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 Post subject: Triankle Killer 1
PostPosted: Wed Oct 02, 2019 12:24 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 693
Location: Saudi Arabia
Triankle Killer 1

Rules
The numbers are from 0 to 9.
There are 8 triangular nonets covering twelve rows and 11 columns.
The two numbers in the centre two grey cells repeat horizontally and vertically. I.e. in row six there are eleven numbers so nine different numbers with the grey number twice and row seven the same. In column 6 there are 12 numbers eight different numbers with the two grey numbers twice.
The other eight numbers do not repeat anywhere.
The repeating numbers are present in every nonet.
Each of the other eight numbers is absent in one and only one nonet.
So Semi-symmetric.
There are five pairs of numbers which are unknown.
If a cell contains a number the opposite cell must contain it or its partner.


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 Post subject: Re: Triankle Killer 1
PostPosted: Mon Aug 23, 2021 5:56 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Back to Triankle Killers after a long break from them. They contain two features which I find makes them harder. Any killer puzzle with candidates 0-9 rather than the usual 1-9 is necessarily longer, often harder, because there are more possible cage combinations. Also I find semi-symmetric hard to visualise at times, as I've commented in my step 5, the key to my solving path.

Here is my walkthrough for Triankle Killer 1:
Numbers are 0 to 9.
The numbers in the grey cells repeat horizontally and vertically and are present in every nonet.
No repeats in any other rows or columns.
Each of the other numbers is missing from one and only one nonet.
Five pairs of unknown numbers are semi-symmetric, with the corresponding cell containing the same number or its partner.

Note that in the steps below locked, placements and killer pairs in R6, R7 and C6 don’t apply for R67C6, as appropriate. However placements and locked for N3 and N6 DO apply for R67C6, as appropriate.

1a. 2(2) cage at R4C5 = {02}, locked for C5 and N3
1b. 23(3) cage at R9C5 = {689}, locked for C5
1c. 9(2) cage at R6C5 = {45} (only remaining combination), locked for C5
1d. 5(2) cage at R5C6 = {14} (only remaining combination), locked for N3
[Not locked for C6 because 1,4 in R6C6 is one of the repeated numbers]
1e. 5(2) cage at R3C8 = [05/23]

2a. 4(2) cage at R8C7 = {04/13}
2b. 5(2) cage at R7C6 = {05/23} (cannot be {14} which clashes with 4(2) cage)
2c. Killer pair 0,3 in 5(2) and 4(2), locked for N6
2d. 3 in C5 only in R23C5, locked for N1
2e. 12(2) cage at R3C6 = {48/57}
2f. 11(3) cage at R1C6 = {038/128/146/236} (cannot be {029/056/245} because R1C5 only contains 1,3,7, cannot be {047} which clashes with 12(2) cage, cannot be {137} which clashes with R3C5), no 5,7,9
2g. 1 of {128/146} must be in R2C5 -> no 1 in R12C6

3a. 16(2) cage at R6C1 = {79}, locked for R6 and N2
3b. 11(2) cage at R6C10 = {38/56}
3c. 15(2) cage at R5C10 = [78/96]
3d. Killer pair 6,8 in R6C9 and 11(2) cage, locked for R6 and N4
3e. 5(2) cage at R4C9 = {05/14/23}
3f. 10(3) cage at R4C3 = {028/046/136/235} (cannot be {145} which clashes with R6C5)
3g. 8 of {028} must be in R5C34 (R5C34 cannot be {02} which clashes with R5C5), no 8 in R4C3

Some more cage combinations.
a. 12(2) cage at R3C4 = [48/57/75/93]
b. 9(2) cage at R6C7 = [09/18/27/36/45/54]
c. 10(2) cage at R7C1 = {19/28/37/46}, no 0,5
d. 15(2) cage at R7C11 = {69/78}
e. 4(2) cage at R8C3 = {04/13}
f. 5(2) cage at R11C6 = {05/14/23}

[Time to look at semi-symmetry.]
4a. R6C12 = {79} correspond with 15(2) cage at R7C10 = {69/78} -> one of 6,8 must correspond with one of 7,9
4b. R6C9 + 11(2) cage at R6C10 correspond with 10(2) cage at R7C1 + R7C3, R6C9 + 11(2) cage at R6C10 contain both of 6,8 for R6 -> 10(2) cage at R7C1 + R7C3 must contain one of 7,9
4c. Killer pair 7,9 in 10(2) cage at R7C1 + R7C3 and 15(2) cage at R7C10, locked for R7, clean-up: no 0,2 in R6C7
4d. Either 4(2) cage at R8C7 = {04}
or 4(2) cage = {13} => 9(2) cage at R6C7 = {45}
-> 4 in R6789C7, locked for C7, clean-up: no 8 in R3C6, no 1 in R11C6
4e. Either 9(2) cage at R6C7 = [18/36] => killer pair 6,8 in R7C7 and 15(2) cage at R7C10, locked for R7
or 9(2) cage at R6C7 = {45} => naked pair {45} in R7C57, locked for R7
-> 10(2) cage at R7C1 = {19/28/37}, cannot be {46}
4f. 10(2) cage and 15(2) cage must contain 9, locked for R7

5a. 5(2) cage at R7C6 = {05/23}, 4(2) cage at R8C7 = {04/13}, both in N6 -> {05}{13}/{23}{04}
5b. 2(2) cage at R4C5 = {02} corresponds with 4(2) cage at R8C7 and 5(2) cage at R5C6 = {14} corresponds with 5(2) cage at R8C7
[Maybe some solvers can visualise these steps; I found it necessary to type out step 5c to be able to see step 5d.]
5c. Either {02} corresponds with {13} and {14} with {05} => 0 corresponds with 1, 2 with 3 and 4 with 5
or {02} corresponds with {04} and {14} with {23} => 1 corresponds with 3, 2 with 4
-> 1 corresponds with one of 0,3, 2 with one of 3,4, 3 with one of 1,2, 4 with one of 2,5
5d. R6C7 corresponds with R7C5, R7C5 = {45} cannot correspond with 1 or 3 -> R67C7 = {45}, locked for C7, clean-up: no 7 in R3C6, no 0 in 4(2) cage at R8C7, no 0 in R11C6
[The key step, the rest is fairly straightforward.]
5e. Naked pair {13} in 4(2) cage at R8C7, locked for C7 and N6, clean-up: no 2 in 5(2) cage at R7C6, no 2,4 in R11C6
5f. R8C5 = 7
5g. Naked pair {45} in R7C57, locked for R7 except for possible repetition at R7C6
5h. Naked pair {45} in R6C57, locked for R6 except for possible repetition at R6C6, clean-up: no 6 in 11(2) cage at R6C11
5i. Naked pair {38} in 11(2) cage at R6C11, locked for R6 and N4, R6C9 = 6 -> R5C10 = 9, clean-up: no 2 in 5(2) cage at R4C10, no 6 in R7C11
5i. Naked pair {05} in 5(2) cage at R7C6, locked for N6
5j. Sub-step 5c now simplifies to 0 is paired with 1, 2 with 3 and 4 with 5, so 6 and 8 must be paired with 7 and 9 (previously step 4a only required one of 6,8 with one of 7,9)
5k. 11(2) cage at R6C11 = {38} corresponds with 10(2) cage at R7C1 -> 10(2) cage = {28/37}, no 1,9
5l. R7C12 = 9 (hidden single in R7) -> R7C11 = 6
5m. 16(2) cage at R6C1 = {79} corresponds with 15(2) cage at R7C11 = [69] -> 6 paired with 7, and 8 with 9 -> 16(2) cage = [97], clean-up: no 3 in R7C1
5n. R8C5 = 7 corresponds with R5C7 -> R5C7 = {67}
5o. 10(3) cage at R4C3 (step 3f) = {046/136/235} (cannot be {028} which clashes with R6C34, ALS block), no 8
5p. Killer triple 0,1,2 in 10(3) cage and R6C34, locked for N4
5q. 11(3) cage at R1C6 (step 2f) = {038/128/146/236}
5r. R11C6 = {35} corresponds with R2C5 -> R2C6 = {24}, R1C6 = {68}
5s. R1C6 = {68} corresponds with R12C6 -> R12C6 = {79}
5t. R6C10 = 6 corresponds with R7C3 -> R7C3 = 7, clean-up: no 3 in R7C1
5u. Naked pair {28} in 10(2) cage at R7C1, locked for R7 and N5
5v. Killer pair 4,5 in 5(2) cage at R4C10 and R6C7, locked for N4
5w. R2C7 corresponds with R11C5 = {689} -> R2C7 = {6789}

6a. R10,11C7 = {02} (hidden pair in C7), locked for N8
6b. Max R10C7 = 2 -> min R910C6 = 12, no 1,2 in R910C6
6c. R2C6 = 2 (hidden single in C6), R3C8 = 0 -> R4C8 = 5, clean-up: no 7 in R3C4, no 0 in R5C9
6d. 0 in R6 only in R6C34, locked for N2
6e. R2C6 = 2 corresponds with R11C6 -> R11C6 = 3 -> R11C7 = 2
6f. R10C7 = 0 -> R910C6 = 14 = [95] (cannot be {68} which clashes with R1C6)
6g. R3C6 = 4 -> R3C7 = 8, R1C6 = 6 -> R2C5 = 3 (cage sum), clean-up: no 8 in R4C4
6h. R4C6 = 8 (hidden single in C6)
6i. R5C6 = 1 -> R6C6 = 4 (cage sum), clean-up: no 4 in R4C9
6j. 4 pair with 5 R7C6 = 5, R8C6 = 0, clean-up: no 4 in R9C3

7a. R5C9 corresponds with R8C3, R5C9 = {45} -> R8C3 = 4, R9C3 = 0
7b. R6C4 = 0 (hidden single in R6)
7c. R7C9 = 0 (hidden single in R7), R4C9 = 1 -> R5C9 = 4, R56C7 = [72], R6C3 = 1, R5C7 = 6
7d. 9(2) cage at R6C5 = [45], 9(2) cage at R6C7 = [54]
7e. 10(3) cage at R4C3 = {235} (only remaining combination), locked for N2 -> R5C2 = 8, 10(2) cage at R7C1 = [82]
7f. R7C1 corresponds with R6C11 -> 11(2) cage at R6C10 = [38]
7g. R4C3 = {23} corresponds with R9C9 -> R9C9 = {23} (can also eliminate 7 by no 6 in N2 -> no 7 in corresponding nonet N7)
7h. 15(3) cage at R8C9 = {258} (only remaining combination) -> R9C9 = 2, R8C910 = {58}, locked for N7
7i. Naked pair {13} in R78C8, 1 locked for C8
7j. 2 missing from N6 -> 3 missing from N3 -> R4C4 = 7, R3C4 = 5, R24C7 = [79]
7k. 9 missing from N1 -> 8 missing from N8
7l. Naked pair {69} in R10,11C5, locked for C5 and N8 -> R9C5 = 8
7m. R5C10 = 9 corresponds with R8C2 -> R8C2 = 9
7n. R2C7 = 7 corresponds with R11C5 -> R11C5 = 6
7o. R5C2 = 8 corresponds with R8C10 -> R8C10 = 8

and the rest is naked singles.

Solution:
Attachment:
Triankle Killer 1.jpg
Triankle Killer 1.jpg [ 63.17 KiB | Viewed 3656 times ]


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