 in r9c8
New 30. 14(3) in n9 (continued from step 27) : {167/356} -> no 4
-> no 7 in r9c8 (Thanks, Andrew)

31. 18(3) in n9 : {279/369/459} -> no 7 in r8c78
32. 23(4) in n5689 : max. (r7c67 + rr6c7) = 8+4+6 = 18 -> min. r6c6 = 5
-> no 1,2,3,4 in r6c6 -> 23(4) = {3578/4568}
33. 13(4) in n2356 : {1246/1345} -> {56} in r4c7 -> r34c6 = {1234}
34. "45" on n5 : r46c46 = 14 -> no 9 -> 9 locked in 31(5) in n5
34a. r6c6 = {5678} (step 32) -> no 7,8 in r4c4
35. 8 locked in r7c46 for r7 -> 8 removed from r7c13 -> 6,9 locked for r7 and n7
36. 19(3) in n7 (step 10) has 6 and 9 -> r9c3 = 4, r9c4 = 5
37. 7 locked in r7c46 for r7 -> nowhere else

EDIT: SudokuEd just pointed out that step 38 is flawed because the elimination of 5,6 is not logical. I couldn't find how I did it, so I will insert a new step (n38). I will leave the old version, maybe someone can tell me how I did it. :-)

n38: hidden triple {123} in r6c349 -> 21(3) in r6 (step 29) has to have 4 -> {489} -> r6c1 = 6 -> r6c7 = 5 -> r6c6 = 7 -> r7c6 = 8 (EDIT: last part added, thanks Andrew)


Old version:
38. from step 32: 5,6 removed from r6c6 (locked in r67c7) -> r6c6 = {78}
38a. naked pair {78} in c6 -> nowhere else in c6
38b. 14(4) in n5 (step 34) : {1238/1247} -> no 1,2 in 31(5) in n5

39. "45" on c6789 -> r258c6 = 14 = {356} -> 3,5,6 locked in c6
39a. 2 locked in r34c6 for c6 and 13(4) in n2356 -> 13(4) = {1246}
-> r4c7 = 6 -> r6c7 = 5 -> r7c7 = 3 (step 32)

Now everything unfolds rapidly.

40. 14(3) in n9 : only {167} left -> r9c8 = 1, r89c9 = {67}
41. r7c9 = 2, r6c9 = 1, r3c9 = 5, r4c9 = 9
42. r8c7 = 9, r78c8 = {45}
43. r2c7 = 7, r5c7 = 2, r23c8 = {69}, r12c9 = {38} -> r5c9 = 4 (EDIT: error in r5c7 corrected, thanks Andrew)
44. 2,3 locked in r6c34 for r6 and 18(4) -> r7c3 = 6, r7c4 = 8
45. r2c8 = 9 (through D/), r3c8 = 6
46. r7c1 = 9, r6c1 = 6, r7c6 = 8, r6c6 = 7
47. r9c9 = 6 (through D\), r8c9 = 7, r6c8 = 8
48. 15(3) has to have 7 -> {357} -> r8c1 = 5, r9c12 = {37} -> r9c5 = 2
49. "45" on c12 -> r257c3 = 10 -> no 8,9
50. r7c2 = 1 -> r8c3 = 2, r8c2 = 8

Now everything should be clear.

I hope everything is clear although my use of brackets might not be fully correct (sorry for that).

Have fun!
Walkthrough by Andrew:
sudokuEd wrote:
So PLEASE everyone - tell Ruud how much you enjoyed the Special Killer X so that I can have one of these regularly

I'll add my support to that. I came late to SKX1 and only solved it in December. I've only just found time to check through my walkthrough which I'm posting now because it has some interesting moves in r78. I also tried to make use of the diagonals as early as I could although I didn't find any clever moves using them, only eliminations.

Many thanks to Ed for reviewing my walkthrough. I've added his comments to steps 22, 27 and 28 which provide useful insight into quicker routes to a solution and included his addition to step 24.

I'm also posting my walkthrough for SKX3 today.


1. R9C67 = {89}, locked for R9

2. 45 rule on N8 4 innies = 29 = {5789} -> R7C46 = {5789}, R9C34 = [27/45]

3. 16(5) cage in N8 = {12346}

4. 31(5) cage in N5 contains 9, locked for N5

5. 22(3) cage in N3 = 9{58/67}, 9 locked for N3

6. 13(4) cage in N2356 contains 1, min of any 3 cells = 6 so max remaining cell 7, no 8,9

7. R67C9 = {12}, locked for C9

8. 10(3) cage in N1, no 8,9

9. 11(3) cage in N7, no 9

10. 45 rule on N9 3 innies = 13 -> R7C7 = {34} (cannot be 2 which would clash with 2,9 in the other two innies)

11. 45 rule on N7 3 innies = 19 -> R7C13 = 15 or 17 = {69/78} or {89} [8/9], R7C46 = [5/7,8/9 because of R9C4 and R9C6 respectively], killer pair 8/9 in R7C1346 for R7

12. R67C1 = {69/78}

13. R7C34 min 11 -> R6C34 max 7, no 7,8,9 in R6C34, R6C34 min 4 (cannot be {12} because of R6C9) -> R7C34 max 14 -> R7C4 = {578}, 9 in N8 locked in R79C6, locked for C6

14. R34C9 = [59]/{68}, 45 rule on N3 3 innies = 10 -> R34C9 = [59/68], R1C67 = {14/23} -> R3C7 = {1234}

15. 22(3) cage in N3 = 9{58/67}, killer pair 5/6 with R3C9 for N3

16. 13(3) cage in N3 must contain 7 or 8 -> {247} or 8{14/23} -> R1C8 = {12}

17. R13C7 = {13} or {14/23}, R7C7 = {34}, killer pair 3/4 in R137C7 for C7

18. R1C67 = {14/23}, R1C8 = {12}, killer pair 1/2 in R1C678 for R1

19. R34C1 = {15/24}

20. R1C34 = {59/68}

21. 45 rule on N1 2 outies R1C4 + R4C1 – 2 = 1 innie R3C3, min R1C4 + R4C1 = 6 -> min R3C3 = 4

22. 45 rule on N6 4 innies = 21, max R4C79 + R6C9 = 18 -> min R6C7 = 3, min R6C7 actually 5 because 3,4 blocked in C7, max R4C9 + R6C79 = 19 -> min R4C7 = 2 but cannot have two 2s in N6 so min R4C7 = 5
[Ed. FYI - can go further here. combo's = [91{56}/81{57}/82{56}] -> R6C7 = {567} and 5 locked for C7, N6 and the 4 innies = [8/9, 6/7..] -> 24(5) = 34{179/269/278} ({12489/12678}blocked by the 4 innies)]

23. 24(5) cage in N6 contains 1 or 2

24. 13(4) cage in N2356, min R4C7 + any two others = 8, max R3C6 or R4C6 = 5
but this would mean two 5s in 13(4) since min R4C7 = 5 -> max R3C6 or R4C6 = 4 [Thanks Ed. I’d missed that last point]

25. 45 rule on C9 2 outies R19C8 + 1 = 1 innie R5C9 -> min R5C9 = 4

26. 45 rule on R9 2 outies R8C19 – 10 = 1 innie R9C5 -> min R8C19 = 11, no 1 in R8C1

27. 23(4) cage in N5689, max R6C7 + R7C67 = 21 -> min R6C6 = 2
[If Ed’s mods to step 22 had been used, this max = {479} = 20 -> min R6C6 = 3 = {3479/3569/3578/4568} = [8/9..] in C6 -> killer pair with R9C6]

28. R9C7 = {89} so R8C789 must have 8 or 9 -> R8C123 must have 8 or 9 -> R7C13 = {69/78}(can’t be {89} which would clash with R8C123) -> R9C3 = 4 (step 11), R9C4 = 5, R1C34 = [59]/{68}
[Ed. One thing you missed from this is that R7C4 = {78} -> R7C13 {78} is blocked -> = {69} only in R7C13. You pick this up in the next couple of steps so no problem]

29. R7C1346 = naked quad {6789}, 6,7 locked for R7

30. 7 in N8 locked in R7C46, locked for R7 -> R7C13 = {69}, locked for R7 and N7, R6C1 = {69}, locked for C1

31. R7C46 = {78} -> R9C6 = 9, R9C7 = 8

32. No 8 in R8C1 because R9C12 cannot total 7 -> 8 in N7 locked in R8C23 -> 11(3) cage = {128}, R7C2 = {12}, R8C23 = 8{12}, R9C12 = {37}, locked for R9, R8C1 = 5

33. R9C9 = 6 (naked single), locked for D\, R3C9 = 5, R4C9 = 9, R13C7 = {14/23} (from step 14), R34C1 = {24}, locked for C1

34. 22(3) cage in N3 = {679}, locked for N3, 8 in N3 locked in R12C9, locked for C9

35. Only valid combination for 14(3) cage in N9 = [716] -> R67C9 = [12], R9C5 = 2

36. R5C9 = 4, R12C9 = {38}, R1C8 = 2

37. R1C67 = {14}, R3C7 = {14}, locked for R1 and C7

38. R5C7 = 2 (hidden single in C7 and N6)

39. R7C7 = 3 (naked single), R7C8 = 5 (hidden single in R7 and N9) -> R8C78 = [94], 3,4 locked for D\

40. R7C5 = 4 (hidden single in N8) -> R8C456 = {136}, R8C23 = {28}, R7C2 = 1

41. 24(5) cage in N6, 3 remaining cells R456C8 = 18 = {378}(subtraction combo), locked for C8 and N6 -> R23C8 = {69}, R2C7 = 7, R46C7 = {56}

42. 45 rule on N1 3 innies R1C3 + R3C13 = 18, no valid combination with R3C1 = 2 -> R3C1 = 4, R13C3 = [59]/[68], R1C34 = [59]/[68], R4C1 = 2

43. R3C7 = 1, locked for D/ and 13(4) cage, R1C7 = 4, R1C6 = 1

44. R4C4 = 1 (hidden single on D\), R8C5 = 1 (hidden single in R8, C5 and N8)

45. 2 in N1 locked in 10(3) cage = 2{17/35}, no 6

46. 2 on D\ can only be in R2C2 or R6C6 -> no 2 in R2C6

47. R5C5 = 5 (hidden single on D/), locked for D\

48. R2C2 = 2 (naked single), R8C23 = [82], 2 locked for D\, 8 locked for D/

49. R2C8 + R7C3 = {69}, killer pair on D/

50. R9C1 = 7 (hidden single on D/), R9C2 = 3

and the rest is naked and hidden singles and simple elimination remembering to use the diagonals and cage sums
Assassin 19 by Ruud (Oct 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:2560:7682:7682:7682:7682:7682:3591:3591:2560:4618:4618:2572:1037:3598:5647:5647:3591:1810:4618:2580:2572:1037:3598:1816:5647:3354:1810:7452:2580:2572:7199:3598:1816:6434:3354:7452:7452:7452:7199:7199:7199:6434:6434:6434:3629:7452:1071:3632:7199:3634:2867:6434:2357:3629:3127:1071:3632:3130:3634:2867:4413:2357:4927:3127:3127:3632:3130:3634:4413:4413:1607:4927:4927:8266:8266:8266:8266:8266:1607:1607:
Solution:
+-------+-------+-------+
| 3 6 7 | 8 9 4 | 2 1 5 |
| 1 5 4 | 2 3 6 | 7 9 8 |
| 2 9 8 | 5 1 7 | 3 6 4 |
+-------+-------+-------+
| 5 7 2 | 3 6 1 | 4 8 9 |
| 4 8 9 | 7 2 5 | 1 3 6 |
| 6 1 3 | 4 8 9 | 5 7 2 |
+-------+-------+-------+
| 8 2 1 | 9 5 3 | 6 4 7 |
| 9 4 6 | 1 7 2 | 8 5 3 |
| 7 3 5 | 6 4 8 | 9 2 1 |
+-------+-------+-------+
Quote:
Andrew: It was good to have a slightly easier puzzle this time but definitely still an Assassin
Ruud: You need almost every trick in the book to break this Assassin. When it finally surrenders, your victory will be total
Andrew: At first I doubted that was true but... I'm now convinced! I'll stick with my "slightly easier" comment; most steps managed to fix at least one cell
Oscar: After the warning by Ruud... do not overlook any detail or opportunity, and ... then it's ok :) . I rate it 1.0
Walkthrough by Andrew:
It was good to have a slightly easier puzzle this time but definitely still an Assassin. I felt I had been 'hitting my head against a brick wall' with #18, which Annette and sudokuEd clearly found a lot easier than I did, and with #17v2.

Since nobody else has posted one, here is my walkthrough. As with my previous ones, it's effectively the way I solved it and some sub-steps are included for completeness of an area before I moved on, even though they may not be used immediately.

Step 1
6(3) cage in N9 = {123}, 45 rule on R9 2 outies = 12 -> R8C19 = [93], R9C89 = {12}, R9C12 = {37/46}, 32(5) cage in R9 = 589{37/46}

Step 2
45 rule on N9 3 innies R7C79 + R9C7 = 22 = 9{58/67} with the 9 in C7, no other 9 in C7, 4 in N9 locked in 17(3) cage

Step 3
R67C1 = {68}, R67C3 = {13}, 45 rule on N7 3 innies R7C13 + R9C3 = 14, R7C13 = 7 or 9 (cannot be 11 because R9C3 = 3 would then clash with R67C3) -> R9C3 = {57}, 12(3) cage in N7 = 2{37/46} -> R9C3 = 5, R67C1 = [68], R67C3 = [31]

Step 4
R34C1 = {25}/[34], R34C3 = {28/46}, 45 rule on N4 2 remaining innies R4C13 = 7, R4C1 must be an odd number since R4C3 is an even number -> R4C1 = 5, R4C3 = 2, R3C1 = 2, R3C3 = 8, 29(5) cage in N4 = {14789}, 2 in N7 locked in R78C2

Step 5
45 rule on C12 3 outies R258C3 = 19, R5C3 = {479}, R8C3 = {467}, only valid combination is {469} -> R1C3 = 7, R2C3 = {469}, R5C3 = {49}, R8C3 = {46}, R78C2 = 2{46}, R9C12 = {37}, 32(5) cage in R9 = {45689} with the 4 locked in N8 -> R78C5 = {57}

Step 6
18(3) cage in N1 = 9{36/45}, 10(3) cage in N1 = 1{36/45}, cannot have R12C1 = {36} or {45} so R12C1 = 1{34}, R1C2 = {56}, 1 in N4 locked in C2, R5C1 = {47}, 45 rule on C1 2outies – 1 innie = 5 -> R19C2 = 9 or 12 -> R19C2 = [57]/[63]

Step 7
7a. 45 rule on N2 3 remaining outies = 6 -> R1C7 + R4C46 = {114}/{123} (the first combination is possible because the outies are in 2 different rows and 3 different columns) -> R4C46 = {134}, R1C7 = {12}
7b. 45 rule on N3 3 innies = 9 = {126}/{135}/{234} -> R3C9 = {456}, R3C7 = {13} (cannot have 4 because {144} not allowed and {234} would clash with the 3 in C9, cannot have 5 because there is a 2 in R4), R34C7 = [16]/[34]
7c. R1C7 cannot be 1 because that would require [34] in R34C7 and {14} in R4C46 (two 4s in R4) -> R1C7 = 2, R4C46 = {13}
7d. R1C456 = {489}, R23C6 = 11 min and cannot be {29} -> R2C4 = 2 (hidden single in N2), R3C4 = {57}, R23C6 = 6{57}, 2 in N8 locked in C6, 2 in N5 locked in C5
[Step 7 edited. Thanks to sudokuEd for pointing out the flaw in my original logic, see step 7a above, and for providing steps 7b and 7c which saved me having to think that out for myself]

Step 8
R34C7 = [16]/[34], R23C5 = {13} so R3C57 = {13}, no other 1/3 in R3, 45 rule on N5 2 remaining innies = 13 -> R6C46 = {49/58}, 28(5) cage in N5 = 267{49/58}, R4C46 = [31] (R4C6 hidden single 1 in C6 because cannot have both 1 and 2 in R78C6) -> R3C4 = 5, R23C6 = {67}, R78C6 = [32] (hidden single 3 in N8), R6C46 = [49], R78C4 = [91], R9C7 = 9, 28(5) cage in N5 = {25678} -> R5C6 = 5 (hidden single in C6), R456C5 = {268}, R5C4 = 7, R5C1 = 4, R5C3 = 9, R456C2 = {178}, R9C12 = [73], R12C1 = {13}, R1C2 = 6, R78C2 = {24}, R8C3 = 6, R2C3 = 4, R23C2 = [59]

Step 9
5 in N3 locked in R1, 45 rule on R1 2 outies = 9 -> R2C9 = {68}, 9 in N3 locked in 22 (3) cage -> {679} -> R2C8 = 9, R2C9 = 8, R12C1 = [31], R3C9 = 4 (hidden single in N3), R4C9 = 9, R1C89 = {15}, R3C7 = 3, R4C7 = 4, R23C5 = [31]

Step 10
R67C9 = {18/27}, 45 rule on N6 2 remaining innies = 7 -> R6C79 = [52], R7C79 = [67], 25(5) cage in N6 = {13678}, R5C8 = 3 (hidden single in N6), R9C89 = [21], R1C89 = [15], R5C9 = 6, 17(3) cage in N9 = {458} -> R8C7 = 8 and carry on … with simple elimination

Ruud wrote:
You need almost every trick in the book to break this Assassin. When it finally surrenders, your victory will be total.

At first I doubted that was true but, after the changes to step 7, I'm now convinced! I'll stick with my "slightly easier" comment; most steps managed to fix at least one cell.

Andrew


Last edited by Ed on Sat Jun 14, 2008 1:01 am, edited 2 times in total.

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PostPosted: Sat Jun 07, 2008 12:12 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
Assassin 20 by Ruud (Oct 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6656:6656:1026:1026:5124:2053:2053:5383:5383:6656:6656:2571:6156:5124:5646:1551:5383:5383:2322:2322:2571:6156:5124:5646:1551:4377:4377:5659:2322:6156:6156:5124:5646:5646:4377:6179:5659:5659:5659:2343:4392:1577:6179:6179:6179:2861:6446:6446:2343:4392:1577:5171:5171:2613:2861:4151:6446:6446:4392:5171:5171:1341:2613:2861:4151:2113:3906:3906:3906:3909:1341:2613:1864:1864:2113:3403:3403:3403:3909:3919:3919:
Solution:
+-------+-------+-------+
| 9 5 1 | 3 7 2 | 6 4 8 |
| 4 8 3 | 6 1 9 | 5 7 2 |
| 6 2 7 | 5 4 8 | 1 9 3 |
+-------+-------+-------+
| 7 1 9 | 4 8 3 | 2 5 6 |
| 5 6 4 | 2 9 1 | 3 8 7 |
| 2 3 8 | 7 6 5 | 9 1 4 |
+-------+-------+-------+
| 8 7 5 | 9 2 6 | 4 3 1 |
| 1 9 6 | 8 3 4 | 7 2 5 |
| 3 4 2 | 1 5 7 | 8 6 9 |
+-------+-------+-------+
Quote:
sudokuEd: Nasenbaer obviously found Assassin 20 way too easy (BTW -great walk-through)
Oscar: I would rate it 1.0
Walkthrough by Nasenbaer:
SudokuEd convinced me that with tiny font there are no spoilers, so here it is:

Walkthrough Assassin 20
EDIT: Thanks to sudokuEd and Andrew for the additional input and corrections!

1. r1 : 4(2) = {13} -> 1,3 locked in r1
2. -> (step 1) r1 : 8(2) = {26} -> 2,6 locked in r1
3. "45" on n1 (2 outies) : r1c4 + r4c2 = 4 -> r4c2 = {13}
4. "45" on n3 (2 outies) : r1c6 + r4c8 = 7 -> r4c8 = {15}
5. "45" on r1234 (2 innies) : r4c19 = 13 = {49}/{58}/{67} -> no 1,2,3
6. "45" on r12345 (3 innies) : r5c456 = 12
7. c7 : 15(2) = {69}/{78} -> no 1,2,3,4,5
8. r9 : 15(2) = {69}/{78} -> no 1,2,3,4,5
9. n9 : 5(2) = {14}/{23} -> no 5,6,7,8,9
10. n7 : 16(2) = {79} -> 7,9 locked in c2 and in n7
11. -> n7 : 8(2) = {26}/{35}
12. "45" on r9 -> r9c37 = 10
12a. -> no 5,6 in r9c3 -> r9c3 = {23} -> no 2,3 in r8c3 -> r8c3 = {56}
12b. -> no 6,9 in r9c7 -> r9c7 = {78} -> no 6,9 in r8c7 -> r8c7 = {78} -> 7,8 locked in c7 and n9
13. -> r9 : 15(2) = {69} -> 6,9 locked in r9 and n9
14. n7 : 7(2) = {34} ({16} not possible because of locked 6 in step 13, {25} not possible because 8(2) in n7 has to have either a 2 or a 5 (step 11)) -> 3,4 locked in r9 and n7
15. -> r9c3 = 2 -> r8c3 = 6
16. -> r9c7 = 8 -> r8c7 = 7
17. -> r8c2 = 9 -> r7c2 = 7
18. r9 : 13(3) = {157} -> 1,5,7 locked in n8
19. "45" on c5 : r89c5 = 8 = {35} (only possible combination left from step 18) -> r9c5 = 5 -> r8c5 = 3
20. -> no 3 in r8c8 -> no 2 in r7c8
21. c1 : look at 11(3) : r78c1 = {18}/{15} = 1{5/8} (leftover from n7)
21a. -> {15} not possible because it would require 5 in r6c1, but already used in 11(3)
21b. -> r78c1 = {18} -> r6c1 = 2
21c. -> no 8 in r4c1 -> no 5 in r4c9 (step 5)
22. -> r7c3 = 5
23. -> r8c9 = 5 -> r67c9 = {14}/{23} (or should it read {[2]3} because 2 is not possible in r6c9 -> no 3 in r7c9 ?)
24. r8 : 15(3) = {348} (only possible combination left)
24a. -> r8c1 = 1 -> r7c1 = 8
24b. -> r8c8 = 2 -> r7c8 = 3
25. -> r67c9 = {14} -> 1,4 locked in c9 (who cares now about my rambling in step 23? :-) )
26. -> no 4 in r4c9 -> no 9 in r4c1 (step 5)
27. r7c7 = {14} (leftover from n9) -> 6(2) in c7 has either 1 or 4 -> 1,4 locked in c7
28. 25(4) in n478 : possible combinations: {3589}/{4579} -> no 1,2,6 -> r7c4 = 9
28a. -> r6c23 = 11 = {38}/{4[7]}
29. "45" on n4 -> r4c23 = 10 -> r4c3 = {79}
30. c3 : 10(2) = {19}/{37}
30a. -> has to have 1 or 3 -> because of r1c3 = {13} 1,3 is locked for c3 -> no 3 in r6c3 -> no 8 in r6c2
30b. -> has to have 7 or 9 -> because of r4c3 = {79} 7,9 is locked for c3 -> no 7 in r6c3 -> r6c3 = 8 -> r6c2 = 3 -> r5c3 = 4
31. -> r9c2 = 4 -> r9c1 = 3
32. -> r4c2 = 1 -> r4c3 = 9 -> r5c3 = 4 -> no 4 or 9 in r4c19 (step 5)
33. -> r1c4 = 3 (step 3) -> r1c3 = 1
34. n1 : 10(2) = {37} -> 3,7 locked in c3 and n1
35. r4c8 = 5 -> r1c6 = 2 (step 4) -> r1c7 = 6
36. no 5 in r4c1 -> no 8 in r4c9 (step 5)
37. c6 : 6(2) = {15} (no 2 -> no 4)
38. r7c6 = 6 -> r7c5 = 2
39. 9(3) in n14 : {126} only possible combination left -> r3c1 = 6 -> r3c2 = 2
39a. no 2 in r3c7 -> no 4 in r2c7
40. r4c1 = 7 -> r4c9 = 6 ->r9c9 = 9 -> r9c8 = 6
41. r5c2 = 6 -> r5c1 = 5
42. r5c6 = 1 -> r6c6 = 5
43. r9c6 = 7 -> r9c4 = 1
44. c4 : 9(2) = {27} (only possible combination left) -> r6c4 = 7 -> r5c4 = 2
45. r5c5 = 9 (from step 6) -> r6c5 = 6 (because of 17(3))
46. r6c7 = 9 (because 1,4 locked in c7 from step 27) -> r6c8 = {14}
47. r4c6 = 3
48. 24 in n245 : {4569} only possible combination left -> r4c4 = 4 -> r3c4 = 5 -> r2c4 = 6
49. r8c4 = 8 -> r8c6 = 4
50. r4c5 = 8 -> r4c7 = 2 -> no 2 in r2c7 -> no 4 in r3c7
51. -> r3c7 = 1 -> r2c7 = 5
52. r7c7 = 4 -> r7c9 = 1 -> r6c9 = 4 -> r6c8 = 1
53. r5c7 = 3 -> r5c89 = {78}
54. c2 : r12c2 = {58} -> r1c2 = 5 -> r2c2 = 8
55. c6 : r23c6 = {89} -> r2c6 = 9 -> r3c6 = 8
56. 17(3) in n36 : {359} only possible combination left -> r3c8 = 9 -> r3c9 = 3
57. r3c3 = 7 -> r2c3 = 3
58. r1c2 = 4 -> r1c1 = 9
59. r3c5 = 4 -> r2c5 = 1 -> r1c5 = 7
60. r1c8 = 4 -> r1c9 = 8
61. r5c9 = 7 -> r5c8 = 8
62. r2c8 = 7 -> r2c9 = 2


Well, if there is still an empty cell then you missed a step. :-)

Please feel free to comment my steps. I know there are very obvious steps included, but it helps me not to overlook anything.

Have fun!
Walkthrough by Andrew:
I've just looked through Peter's walkthrough for Ruud's original version of this puzzle. Nice walkthough.

A lot of it is the same as the way that I solved it but there were some differences that may be of interest so here is my walkthrough. I think I missed seeing a few of Peter's steps and it's possible that he may also have missed some of mine.

Step 1
R1C34 = {13}, 45 rule on N1 2 outies = 4 -> R4C2 = {13}, R1C67 = {26}

Step 2
R9C89 = {69/78}, R89C7 = {69/78}, no other 6, 7, 8 or 9 in N9, R78C2 = {79}, R89C3 = {26/35}, 45 rule on R9 2 innies R9C37 = 10 -> R9C3 = {23}, R9C7 = {78}, R8C3 = {56}, R8C7 = {78}, R9C89 = {69}, R9C12 = {25/34}, killer pair 2/3 in R9C123, no other 2/3 in R9 or N7, 1 in R9 locked in N8 -> 13(3) cage = 1{48/57}

Step 3
45 rule on N7 1 innie – 1 outie = 3, 8 in N7 is in R78C1 or in R7C3
8 cannot be in R7C3 as follows. If R7C3 = 8, R6C1 = 5, R78C1 = {24} which is not possible as these clash with {25/34} in R9C12
-> 8 in R78C1, other two cells of R678C1 = {12}, R7C3 = {45}, R78C1 = {18} (hidden single 1 in N7), R6C1 = 2, R7C3 = 5, R89C3 = [62], R9C12 = {34}, R9C456 = {157}, R89C7 = [78], R78C2 = [79]

Step 4
15(3) cage in N8 must be {348} -> R78C1 = [81], R78C8 = [32], R8C9 = 5 (hidden single in N9), R7C79 = {14}, R6C9 = {14}, R7C456 = {269}

Step 5
R8C456 = {348} from step 4 and R9C456 = {157} from step 3
45 rule on C56789 2 outies R89C4 = 9
45 rule on C12345 2 outies R89C6 = 11 -> R89C5 = 8
R89C6 = 11 -> [47], R89C5 = 8 -> [35] -> R89C4 = [81], R1C34 = [13], R4C2 = 1, R56C6 = {15}, R4C6 = 3 (hidden single in N5), R56C4 = [27]
45 rule on N3 1 innie – 1 outie = 1 R1C7 = {26} (step 1) -> R4C8 = {15} but R4C2 = 1 -> R4C8 = 5, R1C7 = 6, R1C6 = 2, R7C5 = 2, R56C5 = {69}, R4C45 = [48]

Step 6
R23C3 = {37}, R3C12 = [62], 26(4) cage in N1 = {4589}, 7 in N4 locked in R45C1

Step 7
45 rule on C4 1 innie = 1 outie R4C3 = R7C4, R2347C4 = {4569}, R7C4 = R4C3 = 9 (4 and 5 in R7, 6 in C3), R34C4 = [65], R7C6 = 6, R6C23 = [38], R9C12 = [34], R12C2 = {58}, R12C1 = {49}
22(4) cage in N4 = {4567} -> R5C3 = 4, R5C2 = 6, R45C1 = [75], R56C6 = [15]
45 rule on R1234 2 innies R4C19 = 13 -> R4C9 = 6, R4C7 = 2, R23C6 = {89}

Step 8
R56C5 = [96], R9C89 = [69], R123C5 = {147}, R23C7 = [51], R5C7 = 3 (hidden single in C7), R5C89 = {78}, R7C7 = 4, R7C9 = 1, R6C9 = 4, R6C8 = 1 (hidden single in N5), R6C7 = 9

Step 10
R12C2 = [58], R23C6 = [98], R3C89 = [93], 21(4) cage in N3 = {2478} and carry on … with simple elimination

Andrew
Assassin 20 V2 by sudokuEd (Oct 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:6656:6656:1026:1026:5124:2053:2053:5383:5383:6656:6656:2571:6156:5124:5646:1551:5383:5383:2322:2322:2571:6156:5124:5646:1551:4377:4377:5659:2322:6156:6156:5124:5646:5646:4377:6179:5659:5659:5659:2343:4392:1577:6179:6179:6179:2861:6446:6446:2343:4392:1577:5171:5171:2613:2861:4151:6446:6446:4392:5171:5171:1341:2613:2861:4151:2113:7234:7234:7234:3909:1341:2613:1864:1864:2113:7234:7234:7234:3909:3919:3919:
Solution:
+-------+-------+-------+
| 9 5 1 | 3 7 2 | 6 4 8 |
| 4 8 3 | 6 1 9 | 5 7 2 |
| 6 2 7 | 5 4 8 | 1 9 3 |
+-------+-------+-------+
| 7 1 9 | 4 8 3 | 2 5 6 |
| 5 6 4 | 2 9 1 | 3 8 7 |
| 2 3 8 | 7 6 5 | 9 1 4 |
+-------+-------+-------+
| 8 7 5 | 9 2 6 | 4 3 1 |
| 1 9 6 | 8 3 4 | 7 2 5 |
| 3 4 2 | 1 5 7 | 8 6 9 |
+-------+-------+-------+
Quote:
Nasenbaer: Version 2 is MUCH harder. It took me ages to place the first number, even with parts of the solution still in my memory
Ed, Archive lead-in: Everyone, please feel free to do all these early puzzles
Andrew in June '08: OK, here's the first one! :) I had another go at A20 V2. When it first appeared I was still using insertion solving and didn't manage to finish it. I started again, this time using elimination solving. I'll rate A20 V2 as 1.25
Walkthrough by sudokuEd:
Nasenbaer wrote:
do you want to torture me???

Glad you survived the torture chamber :twisted:

And thankyou Ruud for setting V2.

In this tip-toe-through, I've included Nasenbaer's first 10 steps and then let you find your own way back into his steps after the initial placements.

1. r1 : 4(2) = {13} -> 1,3 locked in r1
2. -> (step 1) r1 : 8(2) = {26} -> 2,6 locked in r1
3. "45" on n1 (2 outies) : r1c4 + r4c2 = 4 -> r4c2 = {13}
4. "45" on n3 (2 outies) : r1c6 + r4c8 = 7 -> r4c8 = {15}
5. "45" on r1234 (2 innies) : r4c19 = 13 = {49}/{58}/{67} -> no 1,2,3
6. "45" on r12345 (3 innies) : r5c456 = 12
7. c7 : 15(2) = {69}/{78} -> no 1,2,3,4,5
8. r9 : 15(2) = {69}/{78} -> no 1,2,3,4,5
9. n9 : 5(2) = {14}/{23} -> no 5,6,7,8,9
10. n7 : 16(2) = {79} -> 7,9 locked in c2 and in n7

11. -> n7 : 8(2) = {26}/{35} = [2/5, 3/6 ...]-> no 1
12. 7(2) cage in N7 = {16/34} (not {25} step 11)-> no 2 or 5 -> 7(2) cage = [3/6...]
13. Killer pair on 3 and 6 in N7 in 8(2) and 7(2) cages -> no 3 or 6 elsewhere in N7
14. "45" on N7 -> 1 outie + 3 = 1 innie -> min. r7c3 = 4 -> no 1 or 2. r7c3 = {458} -> r6c1 = {125}
15. 11(3) cage in N47 now {128/245} = 2{18/45} -> 2 in c1 locked in -> no 4 or 6 in r3c2
16. In N9 the two 15(2) cages take {69/78} between them -> no 6,7,8 and 9 elsewhere in N9
17. "45" on N9 -> i innie = 1 outie -> no 6 or 7 in r6c9
18. 10(3) cage in N69 now {145/235} = 5{14/23} -> no 5 elsewhere c9
19. (from step 16) 6 in r7 now locked in N8 -> no 6 in 28(6) cage in N8 -> 28(6) = {123589/124579/134578} = 15{2389/2479/3478} -> no 1 or 5 in r7c456

20. "45" on r89 -> 4 innies = 17. Min. r8c2 = 7 -> max. r8c189 = 10 -> max 6 in any one cell (can't be 7 because it's in r8c2:thanks Andrew) -> no 8 in r8c1.
21. 8 in N7 now locked in r7 -> no 8 elsewhere in r7
22. "45" on N8 -> r7c456 = 17 -> h17(3) cage in N8 must have 6 (step 19) = 6{29/47} -> no 3
23. 3 in N8 locked in 28(6) cage = 1358{29/47}
24. 3 in r7 only in N9 -> no 3 elsewhere in N9 -> no 2 in r7c8
25. from step 22, 8 in 11(3) cage in N47 can only go in r7c1 -> no 1 in r7c1 (since 1 is only in {128} combination in 11(3) cage with 8 now locked in r7c1)
26. 1 in r7 now locked in N9 -> no 1 elsewhere in N9 -> no 4 in r7c8. 5(2) cage in N9 = [32/14]

27. "45" on r89 -> 4 innies = 17 -> 4 cells (r8c1289) = must sum from six candidates (124579). These six sum to 28 -> excess two candidates = 28-17= 11 [Edit:typo] which can only be {29/47} -> must have 1 and 5 -> r8c1 = 1, r8c9 = 5, r67c1 = 10 = [28]
28. "45" on N7 -> r7c3 = 5, r9c12 = 7 = {34} -> 3 and 4 locked for N7 and r9
29. 8(2) cage in N7 = {26} -> 2 and 6 locked for c3
30. r23c3 = 10 = {19/37}(no 4 or 8) = [1/3..] -> Killer pair on 1 and 3 with r1c3 -> 1 and 3 locked for N1 and c3
31. 9(3) cage in N14 = {126/234} (not {135} since 1 and 3 are only in r4c2) ->r3c12 = [62/42] -> r3c2 = 2, r3c1 = {46}
32. r67c9 =5 = {14}/[32]
33. "45" on N9 -> 2 innies = 5 -> r7c79 = {14}/[32]
34. r6c23 + r7c4 = 20 = [389/4{79}] ->r6c2 = {34}, r7c4 = {79}, r6c3 = {789}
35. {34} naked pair r69c2 -> 3 and 4 locked for c2 -> r4c2 = 1, r3c1 = 6
36. "45" on c12 -> 1 outie - 1 = 1 innie -> r5c3 = 4, r6c2 = 3
36. "45" on c1234 -> r89c4 = 9 -> r8c4 = {2478}, r9c4 = {1257}
37. "45" on c5 -> r89c5 = 8 -> r8c5 = {37}, r9c5 = {15}
38. "45" on c6789 -> r89c7 = 11 -> r8c6 = {2349}, r9c6 = {2789}

and the rest goes from there - feeling too tired to go any further - so back to Nasenbaer's every cell walk-through
2008 Walkthrough by Andrew:
Ed wrote:
Everyone, please feel free to do all these early puzzles and let us know on the forum how you found them. Your comments will be added onto here. Wow - FAME!

OK, here's the first one! :) I had another go at A20 V2. When it first appeared I was still using insertion solving and didn't manage to finish it. I started again, this time using elimination solving.

I'll rate A20 V2 as 1.25.

Here is my walkthrough. It's fairly similar to Ed's one in the Assassin Archive & Rating Update thread but I think there's enough different to post it. Step 23a is a fun move but unnecessary after I found step 24; I left it in for interest.

Prelims

a) R1C67 = {17/26/35}, no 4,8,9
b) R1C34 = {13}, locked for R1, clean-up: no 5,7 in R1C67
c) R23C3 = {19/28/37/46}, no 5
d) R23C7 = {15/24}
e) R56C6 = {15/24}
f) R56C4 = {18/27/36} (cannot be {45} which clashes with R56C6), no 4,5,9
g) R89C3 = {17/26/35}
h) R78C2 = {79}, locked for C2 and N7, clean-up: no 1 in R89C3
i) R78C8 = {14/23}
j) R89C7 = {69/78}
k) R9C12 = {16/34} (cannot be {25} which clashes with R89C3)
l) R9C89 = {69/78}
m) 9(3) cage at R3C1 = {126/135/234}, no 7,8,9
n) R678C1 = {128/137/146/236/245}, no 9
o) R678C9 = {127/136/145/235}, no 8,9
p) 26(4) cage in N1 = {2789/3689/4589/4679/5678}, no 1

1. Naked quad {6789} in R8C7 + R9C789, locked for N9

2. Naked pair {26} in R1C67, locked for R1

3. Killer pair 3,6 in R89C3 and R9C12, locked for N7

4. 45 rule on C1234 2 innies R89C4 = 9 = {18/27/36/45}, no 9
4a. 45 rule on C5 2 innies R89C5 = 8 = {17/26/35}, no 4,8,9
4b. 45 rule on C6789 2 innies R89C6 = 11 = {29/38/47/56}, no 1

5. 45 rule on N1 1 outie R4C2 = 1 innie R1C3 -> R4C2 = {13}

6. 45 rule on N3 1 innie R1C7 = 1 outie R4C8 + 1 -> R4C8 = {15}

7. 45 rule on N7 1 innie R7C3 = 1 outie R6C1 + 3, R6C1 = {125}, R7C3 = {458}

8. 45 rule on N9 1 outie R6C9 = 1 innie R7C7, no 6,7 in R6C9

9. R678C9 = {145/235}, 5 locked for C9

10. 45 rule on R1234 2 innies R4C19 = 13 = [49/58/67/76/94], no 1,2,3, no 8 in R4C1

11. 17(3) cage at R3C8 = {179/359/458} (cannot be {269/278/368/467} because R4C8 only contains 1,5), no 2,6
11a. R4C8 = {15} -> no 1,5 in R3C89

12. 45 rule on C12 1 outie R5C3 = 1 innie R6C2 + 1, no 1,8 in R5C3

13. 45 rule on C89 1 outie R5C7 = 1 innie R6C8 + 2, no 1,2 in R5C7, no 8,9 in R6C8

14. 6 in R7 locked in R7C456, locked for N8, clean-up: no 3 in R89C4 (step 4), no 2 in R89C5 (step 4a), no 5 in R89C6 (step 4b)
14a. Hidden killer pair 7,9 in R7C2 and R7C456 for R7 -> R7C456 must contain 7 or 9
14b. 45 rule on N8 3 innies R7C456 = 17 = {269/467}, no 1,3,5,8
14c. 8 in R7 locked in R7C13, locked for N7

15. R678C1 = {128/245}, 2 locked for C1
15a. 8 of {128} must be in R7C1 -> no 1 in R7C1

16. 45 rule on N7 3 innies R7C13 + R8C1 = 14 = {158/248}
16a. 1 of {158} must be in R8C1 -> no 5 in R8C1
16b. 5 of {158} must be in R7C3 (R78C1 cannot be [51] which clashes with R678C1) -> no 5 in R7C1
16c. 5 in N7 locked in R789C3, locked for C3, clean-up: no 4 in R6C2 (step 12)

17. 1,3 in R7 locked in R7C789, locked for N9, clean-up: no 2,4 in R7C8

18. 3 in C7 locked in R4567C7, CPE no 3 in R6C8, clean-up: no 5 in R5C7 (step 13)

19. 45 rule on R89 4 innies R8C1289 = {1259/1457} -> R8C1 = 1, R8C9 = 5, R67C1 = [28] (step 15), R7C3 = 5 (step 16), clean-up: no 3,6 in R5C3 (step 12), no 7 in R5C4, no 4 in R5C6, no 4 in R5C7 (step 13), no 1 in R6C2, (step 12), no 2 in R7C7 (step 8), no 3 in R89C3, no 6 in R9C12, no 8 in R9C4 (step 4), no 3,7 in R9C5 (step 4a)
19a. R8C9 = 5 -> R67C9 = {14/23} (step 9)
19b. 2 of {23} must be in R7C9 -> no 3 in R7C9

20. Naked pair {26} in R89C3, locked for C3, clean-up: no 4,8 in R34C3
20a. Killer pair 1,3 in R1C3 and R23C3, locked for C3 and N1
20b. 4,8 in C3 locked in R456C3, locked for N4, clean-up: no 9 in R4C9 (step 10), no 9 in R5C3 (step 12)

21. 9(3) cage at R3C1 = {126/234} (cannot be {135} because 1,3 only in R4C2), no 5 -> R3C2 = 2, clean-up: no 4 in R2C7

22. Naked pair {34} in R9C12, locked for R9, clean-up: no 5 in R8C4 (step 4), no 7,8 in R8C6 (step 4b)

23. R7C456 (step 14b) = 6{29/47} -> R89C456 = 1358{29/47}
23a. 9 can only be in R89C6 -> 2 can only be in R89C6 (step 4b) -> no 2 in R89C4, clean-up: no 7 in R89C4 (step 4)

24. 25(4) cage at R6C2 = {3589} (only remaining combination, cannot be {4579} because R6C2 only contains 3,6) -> R6C2 = 3, R6C3 = 8, R7C4 = 9, R78C2 = [79], R9C12 = [34], R4C2 = 1, R1C3 = 1 (step 5), R1C4 = 3, R3C1 = 6 (step 21), R4C8 = 5, R5C3 = 4 (step 12), clean-up: no 9 in R23C3, no 7,8 in R4C9 (step 10), no 1,6 in R5C4, no 6 in R6C4, no 4 in R7C56 (step 14b), no 2 in R7C9 (step 19a), no 2 in R89C6 (step 4b), no 6 in R9C7

25. Naked pair {79} in R4C13, locked for R4 and N4 -> R5C12 = [56], R4C1 = 7 (cage sum), R4C3 = 9, R4C9 = 6 (step 10), clean-up: no 1 in R6C6, no 9 in R9C8

26. Naked quad {6789} in R9C6789, locked for R9 -> R89C3 = [62], clean-up: no 9 in R9C7

27. Naked pair {78} in R89C7, locked for C7 and N9 -> R9C89 = [69]

28. Naked triple {134} in R7C789, locked for N9 -> R8C8 = 2, R7C8 = 3

29. R6C56 = [65] (hidden pair in R6), R5C6 = 1, R6C4 = 7, R5C4 = 2, R7C5 = 2, R7C6 = 6, R1C67 = [26]
29a. R5C5 = 9 (hidden single in N5), R5C7 = 3, R6C8 = 1 (step 13), R6C79 = [94], R7C79 = [41], R4C7 = 2

30. Naked pair {78} in R15C9, locked for C9 -> R23C9 = [23], R3C8 = 9 (cage sum), R23C3 = [37]

31. R2C4 = 6 (hidden single in C4)
31a. R2C4 + R4C3 = 15 -> R34C4 = 9 = [18/54], no 4,8 in R3C4

32. R2C6 = 9 (hidden single in C6), R12C1 = [94]
32a. R2C6 + R4C7 = 11 -> R34C6 = 11 = [83], R89C6 = [47], R8C4 = 8, clean-up: no 5 in R9C4 (step 4)

and the rest is naked singles


Last edited by Ed on Sat Jun 14, 2008 1:06 am, edited 4 times in total.

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PostPosted: Sat Jun 07, 2008 12:15 pm 
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Location: Sydney, Australia
Assassin 21 by Ruud (Oct 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3584:3584:4098:1027:4100:4100:5638:5638:5638:3584:4098:4098:1027:4109:4109:4109:3344:5638:3584:4115:2580:2580:4118:4118:792:3344:3344:3867:4115:4637:4637:6943:4118:792:1570:1570:3867:4115:4637:6943:6943:6943:4650:5163:3372:2605:2605:1327:3888:6943:4650:4650:5163:3372:2358:2358:1327:3888:3888:2619:2619:5163:4670:7743:2358:4417:4417:4417:836:4165:4165:4670:7743:7743:7743:2379:2379:836:4165:4670:4670:
Solution:
+-------+-------+-------+
| 2 8 5 | 1 7 9 | 4 6 3 |
| 1 4 7 | 3 2 6 | 8 5 9 |
| 3 9 6 | 4 5 8 | 2 7 1 |
+-------+-------+-------+
| 7 5 9 | 8 6 3 | 1 4 2 |
| 8 2 1 | 7 9 4 | 6 3 5 |
| 4 6 3 | 2 1 5 | 7 9 8 |
+-------+-------+-------+
| 5 1 2 | 9 4 7 | 3 8 6 |
| 6 3 4 | 5 8 1 | 9 2 7 |
| 9 7 8 | 6 3 2 | 5 1 4 |
+-------+-------+-------+
Quote:
Nasenbaer: harder than number 20 (just my opinion). It took some time to get the first number, but then it unfolded nicely
nd: Not a pushover puzzle but actually you can place numbers within a couple of steps using overlapping cages
Walkthrough by nd:
Not a pushover puzzle but actually you can place numbers within a couple of steps using overlapping cages -- there are several instances of 2-cell cages on which you can superimpose another 2-cell cage (via 45 rule).

1. In N2, 4(2) cage = {13}, 16(2) cage = {79}. 45 rule on N3 -> R23C7 = 10 = [91|82] -> R2C56 = {2(5|6)} with 2 locked in R2 and N2 in those cells.

2. 45 rule on N1 -> R3C23 = 15. The difference between this cage & the overlapping 10(2) cage -> R3C4 = R3C2 - 5 -> R3C4 = 4 (because {123} are blocked in N2), R3C23 = [96].

3. 45 rule on N123 -> R4C67 = [31], R4C89 = {24}, R23C7 = [82], R2C56 = [26] (because the 3(2) cage in N8 = {12}), R3C56 = {58}. R3C189 = {137}.

4. There is a naked quad in C1 on {6789} in R4589C1 (since 15(2) and 30(4) must contain only those numbers) -> R3C1 = {13} -> R3C89 = {(1|3)7}. But a 13(3) cage cannot contain {37} (since this would make the 3rd digit a 2nd 3) -> R3C1 = 3, R3C89 = {17}, R2C8 = 5, R2C9 = 9, R1C789 = {346}, R12C4 = [13].

5. 45 rule on N9 -> R7C78 = 11. Use cage superimposition again -> R7C6 = R7C8 - 1 -> R7C6 = 7 and R7C8 = 8 (only possibility not blocked!), R7C7 = 3, R1C56 = [79], R56C8 = {39}, R1C9 = 3. 45 rule on N6 -> R6C6 = 5, R3C56 = [58], R5C6 = 4, R56C9 = [58], R56C7 = {67}, R1C78 = [46], R89C7 = {59}, R8C8 = 2, R89C6 = [12], R4C89 = [42].

6. Mop-up. 45 rule on N5 -> R46C4 = 10 = [82], R5C4 = 7, R56C7 = [67]. 45 rule on N7 -> R78C3 = 6 = [15|24], R8C2 = 3, R7C45 = [94], R7C12 = {15}, R78C3 = [24], R6C3 = 3, R56C8 = [39], R7C9 = 6, R8C45 = [58], R8C9 = 7, R89C7 = [95], R8C1 = 6, R9C45 = [63] & you carry on....
Walkthrough by Andrew:
sudokuEd has persuaded me to post my walkthrough since it differs in some ways from those already posted, mainly I think because of things that nd had spotted but I hadn't. Thanks sudokuEd for the changes that you suggested to me off-forum in private messages. I've also taken the liberty of including a couple of his improvements which would make the solution a lot quicker.

Step 1
R12C4 = {13}, R1C56 = {79}, R34C7 = {12}, 30(4) cage in N7 = {6789}, R89C6 = {12} -> R9C45 = {36/45}

Step 2
45 rule on N3 2 innies = 10 -> R2C7 = {89}, 45 rule on N7 2 innies = 6 = {15/24}, R67C3 = {14/23} -> R7C3 = {124}, R6C3 = {134}, R8C3 = {245}, 3 in N7 locked in 9(3) cage = 3{15/24}, 45 rule on R89 1 innie + 3 = 1 outie -> R8C2 + 3 = R7C9, R8C2 = {12345} -> R7C9 = {45678}

Step 3
45 rule on N1 2 innies = 15 = {69/78}, R3C23 -> [78]/[96] because R3C4 cannot be 1 or 3, R3C4 = {24}

Step 4
45 rule on C123 2 outies – 8 = 1 innie, min. R8C3 = 2 -> min. R34C4 = 10 -> R4C4 = min.6, 5 is only odd digit in R8C3 -> R4C4 cannot be 7 -> R4C4 = {689}

Step 5
45 rule on C12 1 innie + 4 = 1 outie -> R2C2 + 4 = R9C3 -> R2C2 = {2345} -> R12C3 = min.11 -> 1 in N1 locked in 14(4) cage

Step 6
R7C67 = {37/46} (because 1 and 2 locked in C6 and also in C7)
45 rule on N9 2 innies = 11, R7C67 = 10 -> R7C6 + 1 = R7C8 -> R7C8 = {4578}
1 and 2 in N9 must be in R89 (one in each row), killer pair with R89C6 = {12} so no other 1/2 in R89 -> R8C2 = {345}, R8C3 = {45}, R7C3 = {12}, R6C3 = {34}

Step 7
9 in R7 locked in R7C45 -> 15(3) cage in N58 = {249} (cannot be {159} because 1 blocked in C4 and N8) -> R6C4 = 2 -> R7C45 = {49}, R7C67 = {37}, R7C8 = 8 (from second line of step 6, 4 blocked from R7) -> R7C67 = [73], R8C2 = 3, R7C12 = {15}, R7C9 = 6, R7C3 = 2, R6C3 = 3, R8C3 = 4, R3C34 = [64], R3C2 = 9, R9C45 = [63], R8C45 = {58}, R8C1 = 6

Step 8
R45C1 = {78}, R9C1 = 9, 9 in N4 must be in C3 (the 10(2) cage in N4 cannot be [91] because the 1 in C2 is either in 14(4) cage in N1 or in 9(3) cage in N7) -> R4C4 = 8, R45C3 = {19}, R45C2 = {25}, R45C1 = [78], R6C12 = [46], R7C12 = [51], R123C1 = {123}, R1C2 = 8, R9C23 = [78], R2C2 = 4, R12C3 = [57]

Step 9
4 and 6 in N3 locked in R1, rest of 22(4) cage in N3 must be {39} -> R2C9 = 9, R1C789 = {346}, R2C7 = 8, R34C7 = [21], 6 in N2 locked in R2 -> R2C56 = [26], R1C1 = 2, R12C4 = [13], R23C1 = [13], R2C8 = 5, R3C89 = {17}, R3C56 = {58}, R4C6 = 3, R45C3 = [91], R4C89 = {24}, R45C2 = [52], R4C5 = 6

Step 10
45 rule on C89 1 innie + 2 = 1 outie -> R8C8 + 2 = R1C7 -> R8C8 = 2 (4 blocked in R8) -> R1C7 = 4, R1C89 = [63], R89C7 = [95], R89C6 = [12], R8C9 = 7, R9C89 = {14}, R56C9 = [58], R56C8 = [39], R56C7 = [67], R6C6 = 5, 27(5) cage in N5 = {14679} and carry on …. the rest is simple elimination

sudokuEd has also pointed out to me that after my step 3, innies/outies on R123 give direct placements for R3C2 and R4C67 after which R3C34 follow immediately from step 3.

He has also pointed out to me that after my step 6, innies/outies on R789 give direct placements for R6C34 and R7C8.

Alternatively, as I noticed while correcting my walkthrough, the innies/outies from my step 4 can be used after my step 6 to fix R4C4
Assassin 21 V2 by sudokuEd (Oct 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3584:3584:4098:5123:5123:5123:5638:5638:5638:3584:4098:4098:5123:4109:4109:4109:3344:5638:3584:6675:6675:6675:4118:4118:792:3344:3344:3867:6675:4637:4637:6943:4118:792:1570:1570:3867:6675:4637:6943:6943:6943:4650:5163:3372:2605:2605:1327:3888:6943:4650:4650:5163:3372:2358:2358:1327:3888:3888:2619:2619:5163:4670:7743:2358:5185:5185:5185:5185:4165:4165:4670:7743:7743:7743:2379:2379:5185:4165:4670:4670:
Solution:
+-------+-------+-------+
| 2 8 5 | 1 7 9 | 4 6 3 |
| 1 4 7 | 3 2 6 | 8 5 9 |
| 3 9 6 | 4 5 8 | 2 7 1 |
+-------+-------+-------+
| 7 5 9 | 8 6 3 | 1 4 2 |
| 8 2 1 | 7 9 4 | 6 3 5 |
| 4 6 3 | 2 1 5 | 7 9 8 |
+-------+-------+-------+
| 5 1 2 | 9 4 7 | 3 8 6 |
| 6 3 4 | 5 8 1 | 9 2 7 |
| 9 7 8 | 6 3 2 | 5 1 4 |
+-------+-------+-------+
Quote:
Forum thread here
sudokuEd: couldn't solve it. But there are a huge number of logical steps in this V2 - feels like just needs one more - eg maybe the quad etc. I don't have any time to try again at the moment, but maybe someone else can
Andrew: I managed to solve it in less than a day after I started, one of my quickest variants. ..I'll rate my walkthrough at least Hard 1.5
Ed: found an equally interesting alternative way (as Para and Andrew). I think it is a Killer Empty Rectangle.
Walkthrough by Para:
Hi all

One ("unsolvable") down, 9 to go. :wink:

Walk-through Assassin 21V2

1. 14(4) at R1C1 = {1238/1247/1256/1346/2345}: no 9

2. R34C7 = {12} -->> locked for C7

3. R45C1 = {69/78}: no 1,2,3,4,5

4. R4C89 = {15/24}: no 3,6,7,8,9
4a. Killer Pair {12} at R4C7 + R4C89 locked for R4 and N6

5. 20(3) at R5C8 = {389/479/569/578}: no 1,2

6. R56C9 = {49/58/67}: no 3

7. R6C12 = {19/28/37/46}: no 5

8. R67C3 = {14/23}: no 5,6,7,8,9

9. 9(3) at R7C1 = {126/135/234}: no 7,8,9

10. R7C67 = [19/28]/{37/{46}: no 5; R7C6: no 8,9

11. 30(4) at R8C1 = {6789} -->> locked for N7
11a. 9(3) at R7C1 = {135/234} -->> 3 locked for N7
11b. Clean up: R6C3: no 2

12. R9C45 = {18/27/36/45}: no 9

13. 45 on N1: 2 innies: R3C23 = 15 = {69/78}: no 1,2,3,4,5

14. 45 on N3: 2 innies: R23C7 = 10 = [82/91]: R2C7 = {89}

15. 45 on N6: 2 outies and 1 innie: R4C7 + 12 = R6C6 + R7C8: R4C7 = {12} -->> R6C6+R7C8 = 13/14 = {49/58/67/59/68}: no 1,2,3

16. 45 on N7: 2 innies: R78C3 = [15/24/42]: R8C3: no 1

17. 45 on N8: 3 outies: R6C4 + R7C7 + R8C3 = 9 = [432/342/234/162/144/135] -->> R6C4 = {1234}; R7C7 = {346}
17a. Clean up: R7C6 = {467}

18. 45 on N9: 2 innies: R7C78 = 11 = [38/47/65] : R7C8 = {578}

19. 45 on C789: 2 outies and 1 innie: R78C6 = R2C7 + 4: R2C7 = {89} -->> R67C6 = 12/13 = [84/57/94/76/67]: R6C6: no 4

20. 45 on R789: 2 outies and 1 innie: R6C34 + 3 = R7C8: Min R6C34 = 3 -->>Min R7C8 = 6: R7C8 = {78} -->>R6C34 = 4/5 = {13}/{14}/[32]
20a. Clean up: R7C7: no 6; R7C6: no 4
20b. R67C6 = [57/67/76]: no 8,9; 7 locked for C6

21. 45 on R123: 4 outies: R5C2 + R4C267 = 11 = [1]{36}[1]/[1]{45}[1]/[1]{35}[2]/[2]{35}[1]/[2]{34}[2]/[3][431] -->> R5C2 = {123}; R4C26 = {3456}

22. 45 on N23: 1 innie and 2 outies: R3C4 = R4C67: min R4C67 = 4 -->> R3C4: no 1,2,3; Max R4C67 = 8 -->> R3C4: no 9

23. 45 on N1: 3 outies: R3C4 + R45C2 = 11(no duplicates, all in same cage) = [4][61]/[4][52]/[5][42]/[6][41]/[6][32]/[7][31] -->>R3C4 : no 8; R5C2: no 3

24. 16(3) at R2C5 needs one of {89} in R2C7 -->> R2C56 = 7/8 -->> R2C56 no 8,9

25. 20(3) at R5C8 = {39}[8]/{49}[7]/{58][7]/{57}[8]: no 6

26. 45 on R12: R3C189 = 11 = {137/146/236/245}= {1|2..}({128} blocked by R3C7): no 8,9
26a. Killer Pair {1|2..} in R3C189 + R3C7 -->> locked for R3

27. Hidden Killer Pair {89} in R3; R3C23 needs one of {89}; R3C56 needs one of {89}
27a. 16(3) at R3C5 = {349/358}: no 6,7; 3 locked 3 within cage -->> R12C6: no 3

28. R7C678 = [738/647] -->> 7 locked for R7

29. 9(3) at R7C1: [43][2] blocked by R7C7 -->> R8C2: no 2

30. 45 on R89: 1 innie and 1 outie: R8C2 + 3 = R7C9 -->> R8C2 = {135}; R7C9 = {468}

31. R7C6789 = [738]{46}/[6478] -->> 8 locked within R7C89 for R7 and N9

32. 9 in R7 locked within 15(3) at R6C4 -->> 15(3) = [1]{59}/[2]{49}/[4]{29} -->> R6C4 = {124}; R7C45 = {29/49/59}: no 1,3,6; 9 locked for N8
32a. 1 in R7 locked for N7
32b. Clean up: R7C9: no 4(step 30)

33. Only place for {12} in N2 is R2C56 within 16(3) cage at R2C5 or 20(4) at R1C4
33a. 20(4) at R1C4: {1289} blocked by R3C56, so neither cage can contain both {12}, so both need one of {12}
33b. 16(3) at R2C5 = {16}[9]/{25}[9]/[718]/{26}[8]: no 3,4
33c. 20(4) = {1379/1469/1478/1568/2369/2378/2459/2468}

34. 16(3) at R8C7 = {69}[1]/{59}[2]/{457}:{349} blocked by R7C7; {367} blocked by R7C89: no 3; R8C8: no 6,9

35. R7C789 = [478/386]
35a. 18(4) at R7C9 = [6]{129/147}/[8]{136/235}(others blocked by R7C78)

36. 45 on N4: 3 innies and 1 outie: R4C4 + 2 = R45C2 + R6C3; Min. R45C2 + R6C3 = 6 -->> Min. R4C4 = 4: no 3

37. 45 on N2356: 3 innies and 1 outie: R7C8 + 6 = R346C4: R7C8 = {78} -->> R346C4 = 13/14 -->> R346C4 = {47}[2]/{56}[2]/[48][1/2]/{57}[1/2]/[49][1]/[58][1]/{67}[1] -->> R6C4: no 4
37a. Clean up: R7C45: no 2
37b. 2 in R7 locked for N7
37c. Clean up: R7C3: no 4; R6C3: no 1

38. 1 and 8 in N8 contained within 20(5) at R8C3 and 9(2) at R9C4; 9(2) needs either both {18} or neither, so the same goes for 20(5) at R8C3
38a. 20(5) at R8C3 needs one of {45} in R8C3 -->> 20(5) = {12458/23456}: no 7; 2 locked in 20(5) cage for N8
38b. Clean up: R9C45: no 7
38c. R7C6 = 7(hidden); R7C789 = [386]
38d. Clean up: R56C9 = {49/58}= {5|9..}: no 7; R56C8 = {39/57} = {5|9..}: no 4

39. Killer Pair {59} in N6 within R56C8 + R56C9 -->> locked for N6
39a. Clean up: R4C89 = {24} -->> locked for R4 and N6
39b. R34C7 = [21]
39c. Clean up: R56C9 = {58} -->> locked for C9 and N6
39d. R56C7 = {67} -->> locked for C7, N6 and 18(3) cage at R5C7
39e. R6C6 = 5; R56C9 = [58]; R4C6 = 3
39f. R3C4 = 4(step 22); R4C56 = [58](last remaining combo within 16(3) at R3C5)
39g. R8C2 = 3(hidden)
39h. R56C8 = {39} -->> locked for C8

40. R3C23 = {69}(step 13) -->> locked for N1 and R3 and 26(5) at R3C2
40a. R45C2 = [52]

41. 45 on C789: 3 innies: R256C7 = 21 = [8]{67} -->> R2C7 = 8

42. 13(3) at R2C8 = {157} (last remaining combo) -->> R2C8 = 5; R3C89 = {17} -->> locked for R3 and N3
42a. R3C1 = 3; R1C8 = 6(hidden)

43. 20(5) at R8C3 = {12458}: no 6
43a. R8C1 = 6(hidden)
43b. Naked Triple {789} in R9C123 -->> locked for R9
43c. Naked Triple {124} in R9C689 -->> locked for R9
43d. R9C7 = 5

44. R45C1 = {78}(last combo) -->> locked for C1 and N4
44a. R9C1 = 9
44b. Clean up: R6C2: no 1,4
44c. Naked Pair {69} in R4C3 + R6C2 -->> locked for N4

45. 16(3) at R1C3 = {178/457}: no 2
45a. R7C3 = 2(hidden); R6C3 = 3; R8C3 = 4(step 16)

Easy to the end
(note: some corrections identified by Andrew included in this WT. Thanks very much on Para's behalf!)
Walkthrough by Andrew; Feb 2010:
I've been working on some of my unfinished puzzles on Richard's site and decided to have a go at some of the ones on this site too. I only realised later that A21 V2 was one of the ones in the Unsolvables thread.

I found that I hadn't even started this one; not sure why as I managed to solve it in less than a day after I started, one of my quickest variants.

I think there must be a fairly narrow solving path; Para and I used similar key steps.

I'll rate my walkthrough at least Hard 1.5 because of step 24; if I'd found Para's step 37 then it would have simplified my step 24.

Here is my walkthrough for A21 V2.

Prelims

a) R34C7 = {12}
b) R45C1 = {69/78}
c) R4C89 = {15/24}
d) R56C9 = {49/58/67}, no 1,2,3
e) R6C12 = {19/28/37/46}, no 5
f) R67C3 = {14/23}
g) R7C67 = {19/28/37/46}, no 5
h) R9C45 = {18/27/36/45}, no 9
i) 20(3) cage at R5C8 = {389/479/569/578}, no 1,2
j) 9(3) cage in N7 = {126/135/234}, no 7,8,9
l) 14(4) cage in N1 = {1238/1247/1256/1346/2345}, no 9
m) 30(4) cage in N7 = {6789}

Steps resulting from Prelims
1a. Naked pair {12} in R34C7, locked for C7, clean-up: no 8,9 in R7C6
1b. Naked quad {6789} in 30(4) cage, locked for N7
1c. Naked quad {6789} in R4589C1, locked for C1, clean-up: no 1,2,3,4 in R6C2
1d. Killer pair 1,2 in R4C7 and R4C89, locked for R4

2. 45 rule on N1 2 innies R3C23 = 15 = {69/78}
2a. R3C23 = 15 -> R3C4 + R45C2 = 11 = {128/137/146/236/245}, no 9
2b. 8 of {128} must be in R4C2 -> no 8 in R3C4 + R5C2

3. 45 rule on N3 2 innies R23C7 = 10 = [82/91]
3a. 16(3) at R2C5 cannot have both of 8,9 -> no 8,9 in R2C56

4. 45 rule on N7 2 innies R78C3 = 6 = [15/24/42], no 3, no 1 in R8C3, clean-up: no 2 in R6C3

5. 45 rule on N9 2 innies R7C78 = 11 = {38/47}/[65], no 9, no 6 in R7C8, clean-up: no 1 in R7C6

6. 45 rule on R789 1 innie R7C8 = 2 outies R6C34 + 3
6a. Min R6C34 = 3 -> no 3,4,5 in R7C8, clean-up: no 6,7,8 in R7C7 (step 5), no 2,3,4 in R7C6
6b. R7C8 = {78} -> R6C34 = 4,5 = {13/14}/[32], no 5,6,7,8,9
6c. Max R6C4 = 4 -> min R7C45 = 11, no 1
6d. 20(3) cage at R5C8 = {389/479/578} (cannot be {569} because R7C8 only contains 7,8), no 6

7. 9(3) cage in N7 = {135/234}
7a. 2 of {234} must be in R7C12 (R7C12 cannot be {34} which clashes with R7C7), no 2 in R8C2

8. 45 rule on R12 1 innie R2C8 = 1 outie R3C1 +2 -> R2C8 = {34567}

9. 45 rule on R89 1 outie R7C9 = 1 innie R8C2 + 3 -> R7C9 = {4678}
9a. 1 in R7 only in R7C123, locked for N7, clean-up: no 4 in R7C9
9b. Naked triple {678} in R7C689, locked for R7
9c. 8 in R7 only in R7C89, locked for N9

10. 9 in R7 only in R7C45, locked for N8
10a. 15(3) cage at R6C4 = {159/249}, no 3

11. 45 rule on C789 2 outies R67C6 = 1 innie R2C7 + 4
11a. R2C7 = {89} -> R67C6 = 12,13 = [57/67/76], 7 locked for C6

12. 45 rule on N23 2 outies R4C67 = 1 innie R3C4
12a. Min R4C67 = 4 -> min R3C4 = 4
12b. Max R3C4 = 7 -> max R4C67 = 7, max R4C6 = 6

13. R3C4 + R45C2 (step 2a) = {137/146/236/245} (cannot be {128} because 1,2 only in R5C2), no 8
13a. 1,2 only in R5C2 -> R5C2 = {12}
13b. Min R3C4 + R5C2 = 5 -> max R4C2 = 6

14. 45 rule on C89 1 outie R1C7 = 1 innie R8C8 + 2, no 9 in R8C8

15. 45 rule on C12 3(2+1) outies R39C3 + R3C4 = 1 innie R2C2 + 14
15a. Min R39C3 + R3C4 = {67} + 4 = 17 -> min R2C2 = 2

16. 16(3) cage in N9 = {169/259/457} (cannot be {349} which clashes with R7C7, cannot be {367} which clashes with R7C89, Killer ALS block), no 3, clean-up: no 5 in R1C7 (step 14)
16a. 1 of {169} must be in R8C8 -> no 6 in R8C8, clean-up: no 8 in R1C7 (step 14)

17. R67C3 = [14/32/41], R78C3 = [15/24/42] -> R678C3 = [142/324/415], 4 locked for C3

18. Max R4C4 = 9 -> min R45C3 = 9 must contain at least one of 6,7,8,9
18a. Killer quad 6,7,8,9 in R45C1, R45C3 and R6C2, locked for N4
18b. Max R45C3 = {59} = 14 (cannot contain two of 6,7,8,9 which would clash with R45C1 + R6C2, ALS block) -> min R4C4 = 4
18c. 18(3) at R4C3 = {189/279/369/378/459/468/567}
18d. 9 of {189/279/369/459} must be in R4C34 (R4C34 cannot be {45} which clashes with R4C89, cannot be {36} which clashes with R4C2689, Killer ALS block), no 9 in R5C3
[This step was interesting but probably not essential.]
18e. 5 in N4 only in R4C23 + R5C3, CPE no 5 in R4C4

19. 14(4) cage in N1 = {1238/1247/1256/1346/2345}
19a. 6,7,8 of {1238/1247/1256/1346} must be in R1C2 -> no 1 in R1C2

20. 45 rule on N6 2(1+1) outies R6C6 + R7C8 = 1 innie R4C7 + 12
20a. R4C7 = {12} -> R6C6 + R7C8 = 13,14 = [58/68/77] (cannot be [67] which clashes with R67C6, CCC)
20b. R6C6 + R7C8 = [58/68/77], R67C6 = [57/67/76] (step 11a) -> R7C68 = [78/78/67], 7 locked for R7, clean-up: no 4 in R8C2 (step 9)
[Para got this a bit more directly by using the overlap of R7C67 and R7C78.
Ed pointed out that step 20a can be simplified to
R4C7 = {12} -> R6C6 + R7C8 = 13,14 = [58/68/77] (cannot be [67] which clashes with R7C6). Thanks Ed!]

21. 45 rule on R12 3 outies R3C189 = 11 = {137/146/236/245} (cannot be {128} which clashes with R3C7), no 8,9
21a. Killer pair 1,2 in R3C189 and R3C7, locked for R3
[I ought to have spotted this step much earlier.]

22. Hidden killer pair 8,9 in R3C23 and R3C56 for R3, R3C23 contains one of 8,9 -> R3C56 must contain one of 8,9
22a. 16(3) cage at R3C5 = {349/358} (cannot be {367/457} which don’t contain 8 or 9), no 6,7, CPE no 3 in R12C6

23. R4C67 = R3C4 (step 12)
23a. Max R4C67 = 6 (cannot be [52] which clashes with R4C89) -> max R3C4 = 6
23b. Killer triple {345} in R4C2, R4C6 and R4C89, locked for R4
23c. Naked quad {6789} in R45C1, R4C3 and R6C2, locked for N4

24. Hidden killer pair 1,8 in 20(5) cage at R8C3 and R9C45 for N8, R9C45 must contain both of 1,8 or 20(5) cage must contain both of 1,8
24a. 20(5) cage at R8C3 = {12368/12458/23456} (cannot be {12467/13467} which only contain one of 1,8), no 7
24b. {12368/23456}, 6 locked for N8 => R7C6 = 7
{12458} => 3 in N8 only in R9C45 = {36}, locked for N8 => R7C6 = 7
24c. -> R7C6 = 7, R7C7 = 3, R7C89 = [86], clean-up: no 7 in R56C9, no 1 in R8C8 (step 14), no 2 in R9C45

25. 6 in N6 only in R56C7, locked for 18(3) cage at R5C7 -> R6C6 = 5, R56C7 = 13 = {67}, locked for C7 and N6, clean-up: no 8 in R5C9

26. R8C2 = 3 (hidden single in N7)
26a. R2C7 = 8 (hidden single in C7), R2C56 = 8 = [26/62/71], no 3,4, no 1,5 in R2C5

27. 5 in C7 only in R89C7, locked for N9
27a. 16(3) cage in N9 = {259/457}
27b. 2,7 only in R8C8 -> R8C8 = {27}

28. R7C8 = 8 -> R56C8 = 12 = {39} (only remaining combination), locked for C8 and N6, clean-up: no 4 in R56C9
28a. R56C9 = [58], clean-up: no 1 in R4C89, no 2 in R6C1
28b. Naked pair {24} in R4C89, locked for R4 -> R4C2 = 5, R4C6 = 3, R34C7 = [21]

29. R3C4 + R45C2 (step 2a) = {245} (only remaining combination) -> R3C4 = 4, R5C2 = 2, clean-up: no 5 in R9C5

30. 16(3) cage at R3C5 (step 22a) = {358} (only remaining combination) -> R3C56 = [58]

31. R3C23 (step 2) = {69} (only remaining combination), locked for R3 and N1

32. 7 in R3 only in R3C89, locked for N3
32a. 13(3) cage in N3 = {157} (only remaining combination) -> R2C8 = 5, R3C89 = {17}, locked for R3 and N3 -> R3C1 = 3, clean-up: no 7 in R6C2
32b. R1C8 = 6 (hidden single in N3)
32c. Naked pair {69} in R36C2, locked for C2

33. R7C2 = 1 (hidden single in C2), R7C1 = 5 (cage sum)
33a. Naked pair {24} in R78C3, locked for C3

34. 15(3) cage at R6C4 (step 10a) = {249} (only remaining combination) -> R6C4 = 2, R7C45 = [94], R7C3 = 2, R6C3 = 3, R8C3 = 4, R5C3 = 1, R6C1 = 4, R6C2 = 6, R3C23 = [96], R2C23 = [47], R1C23 = [85], R9C2 = 7, clean-up: no 1 in R2C6 (step 26a), no 9 in R45C1, no 5 in R9C4

35. Naked pair {78} in R45C1, locked for C1 and N4 -> R4C3 = 9, R4C4 = 8 (cage sum), clean-up: no 1 in R9C5

36. Naked pair {26} in R2C56, locked for R2 and N2 -> R2C1 = 1, R2C4 = 3, R2C9 = 9, R1C7 = 4
36a. R89C7 = {59} -> R8C8 = 2 (step 27a)

and the rest is naked singles.
Walkthough by Ed; March 2010:
sudokuEd wrote:
If it is solvable - it will certainly spur me on to try again. :)
This has been bugging me for years since I like to only post variants I have solved first. Many thanks to Para and Andrew for solving A21V2. They found a really interesting both/neither move to set up their solutions (Para's step 38; Andrew's step 24). I missed that but found an equally interesting alternative way in another part of the grid (step 11). I think it is a Killer Empty Rectangle. Some extra clarification added to this step (thanks Andrew).

A21 v2
This is an optimised WT so only the essential steps have been included.
Prelims as per Andrew's WT

1. 3(2)n3 = {12}: both locked for c7
1a. no 8,9 in r7c6

2. "45" on n9: 2 innies r7c78 = 11
2a. no 9 in r7c78; no 6 in r7c8
2b. no 1 in r7c6

3. "45" on r789: 2 outies r6c34 + 3 = 1 innie r7c8
3a. -> min r7c8 = 6
3b. -> r6c34 = 4/5 (no 5..9)
3c. r7c7 = (34) (h11(2))
3d. r7c6 = (67)

4. naked quad {6789} in n7: Locked for n7

5. "45" on r89: 1 outie r7c9 - 3 = 1 innie r8c2
5a. r7c9 = (4..8)

6. 9 in r7 only in 15(3)n5: 9 locked for n8
6a. 15(3) = 9{15/24}(no 3,6,7,8)
6b. min. r7c45 = 11 (no 1)

7. hidden triple 6,7,8 in r7c689
7a. r7c9 = (678)
7b. 8 locked for n9

8. "45" on n9: 1 outie r7c6 + 1 = 1 innie r7c8
8a. = [67/78]: 7 locked for r7

9. "45" on n3: 2 innies r23c7 = 10 = [91/82]

10. "45" on c89: 3 outies r189c7 = 18
10a. = {369/378/459/468/567}
10b. no 7 in r1c7 since no 8 in r89c7 for {378} and cannot be [7]{56} in overlapping the 16(3)n9 {can't be {56}[5]}

11. r7c89 = [78/86] = [7/6] -> {67} blocked from 13(2)n6 since there is no 7 in r123c7 (Killer Empty Rectangle?)
11a. another way of seeing this:
i. {67} in r56c9 -> r7c9 = 8 -> r7c8 = 7:
ii. {67} in r56c9 -> 7 in n3 only in c8, 7 locked for c8: but this means two 7s in c8
11b. yet another way of seeing this: (this one courtesy of Andrew)
i. If r7c89 = [78] -> 7 in n3 in c9 -> no 7 in 13(2) at r56c9 -> {67} blocked
ii. If r7c89 = [86] -> {67} blocked from 13(2) at r56c9

12. 20(3)n6 must have 7/8 for r7c8 = {389/479/578}(no 6)

13. 6 in n6 only in c7: locked for c7 and not in r6c6

14. 16(3)n9 = {259/457}(no 1,3,6) ({169} blocked by no 1,6 in r89c7; {349} blocked by r7c7; {367} blocked by [3/7..] in h11(2) at r7c78 step 2; or alternatively, by [6/7..] at r7c89 step 11)
14a. 5 locked for n9

15. "45" on c89: 1 outie r1c7 - 2 = 1 innie r8c8
15a. r1c7 = (49), r8c8 = (27)

16. h18(3)r189c7 = {459} only: all locked for c7

It's cracked.

Cheers
Ed


Last edited by Ed on Mon Mar 15, 2010 8:30 am, edited 4 times in total.

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PostPosted: Sat Jun 07, 2008 12:18 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
Assassin 22 by Ruud (Oct 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:3072:3585:3585:3585:1284:4101:4101:4101:3592:3072:3072:4107:3596:1284:3598:3599:3592:3592:4114:2323:4107:3596:4118:3598:3599:2585:3610:4114:2323:2323:3596:4118:3598:2585:2585:3610:4114:4901:4901:3623:3623:3623:4650:4650:3610:5933:5933:4901:4400:4400:4400:4650:3636:3636:2614:5933:3640:3640:2874:3643:3643:3636:2878:2614:5933:4417:3640:2874:3643:3653:3636:2878:2888:2888:4417:4417:2874:3653:3653:3663:3663:
Solution:
+-------+-------+-------+
| 5 8 2 | 4 3 6 | 1 9 7 |
| 6 1 9 | 7 2 5 | 8 4 3 |
| 4 3 7 | 1 9 8 | 6 2 5 |
+-------+-------+-------+
| 9 2 4 | 6 7 1 | 3 5 8 |
| 3 5 6 | 2 8 4 | 9 7 1 |
| 1 7 8 | 3 5 9 | 2 6 4 |
+-------+-------+-------+
| 8 9 1 | 5 6 7 | 4 3 2 |
| 2 6 5 | 8 4 3 | 7 1 9 |
| 7 4 3 | 9 1 2 | 5 8 6 |
+-------+-------+-------+
Quote:
nd: A very tough one, though no exotic tricks are required
Walkthrough by nd:
A very tough one, though no exotic tricks are required.

1. R23C3 = R34C5 = {79} (eliminating {79} in the rest of R3). 45 rule on C5 -> R56C5 = 13 = {58}. 45 rule on C1234 -> R56C4 = 5 = {14|23}. R56C6 = 17 + 14 - (13 + 5) = 13 = {49|67}.

2. 45 rule on R12 -> R2C3467 = 29 = {5789} -> R23C7 = [86|95], R2C46 = {5789} with 5 locked in those cells in R2/N2.

3. 45 rule on N5 -> R4C456 = 14. 45 rule on N2 -> R1C46 = 10. 45 rule on N1235 -> R3C1289 = 14. 45 rule on R1234 -> R5C19 = 4 = {13} -> R56C4 = [23|41], but placing 4 in R5C4 results in an impossible combo in the 14(3) cage so R56C4 = [23], R56C6 = [49|76], R4C46 = {1(4|6)} with 1 locked in those cells in R4/N5. 45 rule on R5 -> R6C37 = 10 = {28|46} ({19} is disallowed because a 1 can't go in a 19(3) cage).

4. 45 rule on C4 -> R1789C4 = 26 -> the 1 is locked in the 14(3) cage in C4 -> R1C46 = [46|64|73|82|91]. 45 rule on N3 -> R3C89 - R1C6 = 1 -> R1C6 <> 1 -> [91] is eliminated as a possibility from R1C46. We have now eliminated 9 from R1C456 -> X-wing on 9 on R23 in conjunction with R23C3 -> R23C7 = [86], R6C37 = [64|82].

5. 45 rule on R3 -> R3C46 = 9 = {18} (because {567} are blocked) -> R12C5 = {23}, R1C46 = {46}, R3C1289 = {2345} -> R3C1 = {45}, R3C2 = {234} -> R4C23 = {234}.

6. In N1 the 6 must go in the 12(3) cage = {(15|24)6} -> hidden {45} pair in conjunction with R3C1 -> R3C2 = {23} -> 4 is locked in N4/R4 in R4C23 -> R4C46 = {16}, R4C5 = 7, R3C5 = 9, R23C3 = [97], R56C6 = [49], R56C5 = [85], R1C46 = [46], R4C46 = [61], R3C46 = [18], R2C46 = [75], R1C23 = {28}, R3C2 = 3, R12C5 = [32], R1C78 = {19}.

7. Mop-up. 45 rule on N1 -> R3C1 = 4, R45C1 = [93], R78C1 = {28}, R5C9 = 1, R3C89 = [25], R4C9 = 8, R4C78 = {35}, 19(3) cage in N4 has a 5 locked in it -> R5C23 = {56}, R6C3 = 8, R1C23 = [82], R4C23 = [24], R5C78 = {79}, R6C7 = 2, R78C9 = {29} (because the 2 in N9/C9 cannot go in a 14(2) cage!), R9C89 = [86], R6C89 = {46}, R6C12 = {17}, R78C2 = {69}, R5C23 = [56], x-wing on 9 in R78 -> R9C4 = 9, R89C3 = {35}, R78C4 = {58}, R7C3 = 1, and you carry on...


Last edited by Ed on Sat Jun 14, 2008 1:13 am, edited 1 time in total.

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PostPosted: Sat Jun 07, 2008 12:20 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
Special Killer X - October 27 (aka SKX2) by Ruud (Oct 06)
Puzzle pic: 1-9 cannot repeat on the diagonals:
Image
[edit: corrected image supplied by Børge. Many thanks.]
Code: Select, Copy & Paste into solver:
3x3:d:k:5632:5632:5632:4355:4356:4869:5638:5638:5638:4617:5632:4355:4355:4356:4869:4869:5638:2577:4617:4617:2580:4355:4886:4869:4120:2577:2577:4617:6172:2580:4886:4886:4886:4120:3874:2577:6172:6172:6694:6694:6694:6694:6694:3874:3874:6189:6172:2351:4912:4912:4912:1587:3874:6965:6189:6189:2351:3897:4912:6459:1587:6965:6965:6189:4160:3897:3897:1859:6459:6459:5702:6965:4160:4160:4160:3897:1859:6459:5702:5702:5702:
Solution:
+-------+-------+-------+
| 2 9 5 | 1 8 3 | 4 6 7 |
| 1 6 4 | 7 9 2 | 8 5 3 |
| 8 3 7 | 5 4 6 | 9 1 2 |
+-------+-------+-------+
| 6 8 3 | 9 5 1 | 7 2 4 |
| 5 4 2 | 8 3 7 | 6 9 1 |
| 9 7 1 | 6 2 4 | 5 3 8 |
+-------+-------+-------+
| 3 5 8 | 2 7 9 | 1 4 6 |
| 7 2 6 | 4 1 5 | 3 8 9 |
| 4 1 9 | 3 6 8 | 2 7 5 |
+-------+-------+-------+
Quote:
Ruud: I'm looking forward to read your walkthroughs
sudokuEd: Ruud will just have to wait a bit. Am totally stuck - what have I missed? Keep looking for ALSxy - but keep getting tricked :?: :!:
Nasenbaer: nasty (ups, sorry, I mean fantastic ;) ) puzzle. .. After kneeling into it I found some other things.. (I) broke it down in as many substeps as I could so that it is easier to comment on a specific point
sudokuEd: Wow - still some way to go yet! Might have to resort to SumoCue "hints"
Andrew, in SKX4 thread: I haven't yet solved (it) - at Super Assassin level
Andrew (in 2012): Thanks Ruud for a challenging puzzle. SKX-2 was much harder than the other three SKX puzzles.
When I tried this puzzle again at the weekend, I found that I'd got fairly close to the breakthrough in summer 2008; the next couple of steps were hard to find.
There seems to be a very narrow solving path; I worked in the same areas as Ed although our steps were rather different.
Rating 1.75.
Condensed Walk-through by sudokuEd:
Here is the condensed walk-through. Good Luck! Please let me know if anything needs correcting or clarifying. Ta.

Preliminary steps: n36 10(4) = {1234} -> no {1234} in r1c9; n8 7(2) = {16|25|34}

1. r12c5 = 17 = {89} -> 8,9 locked for n2 and c5

2. r34c7 = 16 = {79} -> 7,9 locked for c7

3. 45 on c89 : r19c7 = 6 = {15|24}, r67c7 = 6 = {15|24}
3a. -> 1,2,4,5 locked for c7 in the two 6(2)
3b. -> r258c7 = {368}

4. 45 on n3 (2 outies, 1 innie) : r4c79 - r2c7 = 3
4a. min. r4c79 = [71] = 8 -> min r2c7 = 5 -> r2c7 = {68} -> r4c79 = 9/11 = [72/74/92]
4b. -> r4c9 = {24}
4c. -> 1,3 required in 10(4) only in n3 -> 1,3 locked for n3
4d. -> no 5 in r9c7 (step 3)

5. 45 on n9 (2 outies, 1 innie) : r6c79 - r8c7 = 10
5a.max. r6c79 = [59] = 14 -> max. r8c7 = 4 -> r8c7 = 3
5b. -> r6c79 = 13 = [49|58] (I mean r6c7 = {45}, r6c9 = {89}
5c. -> r7c7 = {12}

6. n69 : 27(4) = {4689|5679} = 69{48/57} with 6 only in n9 -> 6 locked for n9 and no 9 in r9c9
6a. 27(4): 3 cells in n9 = 6{48/57/49} = [4/7...]

7. n9 : 22(4) = {1489/1579/2578} ({2479} blocked - step 6a)
7a. The only 4 possible in 22(4) is in {1489} combination which also takes 9 -> 9 required for 27(4) cage must be in r6c9 -> 4 must be in r6c7 (step 5b) -> no 4 possible in r9c7 -> no 2 in r1c7 (step 3)
7b. -> {12} locked in r79c7 for n9

8. n6 : 15(4) has 1 and 3 locked for n6 -> {1239|1347|1356} (no 8)

9. 45 on n6 : r456c7 + r46c9 = 30(5) = {24789|25689|45678} = [78429/96528/76548]
9a. however, [96528] means that r1c7 = 4 (since r6c7 = 5), and 4 required in 10(4) in n36 must be in n3 (since r4c9 = 2). But this means two 4's in n3! -> [96548] not possible -> no 9 in r4c7 -> r34c7 = [97]
9b. 15(4) in n6 now {1239/1356} (no 4)

10. n89 : 25(4) now 22(3) = {589|679} = 9{58/67} = [5/7..] (no 1,2,3,4)-> 9 locked for n8 and c6

11. 19(4) in n23 has to have 6 or 8 = {1378/1468/1567/2368/2458/2467}

12. Looking at step 11 another way; r2c7 + r123c6 = [8]{137/146/236/245} or [6]{247} ({157} blocked by r789c6 - step 10) -> r123c6 = [1/2,3/4]

13."45" on n2 -> 2 outies r2c37 - 8 = 1 innie r3c5.
13a.Max r3c5 = 7 -> max. r2c36 = 15 ->when 9 in r2c3, must have 6 in r2c7 and 7 in r3c5. 6 in r2c7 means r123c6 can only be {247} (step 12): but this means two 7's in N2 -> no 9 in r2c3. (note: this move does not eliminate 7 from r3c5 since can have [78] = 15 in r2c37)
13b. when 1 or 2 in r2c3, must have 8 in r2c7 (step 13) -> r2c3=r3c5. Because these two cells are in 2 adjacent nonets, the same digit in N3 must be in r1c789. But there are no 1's available in N3 in these cells -> 1 cannot be in r2c3
13c. Further, since 8 must be in r2c7 when r2c3 = 2 (step 13), the 22(4) in n3 must be {4567} -> no 2 available in r1c789 -> no 2 in r2c3

14. 8 in r3 only in n1 -> 8 locked for n1

15. 17(4) in n12 = {1367/1457/2357/2456} = [1/2, ..] with 1 and 2 only in n2 -> Killer pair with r123c6 (step 12) -> no 1&2 in r3c5

16. min. r3c5 = 3 -> min. r2c27 = 11 (step 13). -> when 3 in r2c3, 8 must be in r2c7 -> 3 must be in r1c789 (like step 13b)- Which it isn't -> no 3 in r2c3

17. 3 in r3c5, r2c37 = 11 = [56] only -> must have {247} in r123c6 (step 12) and r2c3 + r123c4 = [5]{147/237/246} but these all clash with {247} in r123c6 -> no 3 in r3c5

18. "45" on n8 means r8c3 + 1 = r7c5
18a. -> max. r8c3 = 6, min. r7c5 = 2 and no 4 (since no 3 in r8c3)
18b.- 7 in r7c5 -> r8c3 = [6] and r789c6 = {589} -> 15(4) = [6]{234} ({135} blocked by r789c6)
18c.- 6 in r7c5 -> r8c3 = [5] and r789c6 = {589} -> 15(4) = [5]{127} ({136} blocked by 6 in r7c5)
18d.- 5 in r7c5 -> r8c3 = [4] and r789c6 = {679} -> 15(4) = [4]{128} ({137/236} blocked by r789c6)
18e.- 3 in r7c5 -> r8c3 = [2] ->.................................15(4) = [2]{148} ({157} blocked r789c6; {346} blocked by 3 in r7c5)
18f.- 2 in r7c5 -> r8c3 = [1] ->..................................15(4) = [1]{347} ({239/248/257} blocked by 2 in r7c5; {356} blocked by r789c6)
18g. In summary, r789c4 = {127/128/148/234/347} = [1/3,1/4,2/4] (no 5 or 6)

19. r123c4 must have 1/2 (step 15). From steps 18cd, when r7c5 = {56}, r789c4 = {128|127}.but this clashes with r123c4 -> no {56} in r7c5 and no {45} in r8c3
19a. -> 15(4) = {1248|1347|2346} = 4{128/137/236} -> 4 locked in r789c4 for n8 and c4
19b.-> {34} not possible for 7(2) in n8
19c. r789c4 = {148/234/347} = 4{18/23/37} with 4 locked in r789c4 for n8 & c4 = [1/3,3/8...]
19d. r89c5 = {16/25}

20. when 6 in r2c3, r123c4 must be {137}, but this is blocked by r789c4 (step 19c)-> no 6 in r2c3

21. "45" on c12 : r19c3 = 14 = {59}|[68] = [5/6..]

22. "45" on c123 : r258c3 = 12(3) = [{47}1/732/4{26}] ({156} blocked by 14(2) (step 21)) (no 5,8 or 9) -> r2c3 = {47}, r5c3 = {23467}, r8c3 = {126}

23. n2: 2 outies - 8 = 1 innie. -> r2c37 = min. 12 -> The only combinations possible for r2c37 = [48/76/78] = 12,13,15 -> no 6 in r3c5

24. From step 23, the only combination possible for 6 in r2c37 is [76]. When r2c3 = 7, r123c4 = {235} ({136} blocked by r789c4 -step 19c). We know from step 12, that 6 in r2c7 means r123c6 = {247} only. But this means two 2's in n2 when r2c37 = [76]-> no 6 in r2c7 = 8

25.r5c7 = 6 -> r6c9 = 8 (single n6) -> r6c7 = 5 (step 5b)-> r4c9 = 4 -> r7c7 = 1 -> r9c7 = 2 -> r1c7 = 4 -> r2c5 = 9 -> r1c5 = 8

26. n3 : 22(4) = 4{567}

27. n9 : 22(4) = 2{578} (have to use the 8)

28. n69 : -> 27(4) = 8{469}

29. naked pair {69} in r78c9 -> r7c8 = 4, 6 and 9 locked for c9

30. n58 : 19(4) = {2467} only possible combination left
30a.{46} locked in r6c456 for r6 and n5
30b. 2 locked in 19(4) in n58 -> no 2 in r45c5

31. Now those diagonals come in handy: r19c9 = {57} -> {57} can be eliminated from r5c5, r1c1 & r9c1 -> r5c5 = 3

32. "45" on r1234 : r4c28 = 10 = {19}/[82] -> r4c8 = {129} -> r6c8 = 3 (h single n6)

33. 45 on r6789-> r6c2 = 7, r6c1 = 9 (single in r6) -> r7c5 = 7 (single in 19(4)) -> r8c3 = 6 (step 18), r789c6 = {589} (5,8,9 locked for n8, c6), r789c4 = {234} (locked for n8c4), r89c5 = [16]
the rest are singles. Yeah!
Walkthrough by Andrew (finished in 2012):
This is a Killer-X. I’ve had another try at this puzzle in 2012 and re-written some steps in the walkthrough style which I use now. I’d got as far as step 25 in summer 2008.

Prelims

a) R12C5 = {89}
b) R34C3 = {19/28/37/46}, no 5
c) R34C7 = {79}
d) R67C3 = {18/27/36/45}, no 9
e) R67C7 = {15/24}
f) R89C5 = {16/25/34}, no 7
g) 10(4) cage at R2C9 = {1234}
h) 27(4) cage at R6C9 = {3789/4689/5679}

Steps resulting from Prelims

1. Naked pair {89} in R12C5, locked for C5 and N2
1a. Naked pair {79} in R34C7, locked for C7, CPE no 7,9 in R4C6 using D/
1b. Naked quad {1234} in 10(4) cage at R2C9, CPE no 1,2,3,4 in R1C9
1d. 27(4) cage at R6C9 = {3789/4689/5679}, CPE no 9 in R9C9

2. 45 rule on R1234 2 innies R4C28 = 10 = {19/28/37/46}, no 5

3. 45 rule on R6789 2 innies R6C28 = 10 = {19/28/37/46}, no 5

4. 45 rule on C12 2 outies R19C3 = 14 = {59/68}

5. 45 rule on C89 2 outies R19C7 = 6 = {15/24}
5a. Naked quad {1245} in R1679C7, locked for C7

6. 45 rule on N1 2 outies R4C13 = 1 innie R2C3 + 5, IOU no 5 in R4C1
6a. 5 in R4 locked in R4C456, locked for N5 and 19(4) cage at R3C5, no 5 in R3C5

7. 45 rule on N3 2 outies R4C79 = 1 innie R2C7 + 3, IOU no 3 in R4C9 (this IOU still works even though R2C7 and R4C7 already have different candidates)
7a. 3 in 10(4) cage locked in R2C9 + R3C89, locked for N3
7b. R2C7 = {68} -> R4C79 = 9,11 = [72/74/92], no 1 in R4C9
[Or, if preferred, R2C7 is even, R4C7 is odd -> R4C9 must be even = [24] ;-)]
7c. 1 in 10(4) cage locked in R2C9 + R3C89, locked for N3, clean-up: no 5 in R9C7 (step 5)
7d. 1 in C7 only in R79C7, locked for N9

8. 45 rule on N7 2 outies R6C13 = 1 innie R8C3 + 4, IOU no 4 in R6C1

9. 45 rule on N9 2 outies R6C79 = 1 innie R8C7 + 10, max R6C79 = 14 -> no 6,8 in R8C7
9a. R8C7 = 3, R6C79 = 13 = [49/58], clean-up: no 4,5 in R7C7, no 4 in R9C5
9b. R8C7 = 3 -> R789C6 = 22 = {589/679}, 9 locked for C6 and N8

10. 1,3 in N6 locked in 15(4) cage = {1239/1347/1356}, no 8, clean-up: no 2 in R4C2 (step 2), no 2 in R6C2 (step 3)

11. 45 rule on N8 1 innie R7C5 = 1 remaining outie R8C3 + 1, no 1,4 in R7C5, no 7,8,9 in R8C3

12. 27(4) cage at R6C9 = {4689/5679}, 6 locked for N9

13. 22(4) cage at R8C8 = {1489/1579/2578} (cannot be {2479} which clashes with 27(4) cage at R6C9)
13a. 2 of {2578} must be in R9C7, no 2 in R8C8 + R9C89
13b. R79C7 = {12} (hidden pair in N9), locked for C7

14. Grouped X-wing for 4 in 10(4) cage at R2C9 and R789C89, no other 4 in C89, clean-up: no 7 in 15(4) cage at R4C8 (step 10), no 3,6 in R4C2 (step 2), no 3,6 in R6C2 (step 3)

15. R4C7 = 7 (hidden single in N6), R3C7 = 9, placed for D/, clean-up: no 3 in R3C3, no 1 in R4C3, no 3 in R4C8 (step 2)
15a. 8 in R3 locked in R3C123, locked for N1, clean-up: no 6 in R9C3 (step 4)

16. 16(3) cage at R8C2 = {1249/1258/1267/1348/1357/1456/2356} (cannot be {2347} because R9C3 only contains 5,8,9)
16a. 9 of {1249} must be in R9C3 -> no 9 in R9C2

17. 45 rule on R1 3 outies R2C258 = 2 innies R1C46 + 16
17a. Min R1C46 = 3 -> min R2C258 = 19, no 1 in R2C2

18. 45 rule on R12 2 outies R3C46 = 2 innies R2C19 + 7
18a. Max R3C46 = 13 -> max R2C19 = 6, no 6,7,9 in R2C1
18b. Min R2C19 = 3 -> min R3C46 = 10, no 1,2 in R3C46

19. 45 rule on R89 2 innies R8C19 = 2 outies R7C46 + 5
19a. Min R7C46 = 6 -> min R8C19 = 11, no 1 in R8C1
19b. Max R8C19 = 17 -> max R7C46 = 12, no 8 in R7C4

20. R7C5 = R8C3 + 1 (step 11)
20a. 45 rule on N8 4 remaining innies R7C45 + R89C4 = 16 = {1258/1267/1348/2347} (cannot be {1357/1456/2356} which clash with R789C6)
20b. 5 of {1258} must be in R7C5 (R7C5 cannot be 2 because R8C3 = 1 would give two 1s in 15(4) cage at R7C4) -> R8C3 = 4 (step 11) -> no remaining combinations in R89C5 -> cannot be {1258} -> no 5 in R7C45 + R89C4, clean-up: no 4 in R8C3 (step 11)
20c. 3 of {1348} must be in R7C5 -> R8C3 = 2 (step 11) -> no 2 in R8C5 (the other remaining combinations have 2), clean-up: no 5 in R9C5
20d. 6 of {1267} must be in R7C5 (R7C5 cannot be 2 because R8C3 = 1 would give two 1s in 15(4) cage at R7C4, R7C5 cannot be 7 because R8C3 = 6 would give two 6s in 15(4) cage at R7C4) -> no 6 in R789C4

[Alternatively, and possibly a bit simpler …
20. R7C5 = R8C3 + 1 (step 11)
20a. 15(4) cage at R7C4 = {1248/1257/1347/2346} (cannot be {1356} because 1{356} clashes with R789C6, 5{136} clashes with R7C5 IOD clash and 6{135} clashes with R89C5)
20b. 5 of {1257} must be in R8C3 (R789C4 cannot be {157/257} which clash with R789C6), no 5 in R789C4
20c. 4 of {1248} must be in R789C4 (cannot be 4{128} which clashes with R89C5), 4 of {1347/2346} must be in R789C4 (cannot be 4{137/236} which together with R7C5 = 5 clash with R789C6) -> no 4 in R8C3, clean-up: no 5 in R7C5
20d. 6 of {2346} must be in R8C3 (cannot be 2{346} which clashes with R7C5, IOD clash), no 6 in R789C4
20e. 15(4) cage = {1248/1257/2346}, CPE no 2 in R8C5, or 15(4) cage = {1347} = 1{347} => R7C5 = 2
-> no 2 in R8C5, clean-up: no 5 in R9C5]

21. 45 rule on C123 3 innies R258C3 = 12 = {129/138/147/237/246/345} (cannot be {156} which clashes with R19C3)
21a. 5 of {345} must be in R8C3 -> no 5 in R25C3

22. 5 in R4 only in R4C456 -> 19(4) cage at R3C5 = {1459/1567/2359/2458/3457}
22a. 7 of {1567} must be in R3C5 -> no 6 in R3C5

23. 45 rule on N2 2 outies R2C37 = 1 innie R3C5 + 8
23a. R3C5 = {12347} -> R2C37 = 9,10,11,12,15, no 6 in R2C3

24. 45 rule on N1 2 innies R23C3 = 1 outie R4C1 + 5
24a. R23C3 cannot total 14 -> no 9 in R4C1

25. 45 rule on R123 3 remaining outies R4C139 = 1 innie R3C5 + 9
25a. R4C139 cannot total 10 -> no 1 in R3C5
25b. Min R3C5 = 2 -> min R2C37 = 10 (step 23), no 1 in R2C3

26. 19(4) cage at R1C6 = {1378/1468/2368/2458/2467} (cannot be {1567} which clashes with R789C6, cannot be {3457} because R2C7 only contains 6,8)
26a. 17(4) cage at R1C4 = {1367/1457/2357/2456} (cannot be {1259/1349} which clash with 19(4) cage), no 9

[Next I looked at 45 rule on C6789 4(3+1) innies R456C6 + R5C7 = 18, remembering that 6 can be repeated in R46C6 and R5C7, and interactions between these innies and 19(4) cage at R1C6. However this didn’t lead anywhere, so then I had another look at N2 …]

27. 19(4) cage at R1C6 (step 26) = {1378/1468/2368/2458/2467}
[Note. The only combination for 19(4) cage with 6 in R2C7 is {247}6.]
27a. R2C37 = R3C5 + 8 (step 23)
27b. Consider placements for R3C5 = {2347}
R3C5 = 2 => R2C37 = 10 = [28] (cannot be [46] because 19(4) cage = {247}6 clashes with R3C5)
or R3C5 = {347} = 11,12,15 = [38/48/78]
-> R2C7 = 8, R12C5 = [89], R5C7 = 6, clean-up: no 4 in R4C2 (step 2), no 4 in R6C2 (step 3)
[Step 27b is really combination analysis for N2 and its outies, but I’ve written it like a forcing chain for clarity.]

28. R6C9 = 8 (hidden single in N6), clean-up: no 2 in R6C8 (step 3), no 1 in R7C3

29. 45 rule on N3 1 remaining outie R4C9 = 4, R6C7 = 5, R7C7 = 1, placed for D\, R1C7 = 4, R9C7 = 2, clean-up: no 4 in R7C3, no 5 in R8C5
29a. Naked triple {123} in R2C9 + R3C89, locked for N3

30. 5 in N8 only in R789C6 (step 9b) = {589} (only remaining combination), locked for C6 and N8

31. R4C5 = 5 (hidden single in C5)

32. Naked quad {1239} in R3456C8, 9 locked for C8 and N6

33. 27(4) cage at R6C9 = {4689} (only remaining combination) -> R7C8 = 4, R78C9 = {69}, locked for C9

34. 19(4) cage at R3C5 (step 22) = {1459/1567/2359/2458} (cannot be {3457} because 4,7 only in R3C5)
34a. 1 of {1567} must be in R4C6 -> no 6 in R4C6

35. 45 rule on C6789 3 innies R456C6 = 12 = {147/237/246}
35a. 1 of {147} must be in R4C6 -> no 1 in R5C6

36. 19(4) cage at R6C4 = {2467} (only remaining combination), no 1,3, 4 locked for R6 and N5, clean-up: no 5 in R7C3
36a. 19(4) cage at R6C4 = {2467}, CPE no 2,7 in R5C5 -> R5C5 = 3, placed for both diagonals, clean-up: no 6 in R6C3, no 4 in R8C5

37. Naked pair {16} in R89C5, locked for C5 and N8

38. Naked quad {2467} in R5C6 + R6C456, locked for N5, 6 also locked for R6 -> R4C6 = 1, placed for D/

39. R4C6 -> R456C6 (step 35) = {147} (only remaining combination) -> R6C6 = 4, placed for D\, R5C6 = 7, R67C5 = [27], R6C4 = 6, placed for D/, R3C5 = 4, R4C4 = 9 (step 34), placed for D\, R5C4 = 8, R4C28 = [82], R5C89 = [91], R6C8 = 3, R6C2 = 7 (step 3), R6C3 = 1, R7C3 = 8, R3C8 = 1, R6C1 = 9

40. R9C3 = 9 (hidden single in N7), R1C3 = 5 (step 4), R1C9 = 7, R2C8 = 5, both placed for D/

and the rest is naked singles without using the diagonals.

I'll rate my walkthrough for SKX2 at 1.75, based on the combination analysis in steps 20 and 26 plus the implied analysis in step 27.


Last edited by Ed on Wed Nov 04, 2009 8:48 pm, edited 4 times in total.

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PostPosted: Sat Jun 07, 2008 12:23 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1043
Location: Sydney, Australia
Assassin 23 by Ruud (Nov 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:7936:7936:4098:4098:5380:4613:3846:3846:3846:5641:7936:7936:4098:5380:4613:4613:4613:3089:5641:5641:7936:4098:5380:5380:4376:4376:3089:2075:5641:5149:2846:4127:4127:4127:4376:3089:2075:2075:5149:2846:2846:3625:3625:6699:6699:4141:5149:5149:6448:6448:3625:3123:6699:6699:4141:4141:6968:6968:6448:3643:3123:3123:3134:3647:3647:6968:6968:6448:3643:3123:4678:3134:3647:3647:1866:1866:3404:3404:3404:4678:4678:
Solution:
+-------+-------+-------+
| 9 8 1 | 5 4 3 | 7 6 2 |
| 3 7 2 | 8 9 6 | 5 4 1 |
| 6 4 5 | 2 7 1 | 9 3 8 |
+-------+-------+-------+
| 1 9 4 | 7 6 8 | 2 5 3 |
| 2 5 7 | 3 1 4 | 8 9 6 |
| 8 6 3 | 9 5 2 | 1 7 4 |
+-------+-------+-------+
| 7 1 8 | 4 3 9 | 6 2 5 |
| 4 2 9 | 6 8 5 | 3 1 7 |
| 5 3 6 | 1 2 7 | 4 8 9 |
+-------+-------+-------+
Quote:
nd: Mostly 45-rule work
Walkthrough by nd:
Mostly 45-rule work.

1. 45 rule on N1 -> R4C2 - R1C3 = 8 -> R4C2 = 9, R1C3 = 1. 45 rule on N4 -> R6C1 = 8. 45 rule on N7 -> R789C3 = 23 = {689} -> R9C34 = [61], R78C3 = {89}, R78C4 = {37|46}, R1C1 = 9.

2. 45 rule on C4 -> R6C4 - R5C5 = 8 -> R5C5 = 1, R6C4 = 9, R4C1 = 1, R6C7 = 1, R8C8 = 1, R9C89 = {89}, R78C6 = {59}. 45 rule on N789 -> R6C5 = 5.

3. Mop-up. 45 rule on R9 -> R8C12 = 6 = {24}, R9C12 = {35}. 45 rule on N9 -> R9C7 = 4, R78C9 = {57}, remaining cells of 12(4) cage = {236}, R9C56 = {27}, R78C4 = [46], R8C7 = 3, R7C12 = [71], R78C9 = [57], R78C5 = [38], R78C3 = [89], R78C6 = [95]. 45 rule on N236 -> R45C7 = 10 = {28}, R7C78 = [62]. 45 rule on N6 -> R4C89 = 8 = [53], R23C9 = {18}, R9C89 = [89], R5C8 = 9, R6C8 = 7, R1C9 = 2, R3C78 = [93]. 45 rule on R4 -> R4C34 = 11 = [47], R5C4 = 3, R5C12 = {25}, R45C7 = [28], R4C56 = [68], R56C6 = [42], and you carry on....


Last edited by Ed on Sat Jun 14, 2008 1:17 am, edited 1 time in total.

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PostPosted: Sat Jun 07, 2008 12:27 pm 
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Location: Sydney, Australia
Assassin 24 Tetris by Ruud (Nov 06)
Puzzle pic:
Image
Code: Select, Copy & Paste into solver:
3x3::k:4352:4352:3842:3842:3332:3845:3845:6919:6919:6921:4352:4352:3842:3332:3845:6919:6919:4625:6921:6921:4116:3842:5398:3845:5144:4625:4625:5147:6921:4116:5398:5398:5398:5144:4625:5923:5147:5147:4116:4116:2600:5144:5144:5923:5923:5677:5147:4399:4399:2600:5938:5938:5923:4405:5677:4399:4399:2361:2600:2363:5938:5938:4405:5677:5677:6209:2361:5699:2363:5189:4405:4405:6209:6209:6209:5699:5699:5699:5189:5189:5189:
Solution:
+-------+-------+-------+
| 5 2 6 | 3 7 4 | 1 8 9 |
| 8 1 9 | 5 6 2 | 3 7 4 |
| 3 7 4 | 1 9 8 | 2 5 6 |
+-------+-------+-------+
| 2 9 5 | 4 1 7 | 6 3 8 |
| 4 8 1 | 6 3 5 | 7 9 2 |
| 7 6 3 | 8 2 9 | 5 4 1 |
+-------+-------+-------+
| 9 4 2 | 7 5 6 | 8 1 3 |
| 1 5 8 | 2 4 3 | 9 6 7 |
| 6 3 7 | 9 8 1 | 4 2 5 |
+-------+-------+-------+
Quote:
Ruud: this Tetris puzzle came out just right. Combined, SumoCue and SudoCue needed more than 160 solving steps (including singles). This Killer is an Assassin and a Nightmare
Andrew: was certainly a challenge. ... it being harder than any previous Assassin
nd: Haven't posted a walkthrough because I couldn't produce one that seemed very satisfactory (i.e. not using ugly T&E steps). Time for me to brush up my midrange-to-advanced techniques in "regular" sudoku, I think...
PsyMar in A25 thread: I tried (A24).. and couldn't get much of anywhere
Andrew, in A55 thread: A24 was the first really difficult one
Andrew: When I first solved this puzzle I wasn't satisfied with how I had completed it so..I'll rate my new walkthrough for A24 at Hard 1.25
Marks pics and solver steps for xyz-Wing and als(xz) by Ruud:
Tall buildings often do not have a 13th floor. I wonder why...

Andrew wrote:
What new techniques were used for this puzzle apart from it being harder than any previous Assassin?


My current - under development - version of SumoCue does all fish-type moves, and also Law of Leftovers for Jigsaws and Jigsaw Killers, which is similar to fish.

Even with these additions, it cannot solve Tetris completely. When it got stuck, I transferred this candidate grid to SudoCue:

Code:
.---------------------.---------------------.----------.---------------------.---------------------.
|(17)                 |(15)                 |(13)      |(15)                 |(27)                 |
| 234589     23458    | 6          35       | 7        | 24         1        | 89         389      |
:----------.          '----------.          |          |          .----------'          .----------:
|(27)      |                     |          |          |          |                     |(18)      |
| 89       | 1234       589      | 135      | 6        | 24       | 379        789      | 45       |
|          '----------.----------:          :----------:          :----------.----------'          |
|                     |(16)      |          |(21)      |          |(20)      |                     |
| 357        357      | 14       | 13       | 9        | 8        | 2        | 456        456      |
:----------.          |          :----------'          '----------:          |          .----------:
|(20)      |          |          |                                |          |          |(23)      |
| 2568     | 69       | 25       | 4          1          7        | 569      | 3        | 25689    |
|          '----------:          '----------.----------.----------'          :----------'          |
|                     |                     |(10)      |                     |                     |
| 1234578    12345789 | 147        6        | 23       | 59         47       | 245789     245789   |
:----------.          :---------------------:          :---------------------:          .----------:
|(22)      |          |(17)                 |          |(23)                 |          |(17)      |
| 67       | 23467    | 34         8        | 23       | 59         59       | 2467     | 1        |
|          :----------'          .----------:          :----------.          '----------:          |
|          |                     |(9)       |          |(9)       |                     |          |
| 469      | 24         234      | 7        | 5        | 36       | 8          1        | 3469     |
|          '----------.----------:          :----------:          :----------.----------'          |
|                     |(24)      |          |(22)      |          |(20)      |                     |
| 145678     145678   | 789      | 2        | 48       | 36       | 469      | 45679      345679   |
:---------------------'          :----------'          '----------:          '---------------------:
|                                |                                |                                |
| 345678     345678     578      | 9          48         1        | 34567      24567      234567   |
'--------------------------------'--------------------------------'--------------------------------'


The first move it found was an XYZ-Wing:

Code:
.---------------------------.---------------------------.---------------------------.
| 234589   23458    6       | 35       7        24      | 1        89       389     |
| 89       1234     589     | 135      6        24      | 379      789      45      |
| 357      357      14      | 13       9        8       | 2        456      456     |
:---------------------------+---------------------------+---------------------------:
| 2568    #69       25      | 4        1        7       |*569      3        2568-9  |
| 1234578  12345789 147     | 6        23       59      | 47       245789   245789  |
| 67       23467    34      | 8        23       59      |#59       2467     1       |
:---------------------------+---------------------------+---------------------------:
| 469      24       234     | 7        5        36      | 8        1        3469    |
| 145678   145678   789     | 2        48       36      | 469      45679    345679  |
| 345678   345678   578     | 9        48       1       | 34567    24567    234567  |
'---------------------------'---------------------------'---------------------------'


Then an ALS-XZ move:
Code:
A=r2c13789
B=r3c1289
X=4
Z=3
r2c2<>3


This enabled a second XYZ-Wing:

Code:
.---------------------------.---------------------------.---------------------------.
| 234589   23-458   6       | 35       7        24      | 1        89       389     |
| 89      *124      589     | 135      6        24      | 379      789      45      |
| 357      357     #14      | 13       9        8       | 2        456      456     |
:---------------------------+---------------------------+---------------------------:
| 2568     69       25      | 4        1        7       | 569      3        2568    |
| 1234578  12345789 147     | 6        23       59      | 47       245789   245789  |
| 67       23467    34      | 8        23       59      | 59       2467     1       |
:---------------------------+---------------------------+---------------------------:
| 469     #24       234     | 7        5        36      | 8        1        3469    |
| 145678   145678   789     | 2        48       36      | 469      45679    345679  |
| 345678   345678   578     | 9        48       1       | 34567    24567    234567  |
'---------------------------'---------------------------'---------------------------'

Followed by a second ALS-XZ move:
Code:
A=r2c269
B=r3467c3
X=1
Z=5
r2c3<>5, r4c9<>5


After this move, a pair {89} in N1/R2 breaks the puzzle.

We may not need the second XYZ-Wing, but the 3 other steps must be executed in sequence to isolate the pair.

Ruud
Walk-through by sudokuEd:
PsyMar wrote:
This was also an Agatean curse, in the Discworld series.
Terry Pratchet wrote:
This is the strongest curse that the people of the ‘Counterweight Continent’ can muster.
Very perceptive!

Well, here is the original solving for this puzzle - before those thought fairies took over! Hopefully, its valid this time. Good luck - not easy reading [edit: but a bit easier after very helpful suggestions from Andrew. Thanks].

1. "45" n8: 1 innie r7c5 = 5, both 9(2) cages in n8 = {18/27/36} (no 4), 22(4) cage = 49{18/27/36}. r56c5 = 5 = {14/23}

2. "45" c6789: 2 innies r49c6 = 8 = {17/26/35} -> r9c6 = {12367}, r4c6 = {12567}

3. "45" c1234: 2 innies r49c4 = 13 = {49/58/67} -> r9c4 = {46789}, r4c4 = {45679}

4. "45" n369: 2 outies - 13 = 1 innie -> min. r56c6 = 14 -> no 1,2,3,4. Max r56c6 = 17 -> max. r1c7 = 4

5. 4 in c6 now only in n2 -> 4 locked for n2 and no 4 in r1c7 -> 13(2) n2 = {67} locked for n2, c5 -> 15(4) in n23 = 4{128} (no 3,5,9) -> 8 locked for n2,c6. Also, 5 in n2 locked in 15(4) in n12: 5 locked for c4 -> no 5 in r1c3 and 15(4) n12 = 5{127/136} = 15{27/36} (no 4,8,9) -> r3c5 = 9 (single n2)

6."45" n2: 2 outies = 7 -> r1c37 = [61]. r12c5 = [76]

7. r123c4 = {135}: locked for c4-> r78c4 = {27}: locked for c4,n8. r123c6 = {248}: locked for c6 -> r78c6 = {36}: locked for c6, n8. r9c6 = 1, r9c4 = 9 (single n8), r89c5 = {48}: locked for c5 -> r56c5 = {23} -> r4c5 = 1. r4c46 = [47] (step 2,3). r56c6 = {59}, r56c4 = {68}

8. 27(4) n3 now {3789}:locked for n3 with 7 locked for r2 -> 18(4) n36 {3456} only ({2349/2358} have 3,8/9 only in r4c8) -> r4c8 = 3 -> now 15(3) n3 = {456}:locked for n3 -> r3c7 = 2

9. "45" n1: 1 outie - 5 = 1 innie -> r4c2 = {689}, r3c3 = {134}. 7 in n1 locked in 27(4) = 79{38/56} (no 1,2,4) and no 9 in r12c2
9a. When 8 in r4c2, must have 3 in r3c3. But the only combo with 8 in 27(4) n12 is {3789} -> two 3's in n1 -> no 8 in r4c2, no 3 in r3c3

10. "45" n1: 4 outies = 21. Min. r4c2 + r5c4 = {66} = 12 -> max. r45c3 = 9 -> no 9 in r4c3, no 8,9 in r5c3
10a. When r45c3 max. = 9, [81] is blocked by 1 in r3c3 (since 6 in r4c2 - step 9) -> no 8 can be in r4c3 = {25}

11. "45" c12 -> 3 outies - 20 = 1 innie -> min. r289c3 = 21 -> no 123. Max. 3 outies = 24 -> max. r7c2 = 4
11a. r289c3 = {489/579/589/789} = 9{48/57/58/78} = [5/8,7/8] = 21, 22 or 24 -> no 3 in r7c2 and 9 locked for c3

12. 16(4) in n145 must have 6/8 (r5c4) and 1/4 (r3c3) and 2/5 (r4c3) = {1258/1267/1456} = 1{258/267/456} with 1 locked for c3, no 3 in r5c3

13. Another way of looking at step 12 is at the combinations available in r345c3 (remembering [6/8] only in r5c4) = {125/127/145} = [5/7]
13a. ->{579} combo. blocked from r289c3 = {489/589/789} = 89{4/5/7} -> 8 locked for c3
13b. r345c3 = [125/152/127/154/451]
13c. another way to see 13b.and step 9: r4c2 + r45c3 = [625/652/627/654/951] = [5/7]: rest of n4 has 5/7 but not both

14. 3 in c3 now locked in 17(4) n547 and must have 6/8 (r6c4) = {1358/1367/2348}

15. "45" n7: r6c1 - 1 = r7c23. Min r7c23 = {12} = 3 -> min r6c1 = 4

16. 17(4) n547: {1358} combo, must have r7c23 = {13} = 4 But this means 5 must be in both r6c1 (step 15) and r6c3 -> no {1358} combo
16a. 17(4) = {1367/2348} (no 5)
16b. r6c34 = [36/76/28/38/48]

17. "45" n7: 3 outies = 18. From 16b. r6c134 = [936/576/828/738/648] ([828] not valid) -> no 4,or 8 in r6c1, no 2 in r6c3
17a. r6c13 = [93/57/73/64]. But from 13c. [57] is blocked -> no 5 in r6c1, no 7 in r6c3

We come now to the key moves to unlock this puzzle. Hopefully its neat and valid this time!
18. "45" r6789 3 innies r6c258 = 12.
18a. r6c5 = {23} -> r6c28 = 9/10. When 5 in either r6c28 -> r6c28 can only = 9 (can't have {55} = 10} -> must have 3 in r6c5 and {45} in r6c28 = {345}. But this is blocked by r6c3 -> no 5 in r6c28

19. "45" n9: 1 outie + 8 = 2 innies
19a. 5 in r6c9 -> r7c78 = 13. -> other 2 cells in 23(4) in n569 must = 10. But this is not possible since 9 must be in r6c6 when r6c9 = 5, and no 1 is available in r6c7 -> no 5 in r6c9

20. 5 in r6 now locked in r6c67 in 23(4) = 5{189/279/369/378/468}

21. "45" n9: 3 outies r6c679 = 15 and must have 5 = [546/564/582/591/951]. r6c7 = {45689}, r6c9 = {1246}

22. When r6c134 = [936] (step 17), r6c5 = 2, however, [9362] is blocked by r6c679 (step 21) -> no 9 in r6c1 = {67}

23. now-from step 13c. r4c2 + r45c3 = [625/652/654/951] ([627] blocked by r6c1 -step 22) -> no 7 in r5c3

24. 16(4) in n145 now = {1258/1456} = 15{28/46} -> 5 locked for c3 and n4

25. from step 13a. r289c3 now {489/789} = 89{4/7} = 21/24 -> no 2 in r7c2 (step 11)

26. from step 16a. 17(4) n547 = {1367/2348} = 3{167/248} -> r7c3 = {27}(only cell in this cage with 2 or 7), {27} locked for r7 in r7c34 and r6c3 = 3 (only 3 left!)

27. n7 2 outies r6c14 = 15 = [78], r7c23 = [42], r78c4 = [72], r5c4 = 6, r4c3 = 5, r35c3 = {14}, 4 in n4 locked in r5

28. 20(4) n365 now 18(3) = {567} -> r56c6 = [59], r456c7 = [675], 23(4) n6 = {2489} -> r6c89 = [41], {36} blocked from r7c78 (r7c6) -> = [81], r4c2 = 9 -> r3c3= 4 (step 9), rest of 27 (4) in n14 = {378} -> r3c2 = 7, r23c1 = {38}.... the rest are singles
April '10 Walkthrough by Andrew:
When I first solved this puzzle I wasn't satisfied with how I had completed it so had kept it in my Unfinished folder to have another go at it. Now that I'm working fairly regularly at puzzles in that folder I've had another go at it.

I'll rate my new walkthrough for A24 at Hard 1.25 because of the combination analysis I used in steps 33 and 34.

Here is my new walkthrough for A24.

Prelims

a) R12C5 = {49/58/67}, no 1,2,3
b) R78C4 = {18/27/36/45}, no 9
c) R78C6 = {18/27/36/45}, no 9
d) 10(3) cage at R5C5 = {127/136/145/235}, no 8,9
e) 27(4) cage at R2C1 = {3789/4689/5679}, no 1,2
f) 27(4) cage at R1C8 = {3789/4689/5679}, no 1,2

1. 45 rule on N8 1 innie R7C5 = 5, R56C5 = 5 = {14/23}, clean-up: no 8 in R12C5, no 4 in R78C4, no 4 in R78C6

2. 45 rule on R9 3 outies R8C357 = 21 = {489/579/678}, no 1,2,3

3. 45 rule on C1234 2 innies R49C4 = 13 = {49/67}[58], no 1,2,3, no 8 in R4C4

4. 45 rule on C6789 2 innies R49C6 = 8 = {17/26}[53], no 4,8,9, no 3 in R4C6

5. 4,9 in N8 only in R8C5 + R9C45 -> R89C5 must contain at least one of 4,9
5a. R12C5 = {67} (only remaining combination, cannot be {49} which clashes with R89C5), locked for C5 and N2

6. 45 rule on N2 1 innie R3C5 = 2 outies R1C37 + 2
6a. Min R1C37 = 3 -> min R3C5 = 5 -> R3C5 = {89}
6b. Max R1C37 = 7, no 7,8,9 in R1C37

7. 4,9 in N8 only in 22(4) cage = {1489/2479/3469}
7a. 1 of {1489} must be in R9C6 -> no 1 in R9C5
7b. 1 in C5 only in R456C5, locked for N5, clean-up: no 7 in R9C6 (step 4)
7c. 2 of {2479} must be in R9C6 -> no 2 in R9C5
7d. 2 in C5 only in R456C5, locked for N5, clean-up: no 6 in R9C6 (step 4)
7e. 3 of {3469} must be in R9C6 -> no 3 in R9C5
7f. 3 in C5 only in R456C5, locked for N5

8. Naked triple {489} in R389C5, locked for C5, 4 also locked for N8, clean-up: no 9 in R4C4 (step 3), no 1 in R56C5
8a. R4C5 = 1 (hidden single in C5)

9. 45 rule on C789 2 outies R56C6 = 1 innie R1C7 + 13
9a. Min R56C6 = 14, no 4
9b. Max R56C6 = 17 -> max R1C7 = 4

10. 4 in C6 only in R123C6, locked for N2 and 15(4) cage at R1C6, no 4 in R1C7
10a. 15(4) cage at R1C6 must contain 4 = {1248} (only remaining combination), 8 locked for C6 and N2 -> R3C5 = 9, CPE no 1 in R1C4, clean-up: no 1 in R78C6

11. Naked pair {48} in R89C5, locked for N8, clean-up: no 5 in R4C4 (step 3), no 1 in R78C4
11a. Naked quad {2367} in R78C46, locked for N8 -> R9C4 = 9, R4C4 = 4 (step 3), R9C6 = 1, R4C6 = 7 (step 4), clean-up: no 2 in R78C6
11b. Naked pair {36} in R78C6, locked for C6 and N8

12. Naked triple {248} in R123C6, locked for N2 and 15(4) cage at R1C6 -> R1C7 = 1
12a. Naked triple {135} in R123C4, locked for C4
12b. R123C4 = {135} = 9 -> R1C3 = 6 (cage sum), R12C5 = [76]

13. 27(4) cage at R1C8 = {3789} (only remaining combination), locked for N3, 7 also locked for R2

14. 45 rule on N1 1 outie R4C2 = 1 remaining innie R3C3 + 5, no 2,5,7,8 in R3C3, no 3,5 in R4C2

15. 45 rule on N3 1 outie R4C8 = 1 remaining innie R3C7 + 1, no 6 in R3C7, no 2,8,9 in R4C8

16. 6 in R3 only in R3C89 -> 18(4) cage at R2C9 = {3456} (only remaining combination) -> R4C8 = 3, R3C7 = 2 (step 15)

17. 20(4) cage at R3C7 = {2459/2567} (cannot be {2468} because R5C6 only contains 5,9), no 8
17a. 4,7 only in R5C7 -> R5C7 = {47}

18. R8C357 (step 2) = {489/678} (cannot be {579} because R8C5 only contains 4,8), no 5, 8 locked for R8
18a. 6 of {678} must be in R8C7 -> no 7 in R8C7

19. 45 rule on C12 3 innies R7C2 + R9C12 = 1 outie R2C3 + 4
19a. Min R7C2 + R9C12 = 6 -> min R2C3 = 2

20. 45 rule on C12 3 outies R289C3 = 1 innie R7C2 + 20
[This was the innie-outie difference which I missed when I originally tried this puzzle. Step 19 isn’t necessary but I’ve kept it in because I think it’s the more obvious of the 45s on C12.]
20a. Min R289C3 = 21, no 2,3 in R29C3
20b. Max R289C3 = 24 -> max R7C2 = 4
20c. R289C3 = 21..24 = {489/579/589/789}, 9 locked for C3
20d. R289C3 cannot be 23 -> no 3 in R7C2

21. 16(4) cage at R3C3 = {1258/1267/1348/1456/2356} (cannot be {1357/2347} because R5C4 only contains 6,8)
21a. R5C4 = {68} -> no 8 in R45C3
21b. 16(4) cage at R3C3 = {1258/1267/1456/2356} (cannot be {1348} because R4C3 only contains 2,5)
21c. 3 of {2356} must be in R3C3 -> no 3 in R5C3

22. R289C3 (step 20c) = {489/589/789} (cannot be {579} which clashes with 16(4) cage at R3C3), 8 locked for C4

23. 45 rule on N6 4 remaining innies R456C7 + R6C9 = 19 = {1459/1468/1567/2458/2467} (cannot be {1279} because 1,2 only in R6C9)
23a. 1,2 only in R6C9 -> R6C9 = {12}

24. 45 rule on N9 3 outies R6C679 = 15 = {159/249/258} (cannot be {168} which clashes with R6C4, cannot be {267} because R6C6 only contains 5,9, cannot be {456} because R6C9 only contains 1,2), no 6,7

25. 45 rule on N9 2 innies R7C78 = 1 outie R6C9 + 8
25a. Max R6C9 = 2 -> max R7C78 = 10, no 8,9 in R7C8

26. 45 rule on R6789 3 innies R6C258 = 12 = {138/237/246} (cannot be {129} which clashes with R6C9, cannot be {147/156} because R6C5 only contains 2,3, cannot be {345} which clashes with R6C679), no 5,9
26a. Killer pair 1,2 in R6C258 and R6C9, locked for R6
26b. R6C679 (step 24) = {159/249} (cannot be {258} which clashes with R6C258), no 8, 9 locked for R6 and 23(4) cage at R6C6, no 9 in R7C7

27. 45 rule on N7 3 outies R6C134 = 18 = {378/468/567}
27a. 4 of {468} must be in R6C3 -> no 4 in R6C1

28. 45 rule on N7 1 outie R6C1 = 2 innies R7C23 + 1
28a. Min R7C23 = 3 -> no 3 in R6C1
28b. Max R6C1 = 8 -> max R7C23 = 7, no 7 in R7C3

29. 23(4) cage at R6C6 must contain 9 = {1589/2489/3479} (cannot be {1679} because 1,6,7 only in R7C78, cannot be {2579} which clashes with R7C4, cannot be {3569} which clashes with R7C6), no 6
29a. 4,9 of {2489/3479} must be in R6C67 -> no 4 in R7C78
29b. 3 of {3479} must be in R7C7 -> no 7 in R7C7

30. 7 in R3 only in R3C12 -> 27(4) cage at R2C1 = {3789/5679}, no 4, CPE no 9 in R12C2 + R45C1
30a. 6,9 of {5679} must be in R2C1 + R4C2 -> no 5 in R2C1
30b. 9 in N4 only in R45C2, locked for C2

31. 45 rule on N1 4 innies R2C1 + R3C123 = 22 (must contain 7) = {1579/3478}
31a. 1,4 only in R3C3 -> R3C3 = {14}, clean-up: no 8 in R4C2 (step 14)
31b. 17(4) cage in N1 = {1259/2348} (cannot be {1349/1358} which clash with R2C1 + R3C123)
31c. 1 of {1259} must be in R2C2 -> no 5 in R2C2

32. 16(4) cage at R3C3 (step 21b) = {1258/1267/1456}, 1 locked for C3

33. R6C134 (step 27) = {378/468/567}
33a. 3 in C3 only in R67C3 -> 17(4) cage at R6C3 = {1367/2348} (cannot be {2357} because R6C4 only contains 6,8, cannot be {1358} which clashes with the combinations for R6C134), no 5
33b. R6C34 = [38/48/76] -> R6C134 = [738/648/576], no 8 in R6C1
[Alternatively R7C23 cannot be [43] (because 2 of 17(4) cage at R6C3 because 2 of {2348} must be in R7C23) -> max R7C23 = 6 -> max R6C1 = 7]

34. R6C134 (step 33b) = [738/648/576] -> R6C13 = [57/64/73]
34a. 8 in N4 only in 20(4) cage = {1289/1568/2378/2468} (cannot be {1478/3458} which clash with R6C13)
34b. Killer pair 2,5 in 20(4) cage and R4C3, locked for N4

35. R6C134 (step 33b) = [738/648] -> R6C4 = 8, R6C3 = {34}, R5C4 = 6

36. R6C258 (step 26) = {237/246}, 2 locked for R6 -> R6C9 = 1
36a. Killer pair 3,4 in R6C258 and R6C3, locked for R6

37. 23(4) cage at R6C6 (step 29) = {1589} (only remaining combination) -> R7C78 = [81]

38. Naked triple {234} in R6C3 + R7C23, 2 locked for R7 and N7, CPE no 4 in R89C3

39. 45 rule on N4 5 innies R4C23 + R5C3 + R6C13 = 25 = {13579/14569/23479/34567} (cannot be {12679/23569} which clash with 20(4) cage)
39a. 1 of {13579} must be in R5C3, 3,4 of {23479/34567} must be in R56C3 -> no 7 in R5C3

40. Naked pair {14} in R35C3, locked for C3 -> R67C3 = [32], R4C3 = 5, R7C2 = 4, R6C1 = 7 (step 34), R56C5 = [32], R6C2 = 6, R6C8 = 4, R4C2 = 9, R3C3 = 4 (step 14), R3C6 = 8, R5C3 = 1, R45C7 = [67], R6C5 = 5 (step 17), R6C67 = [95], R78C4 = [72]

41. Naked pair {56} in R3C89, locked for R3 and N3 -> R2C9 = 4, R3C12 = [37], R2C1 = 8, R2C3 = 9, R4C1 = 2, R5C12 = [48], R4C9 = 8, R2C7 = 3, R1C9 = 9, R12C8 = [87], R12C6 = [42], R2C2 = 1

42. Naked pair {35} in R89C2, locked for C2 and N7 -> R9C1 = 6, R7C1 = 9, R8C1 = 1, R8C2 = 5 (cage sum)

and the rest is naked singles.

Unfortunately because of a re-work with simpler steps, after I realised that I’d missed some eliminations, I was unable to use the interesting
45 rule on R12 2 innies R2C19 = 2 outies R3C46 + 3
R3C4 is odd, R3C6 is even -> R3C46 must be odd -> R2C19 must be even -> no 8 in R2C9 (R2C19 cannot be [88] and all other combinations for R2C19 with 8 in R2C9 are odd)


Last edited by Ed on Sun Apr 11, 2010 8:22 am, edited 2 times in total.

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