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Assassin 73 V1.5

 
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mhparker
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PostPosted: Tue Jan 22, 2008 1:28 pm    Post subject: Assassin 73 V1.5 Reply with quote

The A86 is not very suitable for the creation of variants, so let's wind the clock back a little...

Hi folks,

I've just been looking back through my puzzle archive and found this. I created this variant way back on October 20, just after the original A73 appeared on the scene, and had already labelled it on my file system as "Assassin 73 V1.5". However, I never published it, partially because I had a policy at the time of not publishing any variants, and partially because Para informed me that he was working on a V2 (which he never published, either). I had to work hard back then to resist the temptation of posting this puzzle, because the original A73 was relatively straightforward (rating 1.0), and this attractive cage pattern is very suitable for creation of V2s. It would have thus been a shame to banish this puzzle to the history books without a variant ever having been released for it.

For these reasons, I've decided to publish my variant now. It contains some interesting opportunities that hopefully make it good fun to solve:


Assassin 73 V1.5 (Est. rating: 1.5)



3x3::k:5632:5632:3330:3330:3076:3589:3589:3335:3335:5632:3850:3850:3330:3076:3589:4623:4623:3335:2066:3850:5396:5396:3076:3095:3095:4623:3866:2066:2066:5396:4894:5919:5919:3095:3866:3866:3108:3108:4894:4894:4894:5919:5919:2091:2091:4909:4909:3119:3888:3888:3888:5427:4660:4660:4909:2359:3119:3119:3888:5427:5427:1597:4660:3647:2359:2359:4930:2883:4420:1597:1597:4935:3647:3647:4930:4930:2883:4420:4420:4935:4935:

Enjoy!
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Mike
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mhparker
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PostPosted: Thu Jan 24, 2008 9:26 pm    Post subject: Reply with quote

Looks like this one is destined to become another candidate for the "to do" list... Smile
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PostPosted: Thu Jan 24, 2008 9:46 pm    Post subject: Reply with quote

What a demanding killer! I haven't been able to find a short-cut for this version so there a lot of little moves with no real breakthrough move. As you can see my walkthrough is very long, which shows what I mean with little steps.

A73 V1.5 Walkthrough (old version):

1. R89
a) Innies = 10(4) = {1234} locked for R8, 4 locked in 9(3) for N7; R7C2 <> 4
b) 4 locked in 9(3) = {234} locked for N7
c) Outies = 5(2) = {23} locked for R7
d) 14(3) = 1{58/67} -> 1 locked for R9+N7
e) 6(3) = {123} locked for N9
f) 11(2): R9C5 <> 7,8,9
g) 19(3) @ R8C4: R9C4 <> 7,9 because R8C4+R9C3 <> 2,3,4

2. C6789
a) Outies C89 = 7(2) = [43/52/61]
b) 18(3) @ N3 <> 1,2 because R2C7 = (456)
c) Innies C89 = 17(4): R23C8 <> 3 because Innies would be <= 15
d) 13(3) <> 6 because 6{25/34} blocked by Killer pairs (46,56) of 18(3) @ N3
e) Innies+Outies: 4 = R4C5 - R6C6 -> R4C5 = (56789), R6C6 = (12345)

3. C1234
a) Outies C12 = 9(2) = [54/63/72]
b) 22(3) = 9{58/67} -> 9 locked for N1
c) Innies N1 = 8(3) = 1{25/34} locked for N1
d) 21(3) = 9{48/57} because R3C3 = (45) -> R3C4+R4C3 <> 4,5,6
e) 21(3) = 9{48/57} -> 9 locked between C4+R4 -> R4C4 <> 9
f) 12(3) @ N7: R6C3+R7C4 <> 7,8,9 because R7C3 >= 5
g) Innies N1 = 8(3) must have 4 xor 5 and R3C3 = (45) -> R1C3+R3C1 <> 4,5
h) 4 locked in R456C1 for N4
i) 12(2): R5C1 <> 8
j) 8(3): R4C1 <> 3 because 4 only possible there
k) Innies+Outies: 3 = R5C5 - R6C4 -> R5C5 <> 1,2,3; R6C4 <> 7,8,9

4. C123 !
a) 15(3) = 2{58/67} because 5{37/46} blocked by Killer pairs (56,57) of 22(3) and R2C3 = (567)
-> 2 locked for C2+N1
b) 9(3) = {234} -> R8C3 = 2, R7C2 = 3, R8C2 = 4
c) R7C8 = 2
d) Hidden Single: R3C3 = 4 @ N1
e) Outies C12 = 9(2) = [72] -> R2C3 = 7
f) 15(3) = {267} -> 6 locked for C2+N1
g) 8(3) must have 2 xor 4 and it's only possible @ R4C1 -> R4C1 = (24)
h) 19(3) @ N4: R6C1 <> 7,9 because R6C2+R7C1 <> 2,3,4
i) 21(3) = {489} -> R4C4 <> 8 (step 3e)
j) 12(2): R5C1 <> 9
k) 12(3) = {138/156/345} <> 9

5. C89
a) Outies = 7(2) = [43/61]
b) 8(2): R5C9 <> 6

6. C123
a) Innies N7 = 22(3): R7C1 <> 6 because 7 only possible there
b) Innies N7 = 22(3): R7C1 <> 5 because 5{89} blocked by R4C3 = (89)
c) 19(3) @ N4: R6C1 <> 5,8 because R6C2+R7C1 <> 2,4,6
d) Innies N7 = 22(3): R9C3 <> 6 because R7C3 <> 7,9
e) Coloring 9 in N7: R1C456 <> 9
- i) R9C3 = 9 -> R3C4 @ 21(3) = 9 -> R1C456 <> 9
- ii) R7C1 = 9 -> R1C2 = 9 (HS @ N1) -> R1C456 <> 9
f) 13(3): R2C4 <> 1,3 because R1C4 <> 9

7. R123
a) Outies R12 = 17(3) <> 1,5 because R3C2 = (26)
-> 17(3) = {269/278/368}
b) Killer pair (89) locked in Outies R12 + R3C4 for R3
c) 12(3) @ R3C6 <> 8 because {13}8 blocked by R3C1 = (13)
d) 12(3) @ R3C6 <> 9 because R3C167 = {123} blocked by Killer pair (23) of Outies R12

8. C123 !
a) ! 5,6 locked in R5679C3 for C3 but R79C3 can't have both of (56) because of Innies N7 = 22(3)
-> R56C3 must have 1 of (56)
b) 19(3) @ N4 <> 5 since R6C12 would be [65] -> impossible (step 8a)
c) 19(3) <> 6 because R6C2+R7C1 <> 4
d) 6 locked in 14(3) @ C1 -> 14(3) = {167} locked for N7
e) 22(3): R1C2 <> 5 because R12C1 = {89} blocked by R7C1 = (89)
f) 22(3) = {589} -> 5 locked for C1
g) 12(2): R5C2 <> 7
h) Naked triple (589) locked in R479C3 for C3
i) 12(3) must have 5 xor 8 and R7C3 = (58) -> R7C4 <> 5

9. R123
a) Innies N3 = 14(3) <> 4{19/28} because R3C79 <> 4,8,9
b) Innies N3 <> 9 because 9{23} blocked by Killer pair (23) of Outies R12
c) 18(3) <> 7 because {567} blocked by Killer triple of Innies N3
d) 18(3) = 4{59/68} -> 4 locked for R2+N3
e) Innies N3 <> 8 because {158} blocked by Killer pair (58) of 18(3)
f) 18(3): R2C8 <> 9 because R3C8 <> 4,5
g) 13(3) @ N2: R1C4 <> 6 because R1C3 <> 2,5 and 4 only possible @ R1C4
h) 12(3) @ R1: R1C5 <> 6 because 4 only possible there and R3C5 <> 1,5
i) 6 locked in 14(3) = 6{17/35} @ R1 -> R2C6 <> 6
j) 8,9 in N3 only possible @ 18(3) and 13(3) neither of them can have both
-> 13(3) must have 8 xor 9
k) 13(3) @ N3 = 3{19/28} -> 3 locked for N3
l) 14(3): R1C67 <> 1 because (67) only possible there
m) 12(3) @ R1 <> 8 because {138} blocked by Killer pair (13) of 14(3)
n) 8 locked in R123C4 for C4

10. R789
a) 19(3) @ R8C4: R9C3 <> 5 because R89C4 <> 8
b) Hidden Single: R7C3 = 5 @ N7
c) Innies N9 = 20(3): R9C7 <> 6,8 because R7C79 <> 5
d) 21(3): R6C7 <> 4 because 4{89} blocked by R7C1 = (89)
e) 17(3): R9C6 <> 9 since R8C6+R9C7 <> 2,3

11. R123
a) Innies+Outies N2: -6 = R1C37 - R3C46 -> R3C6 <> 1,2 because R1C37 >= 6
b) 13(3) @ N3: R2C9 <> 9 because {13}9 blocked by R1C3 = (13)
c) 12(3) @ N3: R4C7 <> 7 because R3C6 <> 1,4 and {23}7 blocked by Killer pair (23) of Outies R12
d) Innies+Outies N2: -6 = R1C37 - R3C46 -> R3C6 <> 3 since it forces 14(3) = {356} with 3 in N2
e) 12(3) @ N3: R3C7 <> 7 because R3C6 >= 5
f) 12(3) @ N3: R4C7 <> 2 because R3C67 <> 4 and R3C7 <> 3,7
g) 12(3) @ N3 <> {246} because it's blocked by Killer pair (26) of Outies R12

12. N56
a) Killer pair (13) locked in 12(3) + R8C7 for C7
b) Outies N5 = 12(2+1): R7C5 <> 8 because R5C7 <> 1,3
c) Outies N5 = 12(2+1): R5C7 <> 6,9 since R5C3+R7C5 <> 2,5 and R7C5 <> 3

13. N2 !
a) 13(3) <> 5,7 because 1{57} blocked by Killer triple (567) of 14(3) + R3C6
b) ! 12(3): R1C5 <> 7 since 7{23} blocked by Killer pair (37) of 14(3)
c) 14(3) = {167} -> R2C6 = 1, {67} locked for R1
d) 12(3) = {237/246/345}
e) Killer triple (567) locked in 12(3) + R13C6
f) 9 locked in R123C4 for C4
g) 12(3): R1C5 <> 5 because 4 only possible there

14. R123
a) Hidden Single: R1C1 = 5 @ R1
b) Outies R12 = 17(3) must have 8 xor 9 and it's only possible @ R3C8 -> R3C8 = (89)
c) 18(3) must have 8 xor 9 and R3C8 = (89) -> R2C8 <> 8

15. C456
a) Killer pair (47) of 12(3) @ C5 blocks {47} of 11(2)
b) 19(3) <> 2 because R89C4 <> 8,9

16. R123 !
a) ! Innies+Outies R1: 16 = R2C149 - R1C5 -> R1C5 <> 2 because R2C149 <> 1,5,6,7
b) Killer pair (34) locked in 13(3) @ N2 + R1C5 for R1
c) Hidden Single: R2C9 = 3 @ N3
d) 13(3) @ N3: R1C9 <> 8
e) 8 locked in R13C8 for C8
f) Killer pair (89) locked in 13(3) @ N3 + R1C2 for R1
g) 13(3) @ N2 must have 8 xor 9 and it's only possible @ R2C4 -> R2C4 = (89)
h) Innies+Outies R1: 13 = R2C14 - R1C5 -> R1C5 = 4 because R2C14 = {89}
i) 12(3) = 4[26/53/62]
j) 7 locked in R13C6 for C6

17. C789
a) 8(2): R5C8 <> 5
b) 21(3): R7C6 <> 8 because {678} blocked by R1C7 = (67)

18. N789 !
a) ! Innies+Outies N8: 2 = R9C37 - R7C456
-> R9C7 <> 9 because R7C456 can't be 15(3)
b) 9 locked in 21(3) @ C7 -> 21(3) = 9{48/57}; R7C6 <> 9
c) 21(3) = {489} -> R7C6 = 4, {89} locked for C7
d) Killer pair (89) locked in 19(3) @ N9 + R7C7 for N9
e) Killer pair (67) locked in 19(3) @ N9 + R7C9 for N9
f) 17(3) = 5{39/48} because R9C7 = (45) -> R9C45 <> 5
g) Hidden Single: R9C5 = 2 @ R9 -> R8C5 = 9
h) 17(3) = {458} -> R9C7 = 4, {58} locked for C6+N2
i) 19(3) @ N8 = {379} -> R8C4 = 7, R9C4 = 3, R9C3 = 9
j) 19(3) @ N9 = {568} locked for N9, 8 locked for C9
k) 18(3) = 7{29/56}; R6C9 <> 9
l) 19(3) @ N7 = 8{29/47} -> R7C1 = 8

19. Rest is singles.

Rating: Easy 1.75. Although the moves weren't that difficult (Rating 1.5) there were lots of them with no short-cuts in sight.


Last edited by Afmob on Thu Jan 31, 2008 7:21 am; edited 3 times in total
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Andrew
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PostPosted: Thu Jan 24, 2008 11:34 pm    Post subject: Reply with quote

mhparker wrote:
Looks like this one is destined to become another candidate for the "to do" list... Smile

I must admit that I initially read that statement differently, in the sense of one to be done but not immediately, until I clicked on the link. I wouldn't expect any puzzle with an estimated rating of 1.5 to become an Unsolvable.

I started on it yesterday and got half a dozen placements very quickly. Then I had to start thinking and am currently only able to find little moves, as Afmob said above.
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PostPosted: Sun Jan 27, 2008 7:55 pm    Post subject: Reply with quote

Epic walkthrough, Afmob. Well persevered! Very Happy

Andrew wrote:
I wouldn't expect any puzzle with an estimated rating of 1.5 to become an Unsolvable.

Me, neither! However, it's the only consolidated list we currently have for puzzles that do not yet have a WT posted for them. I wasn't seriously expecting this puzzle to become one of them. With the next Assassin being imminent, and unaware that Afmob had in fact already finished the puzzle, I really just wanted to stimulate you all into action... Wink

Afmob wrote:
What a demanding killer! I haven't been able to find a short-cut for this version so there a lot of little moves with no real breakthrough move. As you can see my walkthrough is very long, which shows what I mean with little steps.

Andrew wrote:
I ... got half a dozen placements very quickly. Then I had to start thinking and am currently only able to find little moves, as Afmob said above.

There was no obvious single shortcut intended for this puzzle. However, there is a quicker route available starting from immediately after Afmob's step 9, as follows:

Grid state after Afmob's step 9:

Code:
.-----------------------.-----------------------.-----------.-----------------------.-----------------------.
| 589         89        | 13          1234578   | 123457    | 3567        567       | 1389        12389     |
|           .-----------'-----------.           |           |           .-----------'-----------.           |
| 589       | 26          7         | 25689     | 123569    | 135       | 46          4568      | 12389     |
:-----------:           .-----------'-----------:           :-----------'-----------.           :-----------:
| 13        | 26        | 4           89        | 23679     | 123567      12567     | 689       | 12567     |
|           '-----------:           .-----------+-----------'-----------.           :-----------'           |
| 24          15        | 89        | 1234567   | 56789       123456789 | 1234567   | 13456789    123456789 |
:-----------------------+-----------'           '-----------.           '-----------+-----------------------:
| 347         589       | 136         12345679    456789    | 123456789   123456789 | 13567       12357     |
:-----------------------+-----------.-----------------------'-----------.-----------+-----------------------:
| 234         789       | 136       | 123456      123456789   12345     | 456789    | 13456789    123456789 |
|           .-----------:           '-----------.           .-----------'           :-----------.           |
| 89        | 3         | 58          146       | 1456789   | 456789      456789    | 2         | 456789    |
:-----------:           '-----------.-----------+-----------+-----------.-----------'           :-----------:
| 67        | 4           2         | 5679      | 56789     | 56789     | 13          13        | 56789     |
|           '-----------.-----------'           |           |           '-----------.-----------'           |
| 167         17        | 589         23456     | 23456     | 23456789    456789    | 456789      456789    |
'-----------------------'-----------------------'-----------'-----------------------'-----------------------'


From this position, the following moves can be made (disguised in Afmob-like WT style... Smile):

Quote:
10. R123
a) R2C9 <> 9 because {13}9 for 13(3) blocked by R1C3
b) AIC (Grouped X-Cycle): (9)R3C4=R4C3-R9C3=R7C1-R2C1=R2C45 -> R3C5 <> 9
c) Hidden pair {89} in R3C48 -> R3C8 <> 6
d) 18(3) can only have 1 of {89} and it must be in R3C8 -> R2C8 <> 8
e) Innies+Outies R12: 9 = R2C278 - R3C5 -> R3C5 <> 7, because max. R2C278 = 15
f) Hidden killer pair {57} in R3C67 + R3C9 -> R3C9 = {57}, R4C7 <> 5,7
g) Hidden killer pair {12} in R3C7 + 13(3) -> R3C7 = {12}
h) -> R3C6 = {57} (step 10f)
i) 12(3): R34C7 cannot have both of {12} because R3C6 <> 9 -> R4C7 <> 1,2
j) 12(3): R4C7 <> 4 because R34C7 = [14] blocked by C89 outies (R28C7, step 5a)
k) Killer pair {36} locked in R4C7 + R28C7 for C7
l) Naked pair {57} locked in R1C7 + R3C9 -> R3C7 = 2 (Innies N3)
...
(Rest of puzzle is trivial now)

[Note: Alternative explanation of step 10b for those unfamiliar with Eureka notation:
(a) R3C4 <> 9 => R4C3 = 9 (strong link, 21(3))
(b) -> R9C3 <> 9 (weak link, C3)
(c) -> R7C1 = 9 (strong link, N7)
(d) -> R2C1 <> 9 (weak link, C1)
(e) -> R2C45 = {9..} (strong link, R2)
=> either R3C4 and/or R2C45 must contain a 9
-> R3C5 <> 9 (common peer)]


Note that one of the reasons for choosing to publish this puzzle was the fact that JSudoku found quite an exotic creature, namely a Grouped Swordfish, which can be applied here immediately after step 10a above. After the removal of the candidate 9 from R2C9, the 9 in R2, N7 and 21(3)n124 is constrained to C1, C3 and N2 -> no 9 elsewhere in C13 and N2. This avoids the explicit use of a chain in step 10b above, and would also have eliminated 9 from R1C456 if they hadn't already been removed by the chain used by Afmob in his earlier step 6e. However, although more elegant, the grouped Swordfish is probably considerably harder to find than the two chains of Afmob's and mine (respectively) that it replaces.

I intend to elaborate on grouped Swordfishes soon, in a separate post. Stay tuned!

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Last edited by mhparker on Thu Jan 31, 2008 8:13 am; edited 1 time in total
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PostPosted: Mon Jan 28, 2008 7:02 pm    Post subject: Reply with quote

After getting some helpful advice from Mike I decided to rewrite my walkthrough to make it shorter. I could remove one chain (old wt: step 6) and I saw that steps 10-12 were unncessary since the cracker (old wt: step 13b, new wt: step 10f) could be applied earlier. Additionaly I optimized the end game to make the walkthrough even shorter.

Nevertheless, this walkthrough is the 7th longest (old wt: 3rd longest Shocked) I've written so far.

Edit: Andrew indirectly showed my a way to simplify step 10f, so that this wt uses no chains at all and is shortened a bit.

A73 V1.5 Walkthrough (improved):

1. R89
a) Innies = 10(4) = {1234} locked for R8, 4 locked in 9(3) for N7; R7C2 <> 4
b) 4 locked in 9(3) = {234} locked for N7
c) Outies = 5(2) = {23} locked for R7
d) 14(3) = 1{58/67} -> 1 locked for R9+N7
e) 6(3) = {123} locked for N9
f) 11(2): R9C5 <> 7,8,9

2. C6789
a) Outies C89 = 7(2) = [43/52/61]
b) 18(3) @ N3 <> 1,2 because R2C7 = (456)
c) Innies C89 = 17(4): R23C8 <> 3 because Innies would be <= 15
d) 13(3) <> 6 because 6{25/34} blocked by Killer pairs (46,56) of 18(3) @ N3

3. C1234
a) Outies C12 = 9(2) = [54/63/72]
b) 22(3) = 9{58/67} -> 9 locked for N1
c) Innies N1 = 8(3) = 1{25/34} locked for N1
d) 21(3) = 9{48/57} because R3C3 = (45) -> R3C4+R4C3 <> 4,5,6
e) 12(3) @ N7: R6C3+R7C4 <> 7,8,9 because R7C3 >= 5
f) Innies N1 = 8(3) must have 4 xor 5 and R3C3 = (45) -> R1C3+R3C1 <> 4,5
g) 4 locked in R456C1 for N4
h) 12(2): R5C1 <> 8
i) 8(3): R4C1 <> 3 because 4 only possible there

4. C123
a) 15(3) = 2{58/67} because 5{37/46} blocked by Killer pairs (56,57) of 22(3) and R2C3 = (567)
-> 2 locked for C2+N1
b) 9(3) = {234} -> R8C3 = 2, R7C2 = 3, R8C2 = 4
c) R7C8 = 2
d) Hidden Single: R3C3 = 4 @ N1
e) Outies C12 = 9(2) = [72] -> R2C3 = 7
f) 15(3) = {267} -> 6 locked for C2+N1
g) 8(3) must have 2 xor 4 and it's only possible @ R4C1 -> R4C1 = (24)
h) 19(3) @ N4: R6C1 <> 7,9 because R6C2+R7C1 <> 2,3,4
i) 21(3) = {489}
j) 12(2): R5C1 <> 9
k) 12(3) = {138/156/345} <> 9

5. C89
a) Outies = 7(2) = [43/61]
b) 8(2): R5C9 <> 6

6. C123
a) Innies N7 = 22(3): R7C1 <> 6 because 7 only possible there
b) Innies N7 = 22(3): R7C1 <> 5 because 5{89} blocked by R4C3 = (89)
c) 19(3) @ N4: R6C1 <> 5,8 because R6C2+R7C1 <> 2,4,6
d) Innies N7 = 22(3): R9C3 <> 6 because R7C3 <> 7,9

7. R123
a) Outies R12 = 17(3) <> 1,5 because R3C2 = (26)
-> 17(3) = {269/278/368}
b) Killer pair (89) locked in Outies R12 + R3C4 for R3
c) 12(3) @ R3C6 <> 8 because {13}8 blocked by R3C1 = (13)
d) 12(3) @ R3C6 <> 9 because R3C167 = {123} blocked by Killer pair (23) of Outies R12

8. C123 !
a) ! 5,6 locked in R5679C3 for C3 but R79C3 can't have both of (56) because of Innies N7 = 22(3)
-> R56C3 must have 1 of (56)
b) 19(3) @ N4 <> 5 since R6C12 would be [65] -> impossible (step 8a)
c) 19(3) <> 6 because R6C2+R7C1 <> 4
d) 6 locked in 14(3) @ C1 -> 14(3) = {167} locked for N7
e) 22(3): R1C2 <> 5 because R12C1 = {89} blocked by R7C1 = (89)
f) 22(3) = {589} -> 5 locked for C1
g) 12(2): R5C2 <> 7
h) Naked triple (589) locked in R479C3 for C3
i) 12(3) must have 5 xor 8 and R7C3 = (58) -> R7C4 <> 5

9. R123
a) Innies N3 = 14(3) <> 4{19/28} because R3C79 <> 4,8,9
b) Innies N3 <> 9 because 9{23} blocked by Killer pair (23) of Outies R12
c) 18(3) <> 7 because {567} blocked by Killer triple of Innies N3
d) 18(3) = 4{59/68} -> 4 locked for R2+N3
e) Innies N3 <> 8 since {158} blocked by Killer pair (58) of 18(3)
f) 18(3): R2C8 <> 9 because R3C8 <> 4,5
g) 13(3) @ N2: R1C4 <> 6 because R1C3 <> 2,5 and 4 only possible @ R1C4
h) 12(3) @ R1: R1C5 <> 6 because 4 only possible there and R3C5 <> 1,5
i) 6 locked in 14(3) = 6{17/35} @ R1 -> R2C6 <> 6
j) Hidden Killer pair (89) @ 13(3) + 18(3) for N3
-> 13(3) = 3{19/28} -> 3 locked for N3

10. R123 !
a) 14(3): R1C67 <> 1 because (67) only possible there
b) 12(3) @ R1 <> 8 because {138} blocked by Killer pair (13) of 14(3)
c) 8 locked in R123C4 for C4
d) Innies+Outies N12: -10 = R1C7 - R3C146; R1C7 = (567)
-> R3C6 <> 1,2,3 because R3C14 <= 12 ([39])
e) 13(3) <> 5,7 because R1C3 = (13) and 1{57} blocked by Killer triple (567) of 14(3) + R3C6
f) ! 12(3): R1C5 <> 7 since 7{23} blocked by Killer pair (37) of 14(3)
g) 7 locked in 14(3) = {167} -> R2C6 = 1
h) 12(3) = {237/246/345}
i) 9 locked in R123C4 for C4

11. R123
a) Hidden Killer pair (89) locked in 13(3) + R3C4 for N2 -> 13(3) <> 6
b) Hidden pair (89) locked in R3C4 + R3C8 for R3 -> R3C8 <> 6
c) 18(3) must have 8 xor 9 and R3C8 = (89) -> R2C8 <> 8
d) Innies+Outies R12: -9 = R3C5 - R2C278
-> R3C5 <> 7 because R2C278 <= 15
e) Outies R12 = 17(3) = 6{29/38} -> 6 locked for R3
f) 12(3) @ R1C5 = 4{26/35} -> R1C5 = 4; R2C5 <> 3
g) 7 locked in R13C6 for C6
h) 12(3) @ R3C6 = {147/156/237/345} because R3C6 = (57); R34C7 <> 5,7
i) 12(3) @ R3C6 = {147/157/237} because R3C7 = (12) -> R4C7 <> 1,2
j) Hidden pair (57) locked in R3C69 for R3 -> R3C9 <> 1,2

12. C789
a) 12(3) <> 4 because 7[14] blocked by Killer pair (14) of Outies R89
b) Killer pair (36) locked in Outies R89 + R4C7 for C7
c) R1C7 = 7, R1C6 = 6, R3C9 = 5
d) 12(3) = {237} -> R3C6 = 7, R3C7 = 2, R4C7 = 3
e) 21(3) = {489}
f) 8(2): R5C8 <> 5

13. C456
a) 12(3) = {345} -> R2C5 = 5, R3C5 = 3
b) 11(2) = {29} -> R9C5 = 2, R8C5 = 9
c) 21(3) = {489} -> 9 locked for C7
d) 17(3) = {458} -> 4 locked for R9

14. C89
a) 19(3) @ N9 = {568} -> R9C8 = 5, {68} locked for C9+N9
b) 8 locked in R13C8 for C8
c) 18(3) @ N6 = {279} -> R6C9 = 2

15. Rest is singles.

Rating: Hard 1.5. I rated it down since I was able to omit both chains but the walkthrough is still quite long.


Last edited by Afmob on Fri Feb 01, 2008 5:40 am; edited 2 times in total
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Andrew
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PostPosted: Thu Jan 31, 2008 6:48 am    Post subject: Reply with quote

Thanks Mike for a very challenging killer! Very Happy It starts fairly easily, which draws one into the puzzle, then gets really difficult.

At one stage I was thinking of giving up but then compared how far I'd got against Mike's state diagram which made me think what I'd been missing and I found steps 20e and 28, which was enough to get me going again. State diagrams like that can be helpful in providing hints while making one think how to reach the position. Thanks Mike!

The way I solved it, the later stages needed a bit more combination analysis and some contradictions but nothing particularly heavy. That includes step 39 which I don't think I used later but have left it in since I'd done it.

I'll go along with Afmob's rating of a hard 1.5.

Here is my walkthrough for A73 V1.5.

Prelims

a) R5C12 = {39/48/57}, no 1,2,6
b) R5C89 = {17/26/35}, no 4,8,9
c) R89C5 = {29/38/47/56}, no 1
d) 22(3) cage in N1 = 9{58/67}, 9 locked for N1
e) 8(3) cage at R3C1 = 1{25/34}, CPE no 1 in R6C1
f) 21(3) cage at R3C3 = {489/579/678}, no 1,2,3
g) 19(3) cage at R6C1 = {289/379/469/478/568}, no 1
h) 21(3) cage at R6C7 = {489/579/678}, no 1,2,3
i) 9(3) cage in N7 = {126/135/234}, no 7,8,9
j) 19(3) cage at R8C4 = {289/379/469/478/568}, no 1
k) 19(3) cage in N9 = {289/379/469/478/568}, no 1
l) 6(3) cage in N9 = {123}, locked for N9

1. 45 rule on N1 3 innies R1C3 + R3C13 = 8 = 1{25/34}, 1 locked for N1
1a. R3C3 = {45} -> no 4,5 in R1C3 + R3C1
1b. 15(3) cage in N1 = {258/267/348} (cannot be {357/456} which clash with 22(3) cage)

2. 45 rule on N7 3 innies R7C13 + R9C3 = 22 = 9{58/67}, 9 locked for N7
2a. 14(3) cage in N7 = {158/167/248/347} (cannot be {257/356} which clash with R7C13 + R9C3)

3. 45 rule on C12 2 outies R28C3 = 9 = {36/45}/[72/81], no 2 in R2C3

4. 45 rule on C89 2 outies R28C7 = 7 = [43/52/61], R2C7 = {456}
4a. Max R2C7 = 6 -> min R23C8 = 12, no 1,2

5. 21(3) cage at R3C3 = {489/579} (cannot be {678} because R3C3 only contains 4,5), no 6
5a. R3C3 = {45} -> no 4,5 in R3C4 + R4C3
5b. R3C4 + R4C3 must contain 9, CPE no 9 in R4C4

6. 12(3) cage at R6C3, min R7C3 = 5 -> max R6C3 + R7C4 = 7, no 7,8,9

7. 45 rule on R89 2 outies R7C28 = 5 = {23}/[41], no 1,5,6 in R7C2
7a. 45 rule on R89 4 innies R8C2378 = 10 = {1234}, locked for R8, clean-up: no 3,4 in R2C3 (step 3), no 7,8,9 in R9C5
7b. 9(3) cage in N7 = {234} (only remaining combination), locked for N7, clean-up: no 8 in R2C3 (step 3)
7c. 1 in R8 locked in R8C78, locked for N9, clean-up: no 4 in R7C2 (step 7)
7d. Naked pair {23} in R7C28, locked for R7
7e. 1 in N7 locked in R9C12, locked for R9
7f. Min R8C6 + R9C7 = 9 -> max R9C6 = 8

8. 15(3) cage in N1 (step 1b) = {258/267} (cannot be {348} because R2C3 only contains 5,6,7), no 3,4, 2 locked in R23C2 for C2 and N1 -> R78C2 = [34], R8C3 = 2, R7C8 = 2, R2C3 = 7 (step 3), R23C2 = 8 = {26}, clean-up: no 5 in R2C7 (step 4), no 8,9 in R5C1, no 6 in R5C9
[I was going to do killer triple 2,3,4 in R23C2 and R78C2, locked for C2 until I realised that step 8 takes away the use of this nice move.]
8a. Naked pair {26} in R23C2, locked for C2 and N1
8b. R3C3 = 4 (step 1 or as hidden single in N1), R3C4 + R4C3 (step 5) = {89}, CPE no 8 in R4C4

9. 12(3) cage at R6C3 = {138/156/345}, no 9

10. R7C13 + R9C3 (step 2) = 9{58/67}
10a. 6 of {679} must be in R7C3 -> no 6 in R7C1 + R9C3

11. 19(3) cage at R6C1 = {289/379/478/568} (cannot be {469} because 4,6 only in R6C1)
11a. 2,3,4,6 only in R6C1 -> R6C1 = {2346}

12. 8(3) cage at R3C1 = 1{25/34}
12a. 2,4 only in R4C1 -> R4C1 = {24}

13. 13(3) cage at R1C3 = {139/148/157/238/346} (cannot be {247/256} because R1C3 only contains 1,3)
13a. 7 of {157} must be in R1C4 -> no 5 in R1C4

14. 45 rule on C89 4 innies R2378C8 = 17 = {1259/1268/2348/2357} (cannot be {2456} because R8C8 only contains 1,3)
14a. 3 of {2348/2357} must be in R8C8 -> no 3 in R23C8
14b. 18(3) cage in N3 = 4{59}/4{68}/[648]/[657]
14c. 13(3) cage in N3 = {139/148/157/238/256} (cannot be {247/346} which clash with 18(3) cage)
14d. R1C89 = {13} clashes with R1C3 -> no 9 in R2C9
14e. Hidden killer pair 1,2 in N3 for 13(3) cage and R1C7 + R3C79 -> R1C7 + R3C79 must contain 1 or 2
14f. 45 rule on N3 3 innies R1C7 + R3C79 = 14 = {149/167/239/248/257} (cannot be {158} which clashes with 13(3) cage and with 18(3) cage)

15. 45 rule on C1234 1 outie R5C5 = 1 innie R6C4 + 3, no 1,2,3 in R5C5, no 7,8,9 in R6C4

16. 45 rule on C6789 1 outie R4C5 = 1 innie R6C6 + 4, no 1,2,3,4 in R4C5, no 6,7,8,9 in R6C6

17. 1 in R7 locked in R7C45, CPE no 1 in R6C4, clean-up: no 4 in R5C5 (step 15)

18. 45 rule on R12 3 outies R3C258 = 17 = {269/278/368} (cannot be {179/359} because R3C2 only contains 2,6), no 1,5, clean-up: no 9 in R2C8 (step 14b)
18a. Killer pair 8,9 in R3C4 and R3C58, locked for R3

19. Hidden killer triple 1,2,3 in R3C1, R3C25 and R3C679 for R3 -> R3C679 must contain one of 1,2,3
19a. Min R3C67 = {15} = 6 -> max R4C7 = 6
19b. Hidden killer pair 6,7 in R3C258 and R3C679 for R3 -> R3C679 must contain one of 6,7
19c. R3C679 must contain 5, one of 1,2,3 and one of 6,7

20. R1C7 + R3C79 (step 14f) = {167/257} (cannot be {149/248} because 4,8,9 only in R1C7, cannot be {239} because R3C79 = {23} clashes with R3C258), no 3,4,8,9, 7 locked for N3
20a. R3C79 cannot be {67} (step 19c) -> no 1 in R1C7
20b. 18(3) cage in N3 (step 14b) = 4{59}/4{68}/[648], 4 locked in R2C78, locked for R2 and N3
20c. 3 in N3 locked in 13(3) cage = {139/238}, no 5,6
[Alternatively killer pair 5,6 in R1C7 + R3C79 and 18(3) cage]
20d. 13(3) cage at R1C3 (step 13) = {139/148/157/238/346}
20e. 4 of {346} must be in R1C4 -> no 6 in R1C4

21. Hidden killer pair 1,3 in R1C3 and R56C3 for C3 -> R56C3 must contain one of 1,3
21a. 45 rule on N4 3 innies R456C3 = 2 outies R37C1 + 6
21b. Min R37C1 = 6 -> min R456C3 = 12 = {138} clashes with R1C3
21c. Min R37C1 = 8 -> min R456C3 = 14 = {158} clashes with R4C2
21d. Min R37C1 = 9, no 5 in R7C1

22. R7C13 + R9C3 (step 2) = 9{58/67}
22a. R79C3 must contain 5 or 6, hidden killer pair 5,6 in R56C3 and R79C3 -> R56C3 must contain one of 5,6
22b. Combining steps 21 and 22a, R56C3 = {1356}, no 8,9

23. Hidden killer triple 7,8,9 in R4C3, R5C12 and R6C2 -> R6C2 = {789}
23a. 19(3) cage at R6C1 (step 11) = {289/379/478}, no 6

24. 6 in N4 locked in R56C3, locked for C3, clean-up: no 7 in R7C1 (step 2)
24a. R7C13 + R9C3 (step 2) = {589} (only remaining combination), locked for N7, 5 locked in R79C3 for C3
[Alternatively 6 locked in R56C3 (step 24) -> no 5 in R56C3 (step 22a).]
24b. R7C67 = {89} clashes with R7C1 -> no 4 in R6C7

25. Naked triple {589} in R127C1, locked for C1, clean-up: no 7 in R5C2
25a. 5 in C1 locked in R12C1, locked for N1

26. 12(3) cage at R6C3 (step 9) = {138/156/345}
26a. 5 of {156/345} must be in R7C3 -> no 5 in R7C4

27. 19(3) cage at R8C4 = {289/379/469/478/568}
27a. 2,3,4 of {289/379/469/478} must be in R9C4 -> no 7,9 in R9C4

28. R123C5 = {129/138/147/156/237/246/345}
28a. 6 of {156} must be in R3C5, 4 of {246} must be in R1C5 -> no 6 in R1C5

29. 6 in R1 locked in R1C67, locked for 14(3) cage -> no 6 in R2C6
29a. 14(3) cage = {167/356}, no 2,4,8,9
29b. 1 of {167} must be in R2C6 -> no 1 in R1C6

30. R123C5 = {129/147/156/237/246/345} (cannot be {138} which clashes with R12C6), no 8
30a. 8 in N3 locked in R123C4, locked for C4

31. 19(3) cage at R8C4 = {289/379/469/478/568}
31a. 8 of {568} must be in R9C3 -> no 5 in R9C3

32. R7C3 = 5 (hidden single in C3)

33. R1C7 + R3C79 (step 20) = {167/257}
33a. R1C7 = {567} -> R3C79 must contain 1 or 2
33b. R3C679 must contain one of 1,2,3 (step 19) -> no 1,2,3 in R3C6
33c. 12(3) cage at R3C6 = {147/156/237/246} (cannot be {345} because 3,4 only in R4C7)
33d. 7 of {147/237} must be in R3C6 -> no 7 in R3C7
33e. 3,4 of {237/246} must be in R4C7 -> no 2 in R4C7

34. 13(3) cage at R1C3 (step 13) = {139/148/157/238/346}
Hidden killer pair 2,4 in N2 for R12C4 and R123C5 -> R123C5 must contain at least one of 2,4 -> R123C5 = {129/147/237/246/345} (cannot be {156} which doesn’t contain 2,4)
34a. 4 of {345} must be in R1C5 -> no 5 in R1C5

35. 45 rule on N9 3 innies R7C79 + R9C7 = 20 = {479/569/578}
35a. 5 of {569/578} must be in R9C7 -> no 6,8 in R9C7

36. 45 rule on N5 3 outies R5C37 + R7C5 = 12 = [38/65]1/[17/35/62]4/[15]6/[14/32]7/{13}8/[12]9, no 6,9 in R5C7

37. 45 rule on R5 3 outies R4C456 = 17 = {179/269/278/359/368/458/467}
Some permutations are eliminated by R4C5 = R6C6 + 4 (step 16)
37a. {269} must have 9 in R4C5 = [296/692]
37b. {359} must have 5 in R4C5 = [359]
37c. {458} must have 5 in R4C5 = [458]
37d. -> no 5 in R4C46

38. R7C79 + R9C7 (step 35) = {479/569/578}
38a. Hidden killer pair 4,6 in R7C456 and R7C79 for R7 -> R7C456 must have at least one of 4,6
38a. 45 rule on N8 2 outies R9C37 = 3 innies R7C456 + 2
38b. Max R9C37 = 17 -> max R7C456 = 15
38c. Only remaining permutation with 9 in R7C456 = {149} => R7C1 = 8 => R7C79 = {67} clashes with R7C79 + R9C7
38d. -> no 9 in R7C56
38e. 9 in N8 locked in R8C456, locked for R8

39. 15(4) cage at R6C4 = {1239/1248/1257/1347/1356/2346}
39a. R6C456 + R7C5 = {234}6 clashes with R6C1
39b. R6C456 + R7C5 = {236}4 => R6C3 = 1 => R7C4 = 6 -> no 1 in R7
39c. -> cannot be {2346}
39d. -> 15(4) cage = {1239/1248/1257/1347/1356}

40. 13(3) cage at R1C3 (step 13) = {139/148/157/238/346}
40a. Cannot be {157} because R3C4 = 8 => 9 must be in R123C5 and cannot then place 4 for N2
40b. -> 13(3) cage = {139/148/238/346}, no 5,7
40c. {139} can be 1{39} or 3{19}
If 1{39} => R3C4 = 8 => R123C5 = 4{26} (step 34) => R123C6 = {157} = [715] (step 29a), R1C7 = 6 clashes with R34C7 = {16}
If 3{19} => R3C4 = 8 => R123C5 = 4{26} (step 34) => R123C6 = {357} = [537] (step 29a), R1C7 = 6 -> cannot place 7 in N3
40d. -> no {139}, no 1,3,9 in R12C4

41. 13(3) cage at R1C3 = 1[48]/3{28}/3[46]
41a. If 1[48] => R3C4 = 9 => R123C5 = {237} (step 34) -> no 3 in R1C6
41b. If 3{28}/3[46], R1C3 = 3 -> no 3 in R1C6
41c. -> no 3 in R1C6
41d. R123C5 (step 34) = {129/147/237/345} (cannot be {246} which clashes with R12C4), no 6

42. 14(3) cage at R1C6 (step 29a) = {167/356}
42a. 3 of {356} must be in R2C6 -> no 5 in R2C6

43. 13(3) cage at R1C3 = 1[48]/3{28}/3[46]
43a. If 1[48] => R3C4 = 9 -> R123C5 cannot be {129}
43b. If 3{28} -> R123C5 cannot be {129}
43c. If 3[46] => R3C4 = 8 => R123C5 = {129} (only place for 9) => R123C6 = {357} = [537] (step 29a), R1C7 = 6 -> cannot place 7 in N3
43d. -> R123C5 cannot be {129}
43e. -> R123C5 (step 34) = {147/237/345}, no 9

44. R3C4 = 9 (hidden single in N2), R4C3 = 8, R9C3 = 9, R7C1 = 8
44a. R1C2 = 8 (hidden single in N1), R2C4 = 8 (hidden single in N2)
44b. R3C8 = 8 (hidden single in R3), R2C78 = 10 = {46}, locked for R2 and N3 -> R23C2 = [26]
44c. R1C6 = 6 (hidden single in R1)

45. 13(3) cage in N3 = {139} (only remaining combination), locked for N3
45a. 9 locked in R1C89, locked for R1 -> R12C1 = [59], R1C7 = 7, R2C6 = 1 (step 29a), R2C9 = 3, R2C5 = 5, R13C5 = [43] (step 43e), R3C1 = 1, R1C34 = [32], R4C2 = 5, R4C1 = 2, R5C2 = 9, R5C1 = 3, R6C12 = [47], R9C2 = 1, R3C6 = 7, clean-up: no 5 in R5C89, no 7 in R5C5, no 6 in R6C4 (both step 15), no 4 in R7C4 (step 9)

46. R3C6 = 7 -> R34C7 = 5 = [23], R3C9 = 5, R8C78 = [13]

47. R9C3 = 9 -> R89C4 = 10 = [64/73], no 5, no 6 in R9C4

48. Naked pair {67} in R8C14, locked for R8 -> R8C9 = 8, R8C5 = 9, R8C6 = 5, R9C5 = 2

49. Killer pair 1,6 in R5C3 and R5C89, locked for R5 -> R5C5 = 8, R6C4 = 5 (step 15)

50. Naked pair {16} in R6C35, locked for R6 -> R6C8 = 9, R6C9 = 2, R1C89 = [19], R6C6 = 3, R4C5 = 7 (step 16), clean-up: no 6 in R5C8

and the rest is naked singles

Near the end I also noticed 45 rule on R12 4 innies R2C2378 = 1 outie R3C5 + 16, but never got to find out if it was useful.


Mike's analysis and alternative path after step 9 of Afmob's original walkthrough was interesting. I look forward to reading Mike's further posts about Grouped Swordfish. I did one use an ordinary Swordfish, a 3 row and 3 column X-Wing, in a walkthrough almost a year ago. Therefore I could understand what Mike said about the Grouped Swordfish in this puzzle and also agree with his comment that it would be difficult to spot; something that's probably easier for a software solver to spot.
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mhparker
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PostPosted: Thu Jan 31, 2008 10:30 am    Post subject: Reply with quote

Andrew wrote:
I look forward to reading...about Grouped Swordfish. I did one use an ordinary Swordfish... Therefore I... agree... that it would be difficult to spot; something that's probably easier for a software solver to spot.

Probably true. At least being able to filter all candidates for a given digit on the grid via software would be very helpful in such cases.

Actually, I realise now that I overlooked something with my AIC in my previous post:

I wrote:
10. R123
a) R2C9 <> 9 because {13}9 for 13(3) blocked by R1C3
b) AIC (Grouped X-Cycle): (9)R3C4=R4C3-R9C3=R7C1-R2C1=R2C45 -> R3C5 <> 9
...
[Note: Alternative explanation of step 10b for those unfamiliar with Eureka notation:
(a) R3C4 <> 9 => R4C3 = 9 (strong link, 21(3))
...
(e) -> R2C45 = {9..} (strong link, R2)
...


What I failed to realize when originally writing that is that the two ends of the chain can be joined up to form a continuous loop, because R2C45 = {9..} implies R3C4 <> 9, taking us back to our starting premise again. So the full loop would now be:

Code:
(9)R3C4=R4C3-R9C3=R7C1-R2C1=R2C45...

[Note: Alternative explanation of step 10b for those unfamiliar with Eureka notation:
(a) R3C4 <> 9 => R4C3 = 9 (strong link, 21(3))
(b) -> R9C3 <> 9 (weak link, C3)
(c) -> R7C1 = 9 (strong link, N7)
(d) -> R2C1 <> 9 (weak link, C1)
(e) -> R2C45 = {9..} (strong link, R2) (grouped node)
(f) -> R3C4 <> 9 (weak link, N2)
...]


Now, because AICs begin and end with a strong link, and because the above AIC is a continuous loop, we can "cut" the loop at any weak link and end up with a bona-fide AIC chain, where one of the two ends must be "true". In other words, one of the ends of each weak link in steps b), d) and f) above must be true, thus eliminating the digit 9 elsewhere in C3, C1 and N2, respectively. (Not just in N2, as I mentioned in my post above... Embarassed )

Note also that what we have above is a continuous grouped X-Cycle on the digit 9 with 3 strong (and 3 weak) links. But this is exactly what a grouped Swordfish is! To be more specific: it's exactly what a grouped Swordfish with the so-called "222" formation is, where each group of the primary set (where the strong links are, namely 21(3), N7 and R2) intersects with 2 groups of the secondary set (where the weak links are, namely C3, C1 and N2), and vice versa. Unfortunately, other Swordfish formations (such as 322, 333, etc.) can only be represented by a net, and not by a simple loop. That explains why JSudoku can only detect generalized/grouped Swordfish and Jellyfish with the 222 formation - namely, because it's using chains (continuous X-Cycles, as above) to find them.

So, generalized/grouped fish (X-Wing, Swordfish, ...) can sometimes be found indirectly by detecting continuous X-Cycles (length 4 = X-Wing, length 6 = Swordfish, ...), from which it's a small step to re-express it in "Fish" terms. As mentioned above, this is exactly what JSudoku is doing internally. Instead of reporting a continuous grouped X-Cycle (aka. "Fishy" cycle), it displays the move using standard Fish terminiology instead. In this case:

JSudoku wrote:
R2, N7, Cage 21/3 in R3C34+R4C3 and C13, N2 forms a Grouped Swordfish (222) on 9 -> not elsewhere in C13, N2

However, viewing Fish moves as using chains and loops is not the most generic way of handling them. I'll leave that for a separate post (again... Smile).
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Mike
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