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Ed
 Post subject: Assassin 207
PostPosted: Thu Mar 03, 2011 10:02 pm 
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Hard to believe it's now three years since Ruud posted his last Assassin. This one is worthy of his legacy except it is not symmetrical. Found a fun way through this one with two key moves. Not complicated but both are a twist on the simplest version of that type of move. Presumably that means there are other ways to solve this. Good luck!

Assassin 207
Image
code: copy and paste into solver:
3x3::k:3585:3585:3585:5634:5634:5634:3843:3076:3076:1029:1029:2566:2566:5634:7175:3843:3076:3336:4105:9738:9738:2566:7175:7175:7175:4363:3336:4105:4105:9738:2828:2828:7175:4363:4363:11533:2830:2830:9738:9738:2063:2063:4363:11533:11533:5136:5136:9738:2833:2833:5906:11533:11533:11533:5136:9738:9738:4627:5906:5906:5906:11533:11533:3860:3860:4627:4627:3349:5906:3350:3863:11533:3352:3352:3352:3349:3349:3349:3350:3863:3863:
solution: 
+-------+-------+-------+
| 8 2 4 | 3 6 9 | 7 1 5 |
| 3 1 7 | 2 4 5 | 8 6 9 |
| 6 5 9 | 1 8 7 | 2 3 4 |
+-------+-------+-------+
| 1 9 2 | 4 7 6 | 5 8 3 |
| 4 7 6 | 8 5 3 | 1 9 2 |
| 5 8 3 | 9 2 1 | 6 4 7 |
+-------+-------+-------+
| 7 4 1 | 6 9 2 | 3 5 8 |
| 9 6 5 | 7 3 8 | 4 2 1 |
| 2 3 8 | 5 1 4 | 9 7 6 |
+-------+-------+-------+
Cheers
Ed
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Andrew
 Post subject: Re: Assassin 207
PostPosted: Sun Mar 06, 2011 6:04 am 
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Thanks Ed for a challenging Assassin! I've no idea whether the steps I used are the ones you referred to but my first key move is certainly a twist on a known technique. I found it harder than your recent Assassins so maybe I missed something.

My walkthrough would have been shorter if I hadn't missed one important step when it first became available; however I didn't feel that it was an obvious step (some may disagree with that) so I haven't re-worked my steps but only made comments about it.

Many thanks to Ed for the comments and one correction, all given in blue in my walkthrough; I've also added a general comment at the end.

Ed told me that his solving method was very different. I'll be interested to see how you did it.

Rating Comment:
It was hard to decide what rating to give my walkthrough for A207; I'll go for Hard 1.5. I used a cloned cage, built up from a single clone plus LOL; also a very short forcing chain and a couple of combo blockers.

Here is my walkthrough for A207:
Prelims

a) R12C7 = {69/78}
b) R2C12 = {13}
c) R23C9 = {49/58/67}, no 1,2,3
d) R4C56 = {29/38/47/56}, no 1
e) R5C12 = {29/38/47/56}, no 1
f) R5C56 = {17/26/35}, no 4,8,9
g) R6C45 = {29/38/47/56}, no 1
h) R8C12 = {69/78}
i) R89C7 = {49/58/67}
j) 10(3) cage at R2C3 = {127/136/145/235}, no 8,9
k) 20(3) cage at R6C1 = {389/479/569/578}, no 1,2
l) 13(4) cage in N8 = {1237/1246/1345}, no 8,9
m) 38(8) cage at R3C2 = {12345689}, no 7
n) And of course 45(9) cage at R4C9 = {123456789}

Steps resulting from Prelims
1a. Naked pair {13} in R2C12, locked for R2 and N1
1b. R23C9 = {49/58} (cannot be {67} which clashes with R12C7), no 6,7
1c. R89C7 = {49/58} (cannot be {67} which clashes with R12C7), no 6,7
1d. Killer pair 8,9 in R12C7 and R23C9, locked for N3
1e. Killer pair 8,9 in R12C7 and R89C7, locked for C7
1f. 13(4) cage in N8 = {1237/1246/1345}, 1 locked for N8

2. 45 rule on N3 2 innies R3C78 = 5 = {14/23}

3. 10(3) cage at R2C3 = {127/145/235} (cannot be {136} because 1,3 only in R3C4), no 6
3a. 1,3 only in R3C4 -> R3C4 = {13}
3b. Killer pair 1,3 in R3C4 and R3C78, locked for R3

4. 45 rule on C789 2 innies R37C7 = 5 = {14/23}

5. 14(3) cage in N1 = {248/257}, no 6,9, 2 locked for R1 and N1
5a. 6,9 in N1 only in R3C123, locked for R3, clean-up: no 4 in R2C9
5b. 10(3) cage at R2C3 (step 3) = {127/145/235}
5c. 2 of {127} must be in R2C4 -> no 7 in R2C4

6. 45 rule on R9 3 outies R8C578 = 9 = {135/234} (cannot be {126} because R8C7 only contains 4,5), 3 locked for R8, clean-up: no 4,5 in R9C7
6a. R8C7 = {45} -> no 4,5 in R8C58

7. 15(3) cage in N9 can only have one of 1,2,3 in R8C8 -> no 1,2,3 in R9C89

8. 2 in C9 only in R45678C9, locked for 45(9) cage at R4C9 -> no 2 in R567C8 + R6C7

9. 45 rule on N1 4 innies R2C3 + R3C123 = 27 = {4689/5679}
9a. 4 of {4689} must be in R2C3 -> no 4 in R3C123

10. 45 rule on N2 3(1+2) outies R2C3 + R3C7 + R4C6 = 15
10a. Max R2C3 = 7 -> min R3C7 + R4C6 = 8 -> min R4C6 = 5 (because R3C7 + R4C6 cannot be [44])

11. 45 rule on N8 3(1+2) outies R6C6 + R7C7 + R8C3 = 9
11a. Min R6C6 + R7C7 = 3 -> max R8C3 = 6
11b. Max R6C6 + R7C7 = 8 -> max R6C6 = 7
11c. Max R8C3 = 6 -> min R78C4 = 12, no 2 in R78C4

12. 45 rule on R12 2 innies R2C69 = 1 outie R3C4 + 13
12a. R3C4 = {13} -> R2C69 = 14,16 = {59}/[68/79], no 2,4, no 8 in R2C6

13. 45 rule on N9 4 innies R7C789 + R8C9 = 17 = {1268/1349/1358/1367/2357} (cannot be {1259/2348} which clash with R89C7, cannot be {1457/2456} which clash with R8C7)

14. 45 rule on C89 3 outies R456C7 = 12 = {147/156/237} (cannot be {246/345} which clash with R37C7)

15. R3C78 = 5 = {14/23} (step 2), R37C7 = 5 = {14/23} (step 5) -> R3C8 = R7C7, R4C78 + R5C7 = R7C89 + R8C9 Law of Leftovers, must contain the same 3 numbers -> 17(4) cage at R3C8 = R7C789 + R8C9 and must contain the same 4 numbers

16. R7C789 + R8C9 (step 13) = {1268/1349/1358/1367/2357} -> 17(4) cage at R3C8 = {1268/1349/1358/1367/2357} (step 15)
16a. 17(4) cage at R3C8 = {1268/1349/1358/1367} (cannot be {2357} because 2,3 cannot be in R3C8 + R4C7 (or R3C8 + R5C7) which would clash with R3C78 and R45C7 cannot be {57} because R456C7 (step 14) cannot contain both of 5,7)
16b. 17(4) cage at R3C8 = {1268/1349/1358/1367}, CPE no 1 in R56C8
16c. 17(4) cage at R3C8 = {1268/1349/1358/1367} -> R7C789 + R8C9 = {1268/1349/1358/1367} (step 15), 1 locked for N9
16d. R7C789 + R8C9 = {1268/1349/1358/1367}, CPE no 1 in R6C7

17. 17(4) cage at R3C8 (step 16a) = {1268/1349/1358/1367}
17a. 8,9 of {1268/1349/1358} must be in R4C8 -> no 2,4,5 in R4C8
17b. 4 of {1349} must be in R45C7 (R45C7 cannot be {13} which clashes with R37C7) -> no 4 in R3C8, clean-up: no 1 in R3C7 (step 2), no 4 in R7C7 (step 4)
17c. 2 of {1268} must be in R3C8 (R45C7 cannot be {26} because R456C7 (step 14) cannot contain both of 2,6) -> no 2 in R45C7

18. 2 in C7 only in R37C7 (step 4) = {23}, locked for C7, clean-up: no 1 in R3C8 (step 2)
18a. Naked pair {23} in R3C78, locked for R3 and N3 -> R3C4 = 1
18b. Naked pair {23} in R38C8, locked for C8
18c. Naked pair {23} in R7C7 + R8C8, locked for N9
18d. R3C4 = 1 -> R2C69 (step 12a) = 14 = [59/68/95], no 7
[There’s also an important CPE but I didn’t spot it for a long time.]

19. 1 in C7 only in R45C7, locked for N6
19a. 1 in 38(8) cage at R3C2 only in R456C3 + R7C23, CPE no 1 in R89C3
19b. 1 in C3 only in R4567C3, locked for 38(8) cage at R3C2, no 1 in R7C2

20. 3 in N2 only in 22(4) cage = {2389/3469/3568} (cannot be {3478} which clashes with R3C56, ALS block), no 7
20a. 7 in N2 only in R3C56, locked for R3 and 28(5) cage at R2C6 -> no 7 in R4C6

21. Hidden killer pair 4,7 in 14(3) cage and R2C3 for N1, 14(3) cage must contain one of 4,7 -> R2C3 = {47}, clean-up: no 4 in R2C4 (step 3)

22. 45 rule on R1 3 outies R2C578 = 18 = {279/459/468} (cannot be {567} which clashes with R2C69)
22a. 2 of {279} must be in R2C5, 9 of {459} must be in R2C7 -> no 9 in R2C5
22b. 7 of {279} must be in R2C8 -> no 7 in R2C7, clean-up: no 8 in R1C7

23. 45 rule on C123 2 innies R28C3 = 1 outie R5C4 + 4
23a. R28C3 cannot be [74] = 11 because no 7 in R5C4 -> no 4 in R8C3
23b. R28C3 = [42/45/46/72/75/76] = 6,9,10,12,13 -> R5C4 = {25689}, no 3,4

24. 3,4 in 38(8) cage at R3C2 only in R456C3 + R7C23, CPE no 3,4 in R9C3

25. 18(3) cage at R7C4 = {279/369/459/468/567} (cannot be {378} because R8C3 only contains 2,5,6)
25a. 5 of {459} must be in R8C3, 5 of {567} must be in R8C34 (R8C34 cannot be {67} which clashes with R8C12) -> no 5 in R7C4

26. 45 rule on R89 4 innies R8C3469 = 21 = {1569/1578/2469/2478} (cannot be {1479/2568} which clash with R8C12)
26a. 1 of {1569/1578} must be in R8C9 -> no 5 in R8C9
[Thanks Ed for pointing out that there was an error in my original step 26a, fortunately not a serious one so I only needed to rewrite this step.]

27. 22(4) cage in N2 (step 20) = {2389/3469/3568}
27aa. 22(4) cage = {2389} => R2C4 = 5 => no 5 in R3C56
27ab. 22(4) cage = {3469}, locked for N2 => R2C6 = 5 => no 5 in R3C56
27ac. 22(4) cage = {3568}, locked for N2 => no 5 in R3C56
27b. -> no 5 in R3C56
[Ed pointed out that this step can be written directly as
27. Killer quad 2,5,6,9 in 22(4) cage, R2C4 and R2C6, locked for N2]

28. R6C6 + R7C7 + R8C3 = 9 (step 11)
28a. R8C3 = {256} -> R6C6 + R7C7 = 3,4,7 = [12/13/43/52], no 2,3,6,7 in R6C6

29. 45 rule on N5 3 innies R46C6 + R5C4 = 15 = {159/168/249/456} (cannot be {258} because the two 11(2) cages in N5 must contain at least one of 2,5,8)
29a. 1,4 only in R6C6 -> R6C6 = {14}

30. R6C6 + R7C7 + R8C3 = 9 (step 11) = [126/135] (cannot be [432] which clashes with R8C38 = [23], combo blocker) -> R6C6 = 1, R8C3 = {56}, clean-up: no 7 in R5C56
30a. R46C6 + R5C4 (step 29) = {159/168}, no 2
30b. Killer pair 5,6 in R46C6 + R5C4 and R5C56, locked for N5
30c. R5C12 = {29/38/47} (cannot be {56} which clashes with R5C56), no 5,6
[Ed pointed out that the [432] permutation in step 30 is actually blocked by R8C8, another difficult to spot effect of R8C8, like the one that I only spotted in step 39.]

31. 2 in 38(8) cage at R3C2 only in R456C3 + R7C23, CPE no 2 in R9C3

32. R8C3469 (step 26) = {1569/1578/2469} (cannot be {2478} because R8C3 only contains 5,6)
32a. 1 of {1578} must be in R8C9 -> no 7,8 in R8C9
32b. 1 of {1569} must be in R8C9, 6 of {2469} must be in R8C3 -> no 6 in R8C9
32c. 2 of {2469} must be in R8C6 -> no 4 in R8C6

33. 45 rule on C123 2 remaining outies R25C4 = 1 innie R8C3 + 5
33a. R8C3 = {56} -> R25C4 = 10,11 = [28/29/56] (cannot be [55]) -> no 5 in R5C4, clean-up: no 9 in R4C6 (step 30a)

34. 45 rule on N7 4 innies R7C123 + R8C3 = 17 = {1259/1457/2456} (cannot be {1268/1367} which clash with R8C12, cannot be {1349/2348} because R8C3 only contains 5,6, cannot be {2357} which clashes with R7C7, cannot be {1358} which clashes with R7C7 + R8C3 = [53/62], combo blocker), no 3,8 in R7C123
34a. 1,2 of {1259} must be in R7C23 -> no 9 in R7C23

35. 3 in N7 only in R9C12, locked for R9
35a. 13(3) cage in N7 = {139/238/346}, no 5,7
35b. R9C3 = {689} -> no 6,8,9 in R9C12

36. 3 in C3 only in R456C3, locked for N4, clean-up: no 8 in R5C12

37. 20(3) cage at R6C1 = {479/569/578}
37a. 4 of {479} must be in R6C12 (R6C12 cannot be {79} which clashes with R5C12) -> no 4 in R7C1

38. Variable hidden killer pair 5,7 in 13(4) cage in N8 and 15(3) in N9 for R9, 13(4) cage in N8 cannot contain more than one of 5,7 -> 15(3) cage in N9 must contain at least one of 5,7 = {258/267/357} (cannot be {249/348} which don’t contain either of 5,7), no 4,9

[The next step has been available since step 18 but I’ve only just spotted it. It probably wouldn’t have made too much difference to my solving path until I reduced R8C3 to {56} in step 30.]

39. Naked pair {23} in R7C7 + R8C8, CPE no 2 in R8C6
39a. R8C58 = {23} (hidden pair in R8), R8C7 = 4 (step 6), R9C7 = 9, R8C9 = 1, clean-up: no 6 in R12C7, no 1 in 13(3) cage in N7 (step 35a)
39b. R12C7 = [78], clean-up: no 5 in R23C9
39c. R23C9 = [94], R2C6 = 5 (step 18d), R2C8 = 6, R1C9 = 5, R1C8 = 1, R2C4 = 2, R2C5 = 4, R2C3 = 7, clean-up: no 7 in R4C4, no 9 in R4C5, no 9 in R5C4 (step 30a), no 3 in R5C5, no 7 in R6C4, no 9 in R6C5
39d. 9 in 45(9) cage at R4C9 only in R56C8, locked for N6

40. Naked pair {78} in R3C56, locked for R3, N2 and 28(5) cage at R2C6 -> R4C6 = 6, R5C4 = 8, clean-up: no 3 in R4C45, no 2 in R5C56, no 3 in R6C45
40a. R5C56 = [53], R1C6 = 9

41. Killer pair 6,8 in R8C12 and R9C3, locked for N7 -> R8C3 = 5

42. R7C3 = 1 (hidden single in R7)
42a. 1 in N4 only in R4C12, locked for R4 -> R4C7 = 5, R6C7 = 6, R5C7 = 1
42b. 16(3) cage at R3C1 must contain 1 = {169} (only remaining combination, cannot be {178} because no 1,7,8 in R3C1) -> R4C12 = {19}, locked for R4 and N4, R3C1 = 6, R3C3 = 9, R3C2 = 5, clean-up: no 2 in R4C5, no 2 in R5C12, no 9 in R8C2
42c. R4C45 = [47]

and the rest is naked singles.

[R7C7 and R8C8 were very important in my solving path, with some of the steps involving them being hard to spot.]


Last edited by Andrew on Tue Apr 12, 2011 1:49 am, edited 1 time in total.

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Ed
 Post subject: Re: Assassin 207
PostPosted: Thu Mar 10, 2011 8:13 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
At the end of his WT Andrew wrote:
[R..C.. and R..C.. were very important in my solving path, with some of the steps involving them being hard to spot.]
Got another step to add to those cells! Plus, a very different way to get to Andrew's first placement. Thanks very much Andrew for finding another way through this one. It shows how long and difficult a puzzle it is without my two key steps (8 & 15), especially the second. I managed to find those two fairly quickly so didn't realise how resistant this puzzle could be.

I've used Andrew's steps up to the end of 7.

A207 alternate breakthrough
8 more steps:
End of Andrew's step 7 here. Select and "Paste Into" A207 in SudokuSolver
.-------------------------------.-------------------------------.-------------------------------.
| 24578     24578     24578     | 13456789  13456789  13456789  | 6789      134567    134567    |
| 13        13        457       | 245       2456789   2456789   | 6789      24567     589       |
| 456789    45689     45689     | 13        24578     24578     | 1234      1234      458       |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 123456789 12345689  | 23456789  23456789  123456789 | 1234567   123456789 123456789 |
| 23456789  23456789  12345689  | 12345689  123567    123567    | 1234567   123456789 123456789 |
| 3456789   3456789   12345689  | 23456789  23456789  123456789 | 1234567   123456789 123456789 |
:-------------------------------+-------------------------------+-------------------------------:
| 3456789   12345689  12345689  | 23456789  23456789  23456789  | 1234      123456789 123456789 |
| 6789      6789      12456789  | 2456789   123       2456789   | 45        123       12456789  |
| 123456789 123456789 123456789 | 1234567   1234567   1234567   | 89        456789    456789    |
'-------------------------------.-------------------------------.-------------------------------'


8. from step 3, 10(3)r2c3 = {127/145/235} = [3->2....] (ie 3 in 10(3)r2c3 -> must have 2 in r2c4)
8a. since 3 in r3 is only in 10(3)r2c3 or in h5(2)r3c78 -> 2 locked for common peers (Locking Cages)
8b. ie, no 2 in r3c56

9. 2 in r3 only in h5(2)r3c78 = {23} only: both locked for n3, 3 locked for r3
9a. h5(2)r37c7 = {23} only: both locked for c7
9b. r3c4 = 1

10. 1 must be in 38(8)r3c2 -> no 1 in r89c3 (CPE)

11. "45" on c89: 1 outie r6c7 + 5 = 2 innies r34c8
11a. -> min r34c8 = 6 -> no 1,2,3 in r4c8 (can't be [33])

12. 2 & 3 in n6 only in 45(9) -> no 2,3 in r7c89 + r8c9

13. hidden pair 2,3 in n9 -> r8c8 = (23)

14. 15(3)n9; {348} blocked by 13(2)n9 = [4/8..]; {456} blocked by r8c7
14a. 15(3)n9 = {249/258/267/357} = [3->5..](no eliminations yet)

15. from step 6, h9(3)r8c578 = {135/234}
15a. but [153] blocked by 15(3)n9 (step 14a) (Blocking Cages)
15b. -> h9(3) = {234} only
15c. r8c7 = 4, 2 locked for r8
15d. r8c9 = 1 (hidden single r8)
15e. r9c7 = 9


Much easier now....but still with some work to do. Sorry I don't have time to go any further.
Cheers
Ed
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Andrew
 Post subject: Re: Assassin 207
PostPosted: Fri Mar 11, 2011 9:17 pm 
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Congratulations Ed for spotting your two steps fairly quickly. :applause: Neither of them are obvious so I don't feel that I missed something that I ought to have seen. Your steps should lead to a shorter solving path than mine. Not a "clone" in sight for you! ;)
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Ed
 Post subject: Re: Assassin 207
PostPosted: Wed Mar 16, 2011 10:40 pm 
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Posts: 1044
Location: Sydney, Australia
Andrew wrote:
Congratulations Ed for spotting your two steps fairly quickly...Neither of them are obvious so I don't feel that I missed something that I ought to have seen.
Thanks! I routinely look for direct-link candidates (if X -> Y) in a cage whenever I look at a list of combinations....perhaps in the way you routinely do cage-placement clean-up in your walk-throughs. In contrast, I don't look for those clean-ups except if I'm stuck. I also look for killer halves routinely in case an easy block/pair is available. I rarely look for killer thirds as a block but do look for a third for if it may lead to a killer triple.

Another way to see my step 15 is using CCC but you have to see the direct linked candidate first.

Alternate way to see my step 15:
15. Since if 3->5 in 15(3)n9 (step 14)
15a. -> a possible hidden 8(2) at r8c8
15b. -> r8c78 <> sum to 8 (CCC)
15c. -> h9(3)r8c578: no 1 in r8c5
Cheers
Ed
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