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 Post subject: Para's Killer-X Revisit
PostPosted: Fri Jan 15, 2021 8:10 pm 
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An X killer so 1-9 cannot repeat on either diagonal


Para's Killer-X Revisit
This should be a nice candidate for a Revisit especially since I love X killers. It didn't get a lot of attention when Para first released it in 2007. Looking for interesting, elegant solutions, especially using the newer techniques that have become the norm. SudokuSolver gives it 1.60, JSudoku uses 4 'complex intersections'.

triple click code:
3x3:d:k:4864:2049:2049:2049:6402:5379:5379:5379:3844:4864:4864:2565:2565:6402:1798:1798:3844:3844:4615:4615:4615:4360:6402:4617:3082:3082:3082:4619:4360:4360:4360:6402:4617:4617:4617:4876:4619:4619:7437:7437:7437:7437:7437:4876:4876:4619:6670:6670:6670:3855:3600:3600:3600:4876:3601:3601:3601:6670:3855:3600:4882:4882:4882:4115:4115:3348:3348:3855:2837:2837:2838:2838:4115:3607:3607:3607:3855:4120:4120:4120:2838:
Solution:
+-------+-------+-------+
| 4 1 5 | 2 3 6 | 7 8 9 |
| 9 6 2 | 8 7 4 | 3 1 5 |
| 7 8 3 | 5 9 1 | 2 6 4 |
+-------+-------+-------+
| 8 4 1 | 7 6 3 | 5 9 2 |
| 2 3 6 | 1 5 9 | 8 4 7 |
| 5 9 7 | 4 2 8 | 1 3 6 |
+-------+-------+-------+
| 1 5 8 | 6 4 2 | 9 7 3 |
| 3 7 4 | 9 1 5 | 6 2 8 |
| 6 2 9 | 3 8 7 | 4 5 1 |
+-------+-------+-------+
Cheers
Ed


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PostPosted: Sat Jan 23, 2021 5:33 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 see that I only solved this puzzle the first time in 2012, when I was blitzing puzzles which I hadn't previously solved.

As I commented then, a fun puzzle, it's a "proper" Killer-X, not just for uniqueness; the interesting steps were fairly late in my solving path.

Thanks Ed for pointing out my minor carelessness in step 6a.
Here is how I solved Para's Killer-X Revisit:
Prelims

a) R2C34 = {19/28/37/46}, no 5
b) R2C67 = {16/25/34}, no 7,8,9
c) R8C34 = {49/58/67}, no 1,2,3
d) R8C67 = {29/38/47/56}, no 1
e) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
f) 8(3) cage at R1C2 = {125/134}
g) 21(3) cage at R1C6 = {489/579/678}, no 1,2,3
h) 19(3) cage at R7C7 = {289/379/469/478/568}, no 1
i) 11(3) cage at R8C8 = {128/137/146/236/245}, no 9
j) 26(4) cage at R6C2 = {2789/3689/4589/4679/5678}, no 1
k) 14(4) cage at R6C6 = {1238/1247/1256/1346/2345}, no 9

1a. 45 rule on C5 1 innie R5C5 = 5, placed for both diagonals
1b. 45 rule on C1234 2 innies R5C34 = 7 = {16/34}
1c. 45 rule on C6789 2 innies R5C67 = 17 = {89}, locked for R5

2a. 8(3) cage at R1C2 = {125/134}, 1 locked for R1
2b. 45 rule on N1 3 innies R1C23 + R2C3 = 8 = {125/134}, 1 locked for N1
2c. 45 rule on N1 2 outie R12C4 = 10 = [19/28/37/46]

3a. 45 rule on R89 2 outies R67C5 = 6 = {24}, locked for C5
3b. 45 rule on R89 2 innies R89C5 = 9 = {18/36}
3c. 45 rule on R12 2 innies R12C5 = 10 = {37}/[91]
3d. 45 rule on R12 2 outies R34C5 = 15 = {69/78}

4a. 45 rule on N3 2 outies R12C6 = 10 = [46/64/73/82/91], no 5, clean-up: no 2 in R2C7
4b. 45 rule on N7 2 outies R89C4 = 12 = {48/57}/[93], no 6 in R8C4, no 1,2,6,9 in R9C4, clean-up: no 7 in R8C3
4c. 45 rule on N9 2 outies R89C6 = 12 = {39/48/57}, no 1,2,6, clean-up: no 5,9 in R8C7
4d. 2 in N8 only in R7C456, locked for R7
4e. 45 rule on R7 3 innies R7C456 = 12 contains 2 = {129/246} (cannot be {237} which clashes with R89C46), no 3,5,7,8

5a. 45 rule on R1234 2 innies R4C19 = 10 = {19/28/37/46}, no 5
5b. 45 rule on R6789 2 innies R6C19 = 11 = {29/38/47/56}, no 1
5c. 1 in R6 only in R6C678, locked for 14(4) cage at R6C6
5d. 14(4) cage contains 1 = {1238/1247/1256/1346}
5e. R7C456 (step 4e) = {246} (only remaining combination), 4,6 locked for R7 and N8, clean-ups: no 9 in R8C3, no 3 in R89C5 (step 3b), no 7 in R8C7
5f. Naked pair {18} in R89C5, locked for C5, 8 locked for N8, clean-ups: no 9 in R1C5 (step 3c), no 5 in R8C3, no 3 in R8C7
5g. Naked pair {37} in R12C5, locked for N2, 7 locked for C5, clean-ups: no 3 in R2C3, no 4 in R2C7
5h. 19(3) cage at R7C7 = {379} (only remaining combination), locked for R7 and N9
5i. Naked triple {158} in 14(3) cage at R7C1, locked for N7, clean-ups: no 5 in R8C4, no 7 in R9C4 (step 4b)
5j. 5 in N2 only in R3C46, locked for R3
5k. 45 rule on R3 3 innies R3C456 = 15 = {159/456} (cannot be {258} because R3C5 only contains 6,9), no 2,8
5l. R3C5 = {69} -> no 6,9 in R3C46
5m. 16(3) cage at R8C1 = {349/367}, no 2, 3 locked for N7
5n. Killer pair 4,6 in 16(3) cage and R8C3, locked for N7
5o. 2 in N7 only in R9C23, locked for R9
5p. 14(3) cage at R9C2 = {239/257}
5q. 16(3) cage at R9C6 = {169/349/457} (cannot be {178} which clashes with R9C5, cannot be {358} which clashes with R9C4, cannot be {367} which clashes with 14(3) cage), no 8
5r. 7,9 only in R9C6 -> R9C6 = {79}, R8C6 = {35} (step 4c), clean-up: no 2,4 in R8C7
5s. Naked triple {279} in R9C236, 7,9 locked for R9
5t. 45 rule on R9 3 innies R9C159 = 15 = {168/348}
5u. 6 of {168} must be in R9C1, 3 of {348} must be in R9C1 -> R9C1 = {36}, no 6 in R9C9
5v. 4 in N7 only in R8C123, locked for R8
5w. 45 rule for N7 1 innie R8C3 = 1 outie R9C4 + 1
5x. Naked pair {35} in R8C6 + R9C4 -> R8C37 = [68/46], 6 locked for R8

6a. 45 rule on R1 3 innies R1C159 = 16 = {349/358/367/457} (cannot be {259/268} because R1C5 only contains 3,7), no 2
[Forgot that I’d already eliminated 5 from R1C19 in step 1a. Doesn’t matter after the next sub-step, which also removed the need to lock 3 for R1.]
6b. 2 in R1 only in 8(3) cage at R1C2 = {125}, 5 locked for R1 and N1, clean-ups: no 6 in R2C4 (step 2c), no 4 in R2C3
6c. Naked triple {125} in R1C23 + R2C3, 2 locked for N1
6d. 21(3) cage at R1C6 = {489/678}, 8 locked for R1
6e. 2 in R3 only in R3C789, locked for N3

7a. R1C159 (step 6a) = {349/367}
7b. 3 of {349/367} must be in R1C5 (cannot be [376/673] which clash with R9C1 using D/) -> R1C5 = 3, R2C5 = 7
[The first fun step.]
7c. 18(3) cage at R3C1 = {378/468} (cannot be {369} which clashes with R3C5), no 9, 8 locked for R3 and N1

8a. 45 rule on N3 1 outie R1C6 = 1 innie R2C7 + 3
8b. 45 rule on N3 3 innies R1C78 + R2C7 = 18 = {189/378/459/468/567} (cannot be {369} = {69}3 which clashes with R1C6 + R2C7 = [63])
8c. 15(3) cage at R1C9 = {159/348/357/456} (cannot be {168} = 6{18} which clashes with R2C34)
8d. 9 of {159} must be in R1C9 -> no 9 in R2C89
8e. 4 of {348} must be in R1C9, 5 of {456} must be in R2C9 -> no 4,6 in R2C9
8f. R1C19 (step 7a) = {49/67}
8g. Consider placements for R9C1 = {36}
8h. R9C1 = 3 => R9C4 = 5 => R8C9 = 5 (hidden single in N9), no 5 in R2C9 => 15(3) cage = {348}
or R9C1 = 6, no 6 in R1C1 and R1C9 + R2C8 using D/ => R1C19 = {49} => 15(3) cage = {159/348}
-> 15(3) cage = {159/348}, no 6,7, clean-up: no 6,7 in R1C1
[Cracked after the second fun step.]

9a. Naked pair {49} in R1C19, locked for R1, clean-ups: no 1,6 in R2C6 (step 4a), no 1,6 in R2C7
9b. 7 in N1 only in 18(3) cage at R3C1 (step 7c) = {378}, 3,7 locked for R3, 3 locked for N1
9c. Naked pair {49} in R1C19, CPE no 4 in R9C9 using D\
9d. 11(3) cage at R8C8 = {128} (only remaining combination), 1,8 locked for N9
9e. R8C7 = 6 -> R8C6 = 5, R8C3 = 4 -> R8C4 = 9, R9C46 = [37], R9C1 = 6, placed for D/, clean-ups: no 4 in R4C9 (step 5a), no 5 in R6C9 (step 5b)
9f. Naked pair {49} in R12C1, locked for C1 and N1, R2C2 = 6, placed for D\, clean-ups: no 1,6 in R4C9 (step 5a), no 2,7 in R6C9 (step 5b)
9g. R3C4 = 5 (hidden single in R3)
9h. R2C9 = 5 (hidden single in C9) -> R1C9 + R2C8 = 10 = [91], both placed for D/, clean-ups: no 1 in R4C1 (step 5a), no 2 in R6C1 (step 5b)
9i. R1C1 = 4, placed for D\, R2C34 = [28], R2C67 = [43], R1C6 = 6, R3C56 = [91], R1C4 = 2, R4C5 = 6, R7C3 = 8, placed for D/, R7C456 = [642], R6C5 = 2
9j. R4C6 = 3, placed for D/, R6C6 = 8, placed for D\, clean-up: no 7 in R4C19 (step 5a)
9k. Naked pair {28} in R4C19, locked for R4
9l. R34C6 = [13] = 4 -> R4C78 = 14 = {59}, locked for N6, 9 locked for R4
9m. R8C2 = 7, placed for D/ -> R6C4 = 4, placed for D/, R5C4 = 1 -> R5C3 = 6 (cage sum), R4C4 = 7, placed for D\
9n. R67C4 = [46] = 10 -> R6C23 = 16 = [97]
9o. R6C67 + R7C6 = [812] -> R6C8 = 3 (cage sum)

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


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