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 Post subject: Assassin 56 V2 Revisit
PostPosted: Thu Apr 15, 2021 6:34 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
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Assassin 56 V2 Revisit. The SudokuSolver score is a little lower than the usual Revisit (at 1.40). However, previous versions of SS give it as high as 1.80. The archive WTs (see blue link) also found it tough. JSudoku used 4 'complex intersections' which usually means a good challenge. So, lets give it another try!
Code: Select, Copy & Paste into solver:
3x3::k:3584:3841:3841:2307:2307:2307:3590:3590:5384:3584:3850:3841:5132:3341:3598:3590:3088:5384:3584:3850:5132:5132:3341:3598:3598:3088:5384:3355:3850:4381:4381:3341:4640:4640:3088:3875:3355:3355:4381:4903:3624:5161:4640:3875:3875:3629:3629:3629:4903:3624:5161:4147:4147:4147:5942:5942:4903:4903:3624:5161:5161:2621:2621:5942:2624:3393:5698:5698:5698:4165:4678:2621:2624:2624:3393:3393:5698:4165:4165:4678:4678:
Solution:
+-------+-------+-------+
| 9 6 1 | 3 2 4 | 5 7 8 |
| 3 7 8 | 9 1 5 | 2 6 4 |
| 2 5 4 | 7 8 6 | 3 1 9 |
+-------+-------+-------+
| 1 3 6 | 2 4 9 | 8 5 7 |
| 4 8 9 | 5 3 7 | 1 2 6 |
| 7 2 5 | 1 6 8 | 9 4 3 |
+-------+-------+-------+
| 8 9 7 | 6 5 1 | 4 3 2 |
| 6 1 3 | 4 9 2 | 7 8 5 |
| 5 4 2 | 8 7 3 | 6 9 1 |
+-------+-------+-------+
Cheers
Ed


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PostPosted: Wed Apr 21, 2021 5:04 pm 
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Maybe not the most elegant start - but quick!
Assassin 56 V2 WT Start:
1. Innies r6 = r6c456 = +15(3)
Outies r6789 = r5c456 = +15(3)
-> r4c456 = +15(3)
-> (789) on different rows in n5
-> Whichever row has 8 cannot be {825} since that leaves no place for 9.

2. Innies c1234 = r18c4 = +7(2)
Innies c6789 = r18c6 = +6(2)
-> r1c4 cannot be one more than r1c6
-> 9(3)n2 cannot be [261]

3. IOD n1 -> r3c3 = r4c2 + 1
IOD n4 -> r4c2 = r4c4 + 1

IOD n3 -> r4c8 = r3c7 + 2
IOD n6 -> r4c6 = r4c8 + 4

-> [r3c7,r4c6,r4c8] from [173], [284], or [395]

4. Trying [r3c7,r4c6,r4c8] = [173]
Puts r4c45 = {26}
which puts r4c2 from (37) contradicting (37) already in r4

5! Trying [r3c7,r4c6,r4c8] = [284]
Puts r4c45 = {16}
But then where are (16) going in n2?
r23c6 = +12(2) - No (16)
13(3)r2c5 cannot contain both (16)
Outies n14 = r234c4 = +18(3) - cannot contain both (16)
-> (16) in n2 in 9(3)n2 = {162}
But r14c4 cannot be {16} since r18c4 is +7(2)
-> 9(3)n2 would have to be [261] contradicting Step 2 Line 4

-> [r3c7,r4c6,r4c8] = [359]

After this the puzzle practically solves itself.

E.g., Continuing
6. -> r4c56 = {24}
-> (Since r4c2 = r4c4 + 1) r4c2 = 3, r4c45 = [24]
-> r3c3 = 4
-> r23c2 = {57}
Also r23c4 = {79}
-> r23c6 = {56}
-> r23c5 = {18}
-> 9(3)n2 = {234}
Also r23c8 = {16}
-> (HS 2 in n3) r2c7 = 2
-> r1c78 = [57]
Also 14(3)n1 = [932]
-> 21(3)n3 = [849]
etc.


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PostPosted: Fri Apr 23, 2021 9:26 pm 
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Incredible shortcut by wellbeback. So many elements to that step. Very impressive. My way is very mundane by comparison. And long.

Much easier time with this puzzle 15 years later! But still no pushover with some advanced steps needed. Similiar start to me and Andrew in the archive WTs but then veers. [Thanks to Andrew for seeing I'd missed part of step 11 and finding some typos]

Start to A56 V2R:
Preliminaries courtesy of SudokuSolver
Cage 23(3) n7 - cells ={689}
Cage 9(3) n2 - cells do not use 789
Cage 21(3) n3 - cells do not use 123
Cage 10(3) n7 - cells do not use 89
Cage 10(3) n9 - cells do not use 89
Cage 20(3) n12 - cells do not use 12

No clean-up done unless stated.
1. "45" on n36: 1 innie r3c7 + 6 = 1 outie r4c6 = [17/28/39]

2. "45" on n6: 1 outie r4c6 - 4 = 1 innie r4c8 = [73/84/95]
2a. -> r3c7 + r4c68 = [173/284/395]

3. 23(3)n7 = {689}: all locked for n7

4. "45" on c12: 3 outies r126c3 = 14 (at most one of 6,8,9). No elimination yet

5. "45" on n14: 3 innies r345c3 = 19
5a. and must have at least two of 6,8,9 for c3 (Hidden killer triple)
5b. = {289/469/568}(no 1,3,7)

6. "45" on n1: 1 innie r3c3 - 1 = 1 outie r4c2
6a. -> no 1,2,6,9 in r4c2

7. h19(3)r345c3 = {289/469/568}
7a. but [6]{49} blocked by 17(3)r4c4 can't be {494}
7b. and [6]{58} blocked by 5 in r4c2 (step 6.)
7c. -> no 6 in r3c3
7d. -> no 5 in r4c2 (iodn1=+1)

8. "45" on n5789: 3 innies r4c456 = 15
8a. -> can't have more than one of 7,8,9
8b. -> no 7,8,9 r4c45

9. "45" on n4: 1 innie r4c2 - 1 = 1 outie r4c4
9a. no 1,4,5 r4c4
9b. no 8 r4c2
9c. -> no 9 r3c3 (iodn1=+1)

10. putting steps 6 & 9 together
10a. -> r3c3 + r4c24 = [432/543/876]

11. "45" on n2: 3 outies r3c37 + r4c5 = 11
11a. [533] blocked by 3 in r4c4 (step 10a)
11b. [524] blocked by 4 in r4c2 (step 10a)
11c. [515] blocked by [33] r4c48 (steps 2a, 10a)
11cc. [416] blocked by [33] in r4c28
11d. = [425/434/821], ie, r3c3 = (48), r3c7 = (23), r4c5 = (145)
11e.-> no 4 in r4c2 (iodn1=+1)
11f. -> no 3 in r4c4 (iodn4=+1)
11g. -> no 7 in r4c6 (iodn36=-6)
11h. -> no 3 in r4c8 (iodn6=-4)

12. h19(3)r345c3 = 19 = {289/469/568}: but {568} as [8]{56} blocked by 17(3)r4c3 can't be {566}
12a. = {289/469}(no 5)
12b. must have 9: locked for c3 and n4
12c. can't have both 4,8 -> no 4,8 in r45c3
12d. -> 17(3) = {269} only -> no 2,6 in r4c1 since it sees all that cage (common peer elimination CPE)

13. 15(3)r2c2 must have 3 or 7 for r4c2
13a. but {48}[3] blocked by r3c3 = (48)
13b. = {267/357}(no 1,4,8,9)
13c. must have 7: locked for c2

14. "45" on n1: 3 innies r2c2 + r3c23 = 16 and must have 4 or 8 for r3c3 = {268/358/457}
14a. no eliminations yet

This was the hardest step to see. A short contradiction chain.
15. "45" on c12: 1 innie r1c2 - 1 = 1 outie r6c3
15a. but [98] forces r3c3 = 4 -> {57} in r23c2 (step 14)
15b. but this means r12c3 <> 6 (cage sum)
15c. -> no 9 in r1c2, no 8 in r6c3

16. r7c2 = 9 (hsingle c2)
16a. r78c1 = {68}: both locked for c1

17. 9 in n1 only in 14(3) = {149/239}(no 5,7)

18. "45" on n4: 3 innies r4c23 + r5c3 = 18 = {279/369}
18a. -> {167/347} blocked from 14(3)n4
18b. = {158/248/257/356}
18c. 8 in {248} must be in r6c2 -> no 4 in r6c2
18d. {248} in 14(3) -> [3]{69} in h18(3) -> 4 in r3c3
18e. -> no 4 in r6c3

Final tricky one. Another short contradiction.
19. if 4 in r5c2 -> r45c1 = 9 = [72] only -> {149} in r123c1 (step 17) -> 8 in r3c3 -> 7 in r4c2 (iodn1=+1)
19a. but can't have two 7s in n4
19b. -> no 4 in r5c2

20. 4 in n4 only in c1, locked for c1
20a. -> 14(3)n1 = {239} only: 2 & 3 locked for c1 and n1

21. h16(3)n1 = {57}[4] only, 5 locked for c2 and n1: 7 for n1
21a. r4c2 = 3
21b. -> r4c4 = 2
21c. -> r45c3 = {69}: 6 locked for c3 and n4

Much more straightforward now but have to do some of the many "45"s I missed. Lots of cage sums too and a cleanup on 12(3)n3. Much longer tail than Wellbeback's.
Cheers
Ed


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PostPosted: Sun May 02, 2021 1:00 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
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Location: Lethbridge, Alberta, Canada
I've been busy so only just got to work on this puzzle before the next one appeared.

My solving path is also long. A lot of work before I spotted the key point about this puzzle.

Here is my walkthrough for Assassin 56V2 Revisited:
Prelims

a) 9(3) cage at R1C4 = {126/135/234}, no 7,8,9
b) 21(3) cage at R1C9 = {489/579/678}, no 1,2,3
c) 20(3) cage at R2C4 = {389/479/569/578}, no 1,2
d) 23(3) cage at R7C1 = {689}
e) 10(3) cage at R7C8 = {127/136/145/235}, no 8,9
f) 10(3) cage at R8C2 = {127/136/145/235}, no 8,9

1a. Naked triple {689} in 23(3) cage at R7C1, locked for N7
1b. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 1 -> no 1,7,9 in R9C4
1c. 45 rule on R89 2 innies R8C19 = 11 = [65/83/92]
1d. 45 rule on R89 4 outies R7C1289 = 22 = {1489/1678/2389/2569/3469/3568} (cannot be {1579/2479/2578/3478/4567} because R7C12 must contain two of 6,8,9) -> R7C89 = {14/17/23/25/35} (cannot be {34} because 10(3) cage at R7C8 cannot be {34}3), no 6

2a. 45 rule on N36 1 outie R4C6 = 1 innie R3C7 + 6 -> R3C7 = {123}, R4C6 = {789}
2b. 45 rule on N36 3 outies R234C6 = 20 = {389/479/569/578}, no 1,2
2c. 45 rule on N3 3 innies R2C8 + R3C78 = 10 = {127/136/145/235}, no 8,9
2d. 45 rule on N6 1 outie R4C6 = 1 innie R4C8 + 4 -> R4C8 = {345}

3a. 45 rule on C1234 2 innies R18C4 = 7 = {16/25/34}, no 7,8,9
3b. 45 rule on C6789 2 innies R18C6 = 6 = {15/24}
3c. 45 rule on C5 3 innies R1C189C5 = 18
3d. Max R1C5 = 6 -> min R89C5 = 12, no 1,2 in R89C5

4a. 45 rule on N2 3(2+1) outies R3C37 + R4C5 = 11
4b. Min R3C37 = 4 -> max R4C5 = 7
4c. 45 rule on R123 3 outies R4C258 = 12
4d. Min R4C8 = 3 -> max R4C25 = 9, no 9 in R4C2
4e. 45 rule on N4 1 innie R4C2 = 1 outie R4C4 + 1, no 1 in R4C2, no 8,9 in R4C4
4f. R4C258 = 12 = {138/147/246/345} (cannot be {156} = [615] which clashes with R4C24 = [65], cannot be {237} = {27}3 which clashes with R4C68 = [73], step 2d)
4g. 1 of {147} must be in R4C5 -> no 7 in R4C5
4h. R4C258 = {147/246/345} (cannot be {138} = [813] because R4C24 = [87] clashes with R4C68 = [73], step 2d), no 8, 4 locked for R4, clean-up: no 5 in R4C2, no 7 in R4C4
4i. 45 rule on N1 1 innie R3C3 = 1 outie R4C2 + 1, R4C2 = {23467} -> R3C3 = {34578}

5a. R4C2 = R4C4 + 1 (step 4e), R4C6 = R4C8 + 4 (step 2d), R4C258 (step 4h) = {147/246/345}
5b. 45 rule on N5789 3 innies R4C456 = 15 = {159/168/249/258/267/348/357} (cannot be {456} because R4C6 only contains 7,8,9)
5c. R4C456 = {168/249/258/267} (cannot be {159} which clashes with R4C68 = [95], cannot be {348} = [348] which clashes with R4C68 = [84], cannot be {357} which clashes with R4C68 = [73]), no 3, clean-up: no 4 in R4C2, no 5 in R3C3 (step 4i)
5d. 4 in R4 only in R4C58 -> R4C456 must contain one of 4,8 = {168/249/258} (locking-out cages), no 7, clean-up: no 3 in R4C8, no 1 in R3C7 (step 2a)
5e. Max R4C4 = 6 -> min R45C3 = 11, no 1 in R45C3
5f. 20(3) cage at R2C4 = {389/479/578} (cannot be {569} because no 5,6,9 in R3C3), no 6
5g. R2C8 + R3C78 (step 2c) = {127/136/235} (cannot be {145} because R3C7 only contains 2,3), no 4

6a. R18C4 = 7 (step 3a), R18C6 = 6 (step 3b) -> R1C4 cannot be 1 greater than R1C6 which would make R8C4 and R8C6 equal
6b. 9(3) cage at R1C4 = {126/135/234}
6c. {126} = [162/612/621] (cannot be [261])
6d. R4C258 (step 4h) = {147/246/345}, R4C2 = R4C4 + 1 (step 4e)
6f. R4C258 = {147} can only be [714] giving R4C2458 = [7614]
6g. Consider combinations for 9(3) cage
9(3) cage = [162] => R8C4 = 6 blocking R4C2458 = [7614]
or 9(3) cage = [612/621] blocking R4C2458 = [7614]
or 9(3) cage = {135/234}, only other place for 1,2 in N2 in R23C5, 13(3) cage cannot contain both of 1,2 => no 1,2 in R4C5
-> R4C258 = {246/345}, no 1,7, clean-up: no 8 in R3C3 (step 4i), no 6 in R4C4 (step 4e)
6h. R4C456 (step 5d) = {168/249/258}
6i. R4C456 = {168} can only be [168]
6j. Consider placement for 6 in N2
6 in R1C4 => R8C4 = 1 blocking [168]
or 6 in R123C5 blocking [168]
or 6 in R23C6 => R23C6 can only be {56} => R3C7 = 3 => R4C6 = 9 (step 2a)
-> R4C456 = {249/258}, no 1,6, 2 locked for R4 and N5, clean-up: no 3 in R3C3 (step 4i)
6k. 20(3) cage at R2C4 (step 5f) = {479/578} (cannot be {389} because R3C3 only contains 4,7), no 3
6l. 45 rule on N14 3 outies R234C4 = 18 = {279/459}, no 8, 9 locked for C4 and N2
6m. R4C4 = {25} -> no 5 in R23C4
6n. 20(3) cage = {479}, CPE no 4,7 in R3C56
6o. 14(3) cage at R2C6 = {257/356} (cannot be {248} = [482] which clashes with R3C7 + R4C6 (step 2a) = [28], cannot be {347} because 4,7 only in R2C6), no 4,8, 5 locked for C6 and N2, clean-up: no 2 in R8C4 (step 3a), no 1 in R18C6 (step 3b)
6p. R3C7 = {23} -> no 3 in R23C6
6q. Naked pair {24} in R18C6, locked for C6
6r. Naked triple {245} in R4C458, 5 locked for R4
6s. R3C37 + R4C5 = 11 (step 4a)
6t. Consider placements for R3C3 = {47}
R3C3 = 4 => R23C4 = {79}, 7 locked for N2 => R23C6 = {56} = 11 => R3C7 = 3 (cage total) => R4C5 = 4
or R3C4 = 7 => R3C7 + R4C5 = 4 = [22]
-> R4C5 = {24}
6u. 8 in N2 only in 13(3) cage at R2C5 = {148/238} -> R23C5 = {18/38}, 8 locked for C5
6v. Max R4C4 = 5 -> min R45C3 = 12, no 2 in R5C3

7a. 9(3) cage = {126/234}, 2 locked for R1
7b. R18C4 (step 3a) = {16/34} (cannot be [25] which clashes with R4C4), no 2,5
7c. R189C5 (step 3c) = 18 = {279/369/459/567}, no 1
7d. 9(3) cage = {126/234} = [162]/{234}, no 6 in R1C4, clean-up: no 1 in R8C4
7e. Consider combinations for 9(3) cage
9(3) cage = [162] => R8C6 = 4, R23C5 = {38} (hidden pair in N2) => R4C5 = 2 (cage sum) => 4 in C5 only in R56C5
or 9(3) cage = {234} => R23C5 = {18} (hidden pair in N2) => R4C5 = 4 (cage sum)
-> 4 in R456C5, locked for C5 and N5

8a. 45 rule on N14 3 innies R345C3 = 19 = {379/469/478} (cannot be {568} because R3C3 only contains 4,7), no 5
8b. 45 rule on N36 3 innies R345C7 = 12 = {129/138/237} (cannot be {147/156} because R3C7 only contains 2,3, cannot be {345} because 4,5 only in R5C7, cannot be {246} = [264] which clashes with R3C7 + R4C8 = [24], combining steps 2a and 2d), no 4,5,6

[Only just spotted this, although I’d seen the 45s a lot earlier but couldn’t see any use for them at that time.]
9a. 45 rule on R6 3 innies R6C456 = 15
9b. 45 rule on R6789 3 outies R5C456 = 15
9c. 1,6,7 in N5 only in R56C456, neither of R5C456 and R6C456 can contain both of 1,7, nor both of 6,7 (because 2 in R4C45) -> one of R5C456 and R6C456 must be {168}, locked for N5
9d. R4C6 = 9 -> R4C45 = [24] (step 6j), R23C5 = 9 = {18}, 1 locked for C5 and N2, R4C8 = 5, R4C2 = 3 (step 4e), R3C3 = 4 (step 4i), R3C7 = 3 (step 2a) -> R23C6 = 11 = {56}, 6 locked for C6 and N2, clean-up: no 6 in R8C4 (step 3a)
9e. Naked triple {234} in 9(3) cage at R1C4, 3,4 locked for R1
9f. Naked pair {34} in R18C4, locked for C4
9g. Naked pair {79} in R23C4, 7 locked for C4
9h. R9C4 = R7C3 + 1 (step 1b) -> R7C3 = {57}, R9C4 = {68}
9i. 19(4) cage at R5C4 = {1567} (only remaining combination) -> R7C3 = 7, R9C4 = 8
9j. R4C4 = 2 -> R45C3 = 15 = [69]
9k. R9C4 = 8 -> R89C3 = 5 = {23}, locked for C3 and N7
9l. Naked triple {234} in R8C346, locked for R8 -> R8C9 = 5, R8C2 = 1, R8C1 = 6 (step 1c)
9m. Naked pair {89} in R7C12, locked for R7
9n. Naked pair {45} in R9C12, locked for R9

10a. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 + 1 -> R7C7 = {24}, R9C6 = {13}
10b. Naked pair {13} in R79C6, locked for C6 and N8 -> R18C4 = [34], R1C5 = 2, R18C6 = [42], R89C3 = [32]
10c. R8C46 = [42] = 6 -> R89C5 = 16 = {79}, 7 locked for C5
10d. R4C8 = 5 -> R23C8 = 7 = {16}, locked for C8 and N3
10e. 21(3) cage at R1C9 = {489} (only remaining combination) -> R2C9 = 4, R13C9 = {89}, locked for C9 and N3, R1C78 = [57], R2C7 = 2
10f. R4C6 = 9 -> R45C7 = 9 = {18}, locked for C7 and N6
10g. R4C9 = 7 -> R5C89 = 8 = [26], R6C9 = 3, R9C9 = 1 -> R89C8 = 17 = [89]
10h. 4 in R5 only in 13(3) cage at R4C1 = {148}, locked for N4, 1 locked for C1
10i. R4C2 = 3 -> R23C2 = 12 = {57}, locked for C2 and N1

and the rest is naked singles.


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