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 Post subject: Assassin 408
PostPosted: Sat Jan 01, 2022 8:09 am 
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a408.png
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X-puzzle so 1-9 cannot repeat on either diagonal. Note the zigzags 21(4)r3c3 and 26(5)r3c7

Assassin 408
Happy New Year! This puzzle drove me bonkers so a perfect welcome to 2022! Finally worked out how to solve it but pretty sure I missed something big. If its too easy for you guys, I have a harder version I couldn't solve. SS gives the posted puzzle 1.30 and JSudoku uses 3 advanced steps. I had to use a very short contradiction.
triple click code:
3x3:d:k:3072:3072:2049:2049:1538:1538:3587:5380:5380:7941:7941:7941:7941:7941:7941:3587:3587:5380:2056:2311:5392:6153:6153:6153:6666:3587:4363:2056:2311:2311:5392:4620:6666:3341:3341:4363:6166:6166:5392:4620:4620:4620:6666:4110:4363:4629:6159:6166:5392:3607:6666:4110:4110:4363:4629:6159:6166:3607:3607:1297:6666:5906:5906:4629:6159:6159:4376:4376:1297:3859:3859:5906:4629:2836:2836:2836:4376:2566:2566:3859:5906:
solution:
+-------+-------+-------+
| 8 4 2 | 6 1 5 | 3 7 9 |
| 7 9 6 | 3 4 2 | 8 1 5 |
| 3 5 1 | 9 8 7 | 4 2 6 |
+-------+-------+-------+
| 5 1 3 | 4 9 8 | 7 6 2 |
| 4 7 9 | 1 2 6 | 5 3 8 |
| 2 6 8 | 7 5 3 | 9 4 1 |
+-------+-------+-------+
| 1 8 5 | 2 7 4 | 6 9 3 |
| 9 3 7 | 8 6 1 | 2 5 4 |
| 6 2 4 | 5 3 9 | 1 8 7 |
+-------+-------+-------+

Cheers
Ed


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 Post subject: Re: Assassin 408
PostPosted: Mon Jan 03, 2022 6:53 am 
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Thanks Ed for your new Assassin.

An easy start, then it became Assassin level. I've managed to simplify my later steps, so now I've only used one forcing chain. Later I added an alternative way to do step 8b, using a forcing chain which I hadn't previously used.

Ed, if you want to post the harder version, provided it's within the Revisit specification, I'll have a try at it when I've got time.

Here's how I solved Assassin 408:
Prelims

a) R1C12 = {39/48/57}, no 1,2,6
b) R1C34 = {17/26/35}, no 4,8,9
c) R1C56 = {15/24}
d) R34C1 = {17/26/35}, no 4,8,9
e) R4C78 = {49/58/67}, no 1,2,3
f) R78C6 = {14/23}
g) R9C67 = {19/28/37/46}, no 5
h) 21(3) cage at R1C8 = {489/579/678}, no 1,2,3
i) 9(3) cage at R3C2 = {126/135/234}, no 7,8,9
j) 24(3) cage at R3C4 = {789}
k) 11(3) cage at R9C2 = {128/137/146/236/245}, no 9
l) 14(4) cage at R1C7 = {1238/1247/1256/1346/2345}, no 9

1a. Naked triple {789} in 24(3) cage at R3C4, locked for R3 and N2, clean-up: no 1 in R1C3, no 1 in R4C1
1b. 45 rule on N3 2 innies R3C79 = 10 = {46}, locked for R3 and N3, clean-up: no 2 in R4C1
1c. 21(3) cage at R1C8 = {579} (only remaining combination), 5,7 locked for N3
1d. 45 rule on R12 1 outies R3C8 = 2, clean-up: no 6 in R4C1
1e. Naked triple {135} in R3C123, locked for N1
1f. R1C12 = {48} (only remaining combination), locked for R1 and N1
1g. R1C56 = {15} (only remaining combination), locked for N2, 5 locked for R1 -> R1C7 = 3, clean-up: no 7 in R9C6
1h. Naked pair {79} in R1C89, locked for N3, 7 locked for R1 -> R2C9 = 5
1i. 45 rule on N6 1 outie R3C9 = 1 innie R5C7 + 1 -> R3C9 = 6, R5C7 = 5, R3C7 = 4, placed for D/, clean-up: no 8 in R4C7, no 8,9 in R4C8, no 6 in R9C6
1j. R3C9 = 6 -> 17(4) cage at R3C9 = {1268/1367}, no 4,9, 1 locked for C9 and N6
1k. R1C6 = 5 (hidden single in C6) -> R1C5 = 1
1l. 4 in C9 only in R789C9, locked for N9
1m. 45 rule on N9 2 innies R79C7 = 7 = {16}, locked for C7 and N9 -> R2C7 = 8, R2C8 = 1, placed for D/, R9C6 = {49}, clean-up: no 7 in R4C8
1n. 45 rule on C1 3 innies R125C1 = 19 = {289/469/478} (cannot be {379} which clashes with R34C1), no 1,3
1o. 7 of {478} must be in R2C1 -> no 7 in R5C1

2a. R35C7 = [45] = 9 -> R46C6 + R7C7 = 17 = {179/368} (cannot be {269} = {29}6 which clashes with R9C67, cannot be {278} because R7C7 only contains 1,6) -> R46C6 = {38/79}, no 1,2,6
2b. 45 rule on C789 3 outies R469C6 = 20 = {389/479}, 9 locked for C6
2c. Killer pair 7,8 in R3C6 and R469C6, locked for C6
2d. Killer pair 3,4 in R469C6 and R78C6, locked for C6
2e. 4 in C6 only in R789C6, locked for N8
2f. R789C6 = {14}9/{23}4
2g. 17(3) cage at R8C4 = {179/278/368} (cannot be {269/359} which clash with R789C6), no 5
2h. Naked pair {26} in R1C3 + R2C6, locked for N2

3a. 45 rule on N8 2 innies R9C46 = 1 outie R6C5 + 9
3b. Max R9C46 = 17 -> max R6C5 = 8
3c. Min R6C5 = 2 -> min R9C46 = 11, no 1 in R9C4
3d. R9C46 cannot total 13 -> no 4 in R6C5

4a. 45 rule on N47 3(2+1) outies R3C12 + R9C4 = 1 innie R5C3 + 4
4b. Min R3C12 = 4 -> R5C3 cannot be lower than R9C4, no 1 in R5C3

5a. 16(3) cage at R5C8 = {268/349/367}
5b. R6C7 = {279} -> no 7,9 in R56C8
5c. 9 in N6 only in R46C7, locked for C7
5d. 15(3) cage at R8C7 = {258/357}, no 9, 5 locked for N9
5e. R8C7 = {27} -> no 7 in R89C8
5f. Combining 16(3) cage with 15(3) cage, they must contain 2 for C7 -> must contain 8 in R5689C8, locked for C8

[The first key step]
6a. 17(3) cage at R8C4 (step 2g) = {179/278/368}
6b. Consider combinations for R46C6 (step 2a) = {38/79}
R46C6 = {38}, locked for C6 => R78C6 = {14}, R9C6 = 9 => 17(3) cage = {278/368}
or R46C6 = {79}, locked for C6 => R3C6 = 8, R3C5 = {79} => 17(3) cage = {278/368} (cannot be {179} = 1{79} which clashes with R3C5)
-> 17(3) cage = {278/368}, no 1,9, 8 locked for N8
6c. R78C6 = {14} (cannot be {23} which clashes with 17(3) cage), locked for N8, 1 locked for C6
6d. R9C6 = 9 -> R9C7 = 1, R7C7 = 6, placed for D\, R46C6 = {38}, locked for N5, 8 locked for C6
6e. 11(3) cage at R9C2 = {236/245}, no 7,8, 2 locked for R9
6f. R9C46 = R6C5 + 9 (step 3a), R9C6 = 9 -> R6C5 = R9C4 = {256}, no 7 in R6C5, no 3 in R9C4
6g. 1 on D\ only in 21(4) cage at R3C3 = {1389/1479/1569/1578}, no 2

7a. 45 rule on N47 6(3+3) outies R3C123 + R469C4 = 25, R3C123 = {135} = 9 -> R469C4 = 16 = {169/259/457}
[Originally seen that way before step 6d reduced 45 rule on N5 3 innies R46C4 + R6C5 to 16 with the same combinations. R46C4 + R6C5 = {169/259/457} is used from here.]
7b. 4 of {457} must be in R4C4 -> no 7 in R4C4
7c. 6 of {169} must be in R9C4 (and R6C5) -> no 6 in R6C4
7d. 6 on D/ only in R8C2 + R9C1, locked for N7
7e. 21(4) cage at R3C3 (step 6g) = {1389/1479/1569} (cannot be {1578} = {15}87 because R46C4 + R6C5 = {457} must have 4 in R4C4)
7f. 8 of {1389} must be in R5C3 -> no 3 in R5C3
7g. 21(4) cage = {1389/1479/1569}, CPE no 9 in R5C45

[Simplified to remove forcing chains from steps 8 and 9]
8a. R46C4 + R6C5 (step 7a note) = {169/259/457} = [196]/{59}2/[475]
8b. R5C56 = [26/76] (cannot be [72] which clashes with R46C4 + R6C5 = {169} because of R1C9 + R6C4 = [79] on D/ and with R46C4 + R6C5 = {259/457}) -> R5C6 = 6, R2C6 = 2, R1C34 = [26]
[An alternative way to do step 8b, using a forcing chain but I find it clearer]
8b. Consider placements for R1C9 = {79}
R1C9 = 7, placed for D/ => R5C56 = [26]
or R1C9 = 9, placed for D/ => R46C5 + R6C5 = [475/952] => R5C56 = [26/76]
-> R5C6 = 6, R2C6 = 2, R1C34 = [26]

[The final key step]
8c. R46C4 + R6C5 = {259/457}, no 1, 5 locked for N5
8d. R46C4 + R6C5 = {259/457}, CPE no 5 in R7C4
8e. R5C4 = 1 (hidden single in N5)
8f. R3C3 = 1 (hidden single on D\), clean-up: no 7 in R4C1
8g. Naked pair {35} in R34C1, locked for C1
8h. 8(2) cage at R3C1, 9(3) cage at R3C2, R3C12 = {35} = 8 -> R4C123 = 9 = {135/234} (cannot be {126} because R4C1 only contains 3,5), no 6, 3 locked for R4 and N5
8i. R4C6 = 8, placed for D/, R6C6 = 3, placed for D\
8j. 3 on D/ only in R7C3 + R8C2, locked for N7
8k. Naked triple {245} in 11(2) cage at R9C2, 4,5 locked for R9, 4 locked for N7

9a. 21(4) cage at R3C3 (step 7e) = {1479} (only remaining combination), no 5,8
9b. 21(4) cage = {1479}, CPE no 7 in R5C5 -> R5C5 = 2, placed for D/
9c. R6C5 = 5, R7C45 = 9 = [27], R9C4 = 5 -> R9C23 = [24]
9d. Naked pair {79} in R5C3 + R6C4, locked for 21(4) cage -> R4C4 = 4, placed for D\, R1C1 = 8, placed for D\, R9C9 = 7, placed for D\
9e. R1C9 = 9, placed for D/ -> R6C4 = 7, placed for D/
9f. R5C3 = 9, R5C1 = 4 -> 24(4) cage at R5C1 = {4578} (only remaining combination) -> R5C2 = 7, R67C3 = [85]

and the rest is naked singles, without using the diagonals.

Happy New Year to all Sudoku Solvers!


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 Post subject: Re: Assassin 408
PostPosted: Tue Jan 04, 2022 7:00 pm 
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Thanks Andrew, will post a next version tomorrow. So, this a408 is a tricky puzzle! I did the start and middle a similiar way but then a different end. Will post my WT now in case it gives some help with the next version. [Thanks to Andrew for checking my WT]
A408 WT:
Preliminaries by SudokuSolver
Cage 5(2) n8 - cells only uses 1234
Cage 6(2) n2 - cells only uses 1245
Cage 8(2) n12 - cells do not use 489
Cage 8(2) n14 - cells do not use 489
Cage 12(2) n1 - cells do not use 126
Cage 13(2) n6 - cells do not use 123
Cage 10(2) n89 - cells do not use 5
Cage 24(3) n2 - cells ={789}
Cage 9(3) n14 - cells do not use 789
Cage 21(3) n3 - cells do not use 123
Cage 11(3) n78 - cells do not use 9
Cage 14(4) n3 - cells do not use 9

This is a highly optimised solution so any clean-up needed is stated.

1. "45" on r12: 1 outie r3c8 = 2

2. 24(3)n2 = {789}: all locked for n2 and r3

3. "45" on n3: 2 innies r3c79 = 10 = {46} only: both locked for n3 and r3

4. 6 in r1 only in 8(2)r1c3 = {26}: 2 locked for r1
4a. -> 6(2)n2 = {15}: both locked for r1 and n2

5. naked triple {135} in r3c123: all locked for n1
5a. -> 12(2)n1 = {48}: both locked for n1, 8 for r1

6. 21(3)n3 = {79}[5]: 7 locked for n3
6a. r1c7 = 3

7. "45" on n36: 2 innies r35c7 = 9 = [45], 4 placed for d/
7a. r3c9 = 6

8. "45" on n9: 2 innies r79c7 = 7 = {16} only: both locked for n9 and c7
8a. r2c78 = [81]: 1 placed for d/
8b. r9c67 = [91/46]

9. r1c6 = 5 (hsingle c6), r1c5 = 1

10. "45" on c789: 3 outies r469c6 = 20 = {389/479}(no 1,2,6) = 3 or 4
10a. 9 locked for c6

11. killer pair 3,4 in c6 between h20(3) and 5(2)n8: both locked for c6
11a. 4 locked for n8

12. 17(3)n8 = {179/269/278/359/368} = 1 or 2 or 3
12a. -> killer single 1 with 5(2): 1 locked for n8
12b. note: if 17(3) has 1 -> r89c5 = {79} -> r3c5 = 8

Key step
13. "45" on c6789: 3 remaining innies r235c6 = 15
13a. but {168} as [681] only, blocks r3c5 = 8 -> 1 cannot be in r8c4 (step 12b)-> no 1 left for c4
13b. = {267} only
13c. -> r3c6 = 7
13d. r25c6 = {26}: 2 locked for c6

14. 5(2)n8 = {14}: both locked for n8
14a. r9c67 = [91]
14b. r7c7 = 6: placed for d\
14c. r46c6 = {38}: both locked for n5

15. 11(3)r9c2 = {236/245}(no 7,8)
15a. 2 locked for r9

16. "45" on c9: 2 outies r1c17 = 16 = {79}: both locked for c8

Final cracking step, was very slow to see this
17. 7 in r9 in r9c159
17a. r5c5 sees all those -> no 7 in r5c5

18. 18(4)n5 must have two of {269} for r5c56
18a. = {1269} only
18b. -> r5c4 = 1
18c. 2,6,9 locked for n5

19. "45" on n8: 1 outie r6c5 = 1 remaining innie r9c4
19a. -> both are [5]
19b. r46c4 = [47]: both placed for their D
19c. -> r35c3 = 10 (cage sum) = [19] [alternatively, 1 is a hidden single on d\]

20. 8(2)r3c1 = {35}: both locked for c1

Much easier now. Don't forget the diagonals
Cheers
Ed


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 Post subject: Re: Assassin 408
PostPosted: Wed Jan 05, 2022 6:26 pm 
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And here's v1.85. Same solution and still an X, so 1-9 cannot repeat on either diagonal. SS gives it 1.85 and JSudoku uses 4 advanced steps. Look forward to finding out what I missed.
puzzle pic for a408v1.85:
Attachment:
a408v1.85.png
a408v1.85.png [ 82.24 KiB | Viewed 128 times ]
triple click code:
3x3:d:k:3072:3072:2049:2049:1538:1538:3587:5380:5380:7941:7941:7941:7941:7941:7941:3587:3587:5380:2056:2311:5399:6153:6153:6153:6666:3587:4363:2056:2311:2311:5399:4620:6666:3341:3341:4363:6166:6166:5399:4620:4620:4620:6666:4110:4363:4629:6159:6166:5399:7952:6666:4110:4110:4363:4629:6159:6166:7952:7952:1297:6666:5906:5906:4629:6159:6159:7952:7952:1297:3859:3859:5906:4629:2836:2836:2836:7952:2566:2566:3859:5906:
Cheers
Ed


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 Post subject: Re: Assassin 408
PostPosted: Thu Jan 06, 2022 9:47 pm 
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Happy New Year!

I think both your solutions were cleaner than mine. Here's how I did both of them.
Corrections thanks to Ed.
Assassin 408 WT:
1. Outies r12 = r3c8 = 2
24(3)n2 = {789}
-> Innies n3 = r3c79 = +10(2) = {46}
-> r3c123 = {135}
-> 12(2)n1 = {48}
-> 6(2)n2 = {15}
-> 8(2)r1 = {26}
-> r1c7 = 3
-> 21(3)n3 = [{79}5]
-> r2c78 = {18}

2. Innies n36 = r35c7 = +9(2) = [45]
-> r3c9 = 6
-> Innies n9 = r79c7 = +7(2) = {16}
-> r2c78 = [81]

3. HS 5 in c6 -> 6(2)n2 = [15]

4. Outies c9 = r17c8 = +16(2) = {79}
-> 5 in c8 in r89c8

5. Remaining cells c7 = r468c7 = {279}
-> 13(2)n6 = [76] or [94]
Also 15(3)n9 = [2{58}] or [7{35}]
-> 9 in c7 in r46c7

6! Short/long contradiction chains for original/later versions of puzzle.
Consider case where r79c7 = [16].
This puts r9c7 = 4, r46c6 = {79}, 5(2)n8 = {23}, 24(3)n2 = [{79}8]

(For original puzzle this leaves no solution for 17(3)n8 since it would have to be {179} contradicting r13c5 = [1(7|9)]. -> r79c7 = [61])

Also puts r25c6 = [61] and 8(2)r1 = [62]
Since r4c67 = {79} also puts 1 in r3 only in r3c2
This puts 8(2)r3c1 = {35}, r4c23 = {26}, 13(2)r4 = [94]
Also puts (HS 6 in D\) r5c5 = 6 which leaves no solution for remaining Innies n5 = {245}. (Both of (24) would have to go in r6c5.)

-> r79c7 = [61]

7. -> r9c6 = 9
-> r46c6 = {38}
-> 24(3)n2 = [{89}7]
Also -> 5(2)n8 = {14}
-> r25c6 = {26}
-> r2c45 = {34}

8! 1 in D\ only in r3c3 or r4c4
-> 2 not in 21(4)r3c3
-> 2 in D/ only in n7 or in r5c5
11(3)r9 = {236} or {245}
Also IOD n8 -> r6c5 = r9c4

Either 2 in r9c23 -> 2 in D/ in r5c5
or 2 in r9c4 -> 2 in r6c5
-> 2 in n5 in r56c5
-> r5c6 = 6
-> r2c6 = 2
-> 8(2)r1 = [26]

9. Since IOD n8 -> r6c5 = r9c4 -> r9c4,r6c5 from (25)
Remaining Innies n5 = +16(3)
-> Min r46c4 = +11(2)
-> (HS 1 in D\) r3c3 = 1
-> Remaining IOD n5 -> r5c3 = r6c5 + 4
Since r5c3 cannot be 6 -> r6c5 cannot be 2
-> r6c5 = 5
-> r5c3 = 9
-> r4c5 = 9
Also r9c4 = 5
-> 11(3)r9 = [245]
-> (HS 2 in D/) r5c5 = 2
-> r5c4 = 1
-> r46c4 = [47]
-> r1c89 = [79]
-> r7c8 = 9

10. Also 13(2)n6 = [76]
Also r3c12 = {35}
-> r4c123 = [5{13}] or [3{15}]
-> r4c9 = 2
Also r46c6 = [83]
Also 16(3)n6 = [394]
etc.


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 Post subject: Re: Assassin 408
PostPosted: Mon Jan 24, 2022 10:02 pm 
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Combining the 14(3) and 17(3) cages in a 31(6) cage made this variant a lot harder. My new step 7 was particularly heavy going.

Here's how I solved Assassin 408 v1.85:
Prelims

a) R1C12 = {39/48/57}, no 1,2,6
b) R1C34 = {17/26/35}, no 4,8,9
c) R1C56 = {15/24}
d) R34C1 = {17/26/35}, no 4,8,9
e) R4C78 = {49/58/67}, no 1,2,3
f) R78C6 = {14/23}
g) R9C67 = {19/28/37/46}, no 5
h) 21(3) cage at R1C8 = {489/579/678}, no 1,2,3
i) 9(3) cage at R3C2 = {126/135/234}, no 7,8,9
j) 24(3) cage at R3C4 = {789}
k) 11(3) cage at R9C2 = {128/137/146/236/245}, no 9
l) 14(4) cage at R1C7 = {1238/1247/1256/1346/2345}, no 9

1a. Naked triple {789} in 24(3) cage at R3C4, locked for R3 and N2, clean-up: no 1 in R1C3, no 1 in R4C1
1b. 45 rule on N3 2 innies R3C79 = 10 = {46}, locked for R3 and N3, clean-up: no 2 in R4C1
1c. 21(3) cage at R1C8 = {579} (only remaining combination), 5,7 locked for N3
1d. 45 rule on R12 1 outies R3C8 = 2, clean-up: no 6 in R4C1
1e. Naked triple {135} in R3C123, locked for N1
1f. R1C12 = {48} (only remaining combination), locked for R1 and N1
1g. R1C56 = {15} (only remaining combination), locked for N2, 5 locked for R1 -> R1C7 = 3, clean-up: no 7 in R9C6
1h. Naked pair {79} in R1C89, locked for N3, 7 locked for R1 -> R2C9 = 5
1i. 45 rule on N6 1 outie R3C9 = 1 innie R5C7 + 1 -> R3C9 = 6, R5C7 = 5, R3C7 = 4, placed for D/, clean-up: no 8 in R4C7, no 8,9 in R4C8, no 6 in R9C6
1j. R3C9 = 6 -> 17(4) cage at R3C9 = {1268/1367}, no 4,9, 1 locked for C9 and N6
1k. R1C6 = 5 (hidden single in C6) -> R1C5 = 1
1l. 4 in C9 only in R789C9, locked for N9
1m. 45 rule on N9 2 innies R79C7 = 7 = {16}, locked for C7 and N9 -> R2C7 = 8, R2C8 = 1, placed for D/, R9C6 = {49}, clean-up: no 7 in R4C8
1n. 45 rule on C1 3 innies R125C1 = 19 = {289/469/478} (cannot be {379} which clashes with R34C1), no 1,3
1o. 7 of {478} must be in R2C1 -> no 7 in R5C1
[And to simplify things a bit]
1p. 45 rule on C9 2 outies R17C8 = 16 = {79}, locked for C8
1q. 9 in N6 only in R46C7, locked for C7

2a. R35C7 = [45] = 9 -> R46C6 + R7C7 = 17 = {179/368} (cannot be {269} = {29}6 which clashes with R9C67, cannot be {278} because R7C7 only contains 1,6) -> R46C6 = {38/79}, no 1,2,6
2b. 45 rule on C789 3 outies R469C6 = 20 = {389/479}, 9 locked for C6
2c. Killer pair 7,8 in R3C6 and R469C6, locked for C6
2d. Killer pair 3,4 in R469C6 and R78C6, locked for C6
2e. 4 in C6 only in R789C6, locked for N8
2f. R789C6 = {14}9/{23}4
2g. Naked pair {26} in R1C3 + R2C6, locked for N2

3a. 45 rule on N8 2 innies R9C46 = 1 outie R6C5 + 9
3b. Max R9C46 = 17 -> max R6C5 = 8
3c. Min R6C5 = 2 -> min R9C46 = 11, no 1 in R9C4
3d. R9C46 cannot total 13 -> no 4 in R6C5
3e. 31(6) cage at R6C5 = {125689/135679/235678}

4a. 45 rule on N47 3(2+1) outies R3C12 + R9C4 = 1 innie R5C3 + 4
4b. Min R3C12 = 4 -> R5C3 cannot be lower than R9C4, no 1 in R5C3

[Steps 1p and 1q added to the early part of my Assassin 408 WT; now for new steps for this variant.]

5a. 18(4) cage at R4C5 = {1269/1368/1458/1467/2358/2457/3456} (cannot be {1278/1359/2349/2367} which clash with R46C6)
5b. Consider combinations for R46C6 + R7C7 (step 2a) = {38}6/{79}1
R46C6 + R7C7 = {38}6, 3,8 locked for N5 => 18(4) cage at R4C5 = {1269/1467/2457}
or R46C6 + R7C7 = {79}1, 7,9 locked for N5, 1 placed for D\ => 18(4) cage must contain 1 for N5 = {1368/1458}
-> 18(4) cage = {1269/1368/1458/1467/2457}
5c. Combined cages 18(4) cage + R46C6 = {1269/1467/2457}{38}/{1368/1458}{79}, 8 locked for N5
5d. R9C46 = R6C5 + 9 (step 3a)
5e. Consider placements for R6C5 = {23567}
R6C5 = {257} => 18(4) cage = {1269/1368/1458/1467} (cannot be {2457} which clashes with R6C5)
or R6C5 = 3 => R46C6 = {79}, locked for N5 => 18(4) cage = {1368/1458}
or R6C5 = 6 => R9C46 = 15 = [69], R9C7 = 1 => 11(3) cage at R9C2 = {23}6 => 7 in R9 only in R9C159, CPE no 7 in R5C5 using diagonals => 18(4) cage = {1269/1368/1458/1467} (cannot be {2457} because 4,5,7 only in R4C5 + R5C4)
-> 18(4) cage = {1269/1368/1458/1467}, 1 locked for R5 and N5

6a. 45 rule on N47 6(3+3) outies R3C123 + R469C4 = 25, R3C123 = {135} = 9 -> R469C4 = 16 = {259/358/367/457} (cannot be {268} which clashes with R1C4, cannot be {349} which clashes with R2C4)
6b. 6 of {367} must be in R46C4 (R46C4 cannot be {37} which clashes with R46C6), no 6 in R9C4
6c. 6 in N8 only in R78C4 + R789C5, locked for 31(6) cage at R6C5, no 6 in R6C5
6d. R9C46 = R6C5 + 9 (step 3a)
6e. R469C4 = {259/367/457} (cannot be {358} = {35}8 which clashes with R6C5 + R9C46 = [384]), no 8 in R9C4
6f. 8 in N8 only in 31(6) cage at R6C5 (step 3e) = {125689/235678}
6g. 45 rule on C6789 3 remaining innies R235C6 = 15 = {168/267}
6h. Consider combinations for R235C6
R235C6 = {168} = [681] => R1C4 = 2 => R469C4 = {367/457}, 7 locked for C4
or R236C6 = {267} => R3C6 = 7
-> no 7 in R3C4

7a. R46C6 + R7C7 (step 2a) = {179/368} = {38}6/{79}1, R469C4 (step 6e) = {259/367/457}
7b. Consider placement for 1 on D\
R3C3 = 1 => 21(4) cage at R3C3 = {1479/1569} (cannot be {1389} because R46C4 = {39} cannot contain both of 3,9, cannot be {1578} because no 4 in R9C4)
or R3C3 = 3 => 21(4) cage = {3459/3567} (cannot be {2379} which clashes with R46C6 + R7C7 = {79}1, cannot be {3468} because R469C4 cannot contain both of 4,6)
or R3C3 = 5 => 21(4) cage = {3567} (cannot be {2568} because R469C4 cannot contain both of 2,6, cannot be {3459} because R469C4 cannot contain two of 3,4,9)
-> 21(4) cage = {1479/1569/3459/3567}, no 2,8
7c. Consider placement for R5C6 = {126}
R5C6 = 1 => R78C6 = {23}, locked for N8 => R469C4 = {367/457} => 21(4) cage = {1479/3459/3567} (cannot be {1569} because R46C4 cannot contain both of 5,6)
or R5C6 = 2 => 2 in 31(6) cage at R6C5 (step 6f) in N8, locked for N8 => R469C4 = {367/457} => 21(4) cage = {1479/3459/3567} (same reason)
or R5C6 = 6 ‘sees’ R46C4 + R5C3 => 21(4) cage = {1479/3459}
-> 21(4) cage = {1479/3459/3567}
7d. Consider combinations for 21(4) cage
21(4) cage = {1479} => R469C4 = {457} (only way to include two of 4,7,9)
or 21(4) cage = {3459} => R46C6 + R7C7 = {79}1, locked for N5 => R46C4 = {45} => R469C4 = {457}
or 21(4) cage = {3567}, no 9 => R469C4 = {367}
-> R469C4 = {367/457}, no 2,9, 7 locked for C4
7e. Killer pair 3,4 in R2C4 and R469C4, locked for C4
7f. Hidden killer pair 8,9 in 18(4) cage at R4C5 and R46C6 for N5, R46C6 contains one of 8,9 -> 18(4) cage (step 5e) must contain one of 8,9 = {1269/1368/1458}, no 7
7g. 18(4) cage = {1269/1368} (cannot be {1458} because 4,5 only in R4C5), no 4,5

[After that combination analysis, it gets easier from here]
8a. R4C4 = 4 (hidden single in N5), placed for D\ -> R1C1 = 8, placed for D\, R1C2 = 4, R2C45 = [34]
8b. R4C4 = 4 -> R69C4 (step 7d) = {57}, 5 locked for C4
8c. 5 in N5 only in R6C45, 5 locked for R6
8d. R4C4 = 4 -> 21(4) cage at R3C3 (step 7c) = {1479/3459} -> R5C3 = 9, R3C3 = {13}
8e. R1C1 = 8 -> R25C1 (step 1n) = 11 = [74/92]
8f. R4C8 = 6 -> R4C7 = 7, clean-up: no 1 in R3C1
8g. R8C7 = 2 -> R89C8 = 13 = [58], R6C7 = 9
8h. R5C2 = 7 (hidden single in R5) -> 24(4) cage at R5C1 = {4578} (only remaining combination containing one of 2,4 for R5C1) -> R5C1 = 4, R67C3 = [85], 5 placed for D/, R2C1 = 7
8i. Naked pair {26} in R12C3, locked for C3 and N1 -> R2C2 = 9, placed for D\
8j. R6C4 = 7, placed for D/, R6C6 = 3, placed for D\ -> R3C3 = 1, placed for D\, R7C7 = 6, placed for D\ -> R5C5 = 2, placed for D/, R1C9 = 9, placed for D/, R4C6 = 8, placed for D/
8k. R78C6 = {14} (only remaining combination), locked for N8, 1 locked for C6
8l. R9C4 = 5 -> R9C24 = 6 = [24]

and the rest is naked singles, without using the diagonals.


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