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 Post subject: Triankle Killer 7
PostPosted: Mon Nov 25, 2019 7:01 pm 
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Grand Master
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 656
Triankle Killers 7

This one is hard: probably low assassin level.

Rules
The numbers are from 0 to 9.
There are 8 triangular nonets covering twelve rows and 11 columns.
The two different 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.


The Diagram was wrong, I have now corrected it. The cage in nonet 5 is 13/3 not 12/3.
Apologies to Andrew and anyone else who has tried it.


Triankle Killer 7
The red 22 cage at r7c5 may repeat.

Image

If you wish to solve in JSudoku:
open as a 12 by 12 Latin Square from 0 - B
enter the eight nonets as killer cages with no sum (c, then choose operator "none")
select a cell in C6: ctrl right click select remove and then C6
select a cell in R6: ctrl right click select remove and then R6
select a cell in R7: ctrl right click select remove and then R6 (JS has renumbered the rows)
select all the nonet cells shift A then shift B to remove the A & B pencilmarks
do a set of solves to put A B in every cell around the diamond.
select r67c6 as a twin killer cage with no sum (twin killer is easier to see)
select r6 c1-5 & c7-11 as a twin killer cage 45/10
select r7 c1-5 & c7-11 as a twin killer cage 45/10
select r1-5&r8-12 c6 as a twin killer cage 45/10
Save as your TRIANKLE BASE


Last edited by HATMAN on Wed Dec 22, 2021 1:54 pm, edited 1 time in total.

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 Post subject: Re: Triankle Killer 7
PostPosted: Sun Nov 07, 2021 9:07 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1804
Location: Lethbridge, Alberta, Canada
Thanks HATMAN for correcting the puzzle after I asked him why I couldn't make the cage at R7C1 total 12. Fortunately that came right at the end of my solving path so I didn't need any rework.

Another interesting Triankle Killer, definitely Assassin level. All killer puzzles using 0-9 are harder because of the increased number of combinations. However it's the semi-symmetric aspect, where one has to work out matching pairs, which I find particularly hard so I'm pleased that I've reached the end of the Triankle Killers.

Here is my walkthrough for Triankle Killer 7:
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.

The 22(3) cage at R7C5 crosses the lower grey cell and may have repeats.

Prelims

a) 6(2) cage at R6C2 = {06/15/24}
b) 10(2) cage at R6C4 = {19/28/37/46}
c) 15(2) cage at R6C7 = {69/78}
d) 5(2) cage at R6C10 = {05/14/23}
e) 8(2) cage at R7C2 = {08/17/26/35}
f) 7(3) cage at R1C6 = {016/025/034/124}
g) 9(3) cage at R2C5 = {018/027/036/045/126/135/234}
h) 24(3) cage at R2C6 = {789}
i) 20(3) cage at R4C4 = {389/479/569/578}
j) 7(3) cage at R4C8 = {016/025/034/124}
k) 22(3) cage at R7C5 (may contain repeats) = {499/589/679/688/778}
l) 5(3) cage at R8C5 = {014/023}
m) 9(3) cage at R9C4 = {018/027/036/045/126/135/234}
n) 24(3) cage at R9C5 = {789}
o) 8(3) cage at R10C4 = {017/026/035/125/134}

1a. 5(3) cage at R8C5 = {014/023}, CPE no 0 in R89C7
1b. Naked triple {789} in 24(3) cage at R9C5, CPE no 7,8 in R10C5, no 7 in R11C5

2a. 45 rule on N68 2 missing numbers = 16 must be 7 and 9
2b. 24(3) cage at R9C5 = {789} -> no other 7,9 in N68
2c. 7,9 in N6 only in R9C5 -> R9C5 = {79}
2d. 24(3) cage at R9C5 = {789}, 8 locked for N8
2e. 15(3) cage at R7C6 = {168/258/348/456}, no 0
2f. 13(3) cage at R8C6 = {058/148/238/256/346}
2g. {058/148} cannot be {58}0/{48}1 which clash with 15(3) cage, no 0,1 in R10C8
2h. 22(3) cage at R7C5 = {499/589/679/688/778}
2i. 4 of {499} must be in R7C6 -> no 4 in R7C57
2j. 9(3) cage at R9C4 = {036/045/126/135/234} (cannot be {018} = 8{01} which clashes with 8(3) cage at R10C4), no 8
2k. 45 rule on N13 2 missing numbers = 3 must be 0,3 or 1,2

[Time to look at semi-symmetry and I remembered that, in addition to pairs, the numbers missing from corresponding nonets must be paired.]
3a. 0,3 or 1,2 missing from N13, 7,9 missing from N68 -> 7,9 must be paired with either 0,3 or 1,2
3b. 24(3) cage at R2C6 corresponds with 9(3) cage at R9C4 -> 7,9 paired 0,3 or 1,2 -> 9(3) cage must be {036/126}, 8 paired with 6
[Amusingly that was almost the exact opposite of a step I used in Triankle Killer 6 which prevented 9(3) innies in C5 of that puzzle being {036/126}.]
3c. 9(3) cage at R2C5 corresponds with 24(3) cage at R9C5 -> 9(3) cage at R2C5 = {036/126}
3d. 8 in R10C6 + R11C7 -> 6 in R2C5 + R3C6, locked for N1 and 9(3) cage, no 6 in R4C7
3e. 7(3) cage at R1C6 corresponds with 8(3) cage at R10C4, no 6,8 in 7(3) cage -> no 6 in 8(3) cage
3f. Looking at which of 0,1,2,3 is missing from N1
0 missing, 3 must be in 9(3) cage = {36}0, 7(3) cage = {124}
or 1 missing, 2 must be in 9(3) cage = {26}1, 7(3) cage = {034}
or 2 missing, 1 must be in 9(3) cage = {16}2, 7(3) cage = {034}
or 3 missing, 0 must be in 9(3) cage = {06}3, 7(3) cage = {124}
-> 7(3) cage = {034/124}, 4 locked for N1
3g. 5,7,8,9 all in N1 -> R3C45 = {5789}, CPE no 5 in R4C5

4a. 6 paired with 8
4b. 16(3) cage at R3C5 corresponds with 5(3) cage at R8C5, no 6,8 in 5(3) cage -> no 6,8 in 16(3) cage = {259/349/457} (cannot be {079} because if 7,9 paired with 0,3 then 0 cannot be paired with 2, and 7,9 cannot then be paired with 1,2 because 5(3) cage cannot contain both of 1,2), no 0,1
4c. R67C2 corresponds with 5(2) cage at R6C10, no 6,8 in 5(2) cage -> no 6,8 in R67C2, clean-up: no 0 in R6C3, no 0,2 in R7C3

5a. 7,9 paired with 0,3 or 1,2, 6 paired with 8
5b. R6C7 corresponds with R7C5
5c. 22(3) cage at R7C5 (step 2h) = {589/679/688} (cannot be {499/778} = [949/787] because no 0,1,2,3 in R6C7), no 4
5d. {589} cannot be [589] (because no 5 in R6C7), no 5 in R7C5
5e. 22(3) cage at R7C5 = {589/688} (cannot be {679} = [769/967] because R67C7 = {79} clash with R311C7, ALS block), no 7
5f. {589} must be [859/985] (cannot be [958] because R67C7 = {89} clash with R311C7, ALS block), no immediate candidate elimination
5g. R6C7 corresponds with R7C5 -> R6C7 = {689}, clean-up: no 8 in R6C8

6a. 6 paired with 8
6b. 11(3) cage at R4C4 corresponds with 15(3) cage at R7C6
6c. 15(3) cage at R7C6 (step 2e) = {258/348/456} (cannot be {168} because 11(3) cage cannot contain both of 6,8), no 1
6d. 11(3) cage must contain one of 6,8 = {038/056/128/146/236}, no 7,9
6e. 20(3) cage at R3C4 corresponds with 13(3) cage at R8C6
6f. 13(3) cage (step 2f) contains one of 6,8 -> 20(3) cage must contain one of 6,8 = {389/569/578}, no 4
6g. R7C6 = {568}, R6C6 cannot repeat in R7C6 -> no 5 in R6C6

7a. 7,9 paired with 0,3 or 1,2
7b. 9(3) cage at R2C5 corresponds with 24(3) cage at R9C5, 24(3) cage at R2C6 corresponds with 9(3) cage at R9C4
7c. 7(3) cage at R1C6 = {034/124} corresponds with 8(3) cage at R10C4 = {035/125/134}
7d. 7,9 paired with 0,3, 9(3) cages = {036} => 7(3) cage = {124}, 8(3) cage = {125} (cannot be {134} because 2 cannot be paired with 3)
or 7,9 paired with 1,2, 9(3) cages = {126}, 7(3) cage = {034}, 8(3) cage = {035} (cannot be {134} because 0 cannot be paired with 1)
-> 8(3) cage = {035/125}, no 4, 5 locked for N8
7e. 4 in N8 only in R10C78, CPE no 4 in R9C7
7f. 7(3) cage = {034/124}, 8(3) cage = {035/125} -> 4 paired with 5
7g. 4 paired with 5, 6 with 8, R7C6 = {568} -> R6C6 = {468} and either 4,5 or 6,8 must be the repeated numbers in C6, one of each in R6 and R7
7h. 11(3) cage at R4C4 (step 6d) = {038/056/128/146/236}
7i. 8 of {038/128} must be in R6C6 -> no 8 in R4C4 + R5C5
7j. 7,9 paired with 0,3 or 1,2 -> either 0,3 or 1,2 must be paired

8a. 4 paired with 5
8b. 16(3) cage at R3C5 corresponds with 5(3) cage at R8C5
8c. 16(3) cage (step 4b) = {259/349} (cannot be {457} because 5(3) cage cannot have more than one of 4,5), no 7
8d. X-Wing for 9 in 24(3) cage at R2C6 and 16(3) cage at R3C5 for N13, no other 9
8e. 20(3) cage at R3C4 = {578}, CPE no 5 in R4C4
8f. X-Wing for 7,8 in 24(3) cage at R2C6 and 20(3) cage at R3C4 for N13, no other 8 in N3
8g. R6C6 = {46} -> R7C7 = {58}
8h. 11(3) cage at R4C4 (step 6d) = {056/146} (cannot be {236} which clashes with 16(3) cage), no 2,3
8i. 5 of {056} must be in R5C5 -> no 0 in R5C5
8j. 11(3) cage corresponds with 15(3) cage at R7C6
8k. 15(3) cage (step 6c) = {258/348} (cannot be {456} because 11(3) cage only contains one of 4,5), no 6, 8 locked for N6
8l. 13(3) cage at R8C6 (step 2f) = {256/346}, no 0,1
8m. R4C5 = {78} corresponds with R9C7, no 4,5 in R4C5 -> R9C7 = {236}

9a. 4 paired with 5, 7,9 paired with 0,3 or 1,2 -> either 0,3 or 1,2 must be paired
9b. 16(3) cage at R3C5 corresponds with 5(3) cage at R8C5
9c. 16(3) cage (step 8c) = {259/349} contains one of 4,5 -> 5(3) cage must contain 4 = {014}, CPE no 4 in R8C7
9d. 16(3) cage = {259} (cannot be {349} because 3,9 cannot be paired with 0,1), 2 locked for N3
9e. X-Wing for 5 in 20(3) cage at R3C4 and 16(3) cage at R3C5 for N13, no other 5 in N13
9f. 11(3) cage at R4C4 = {146}, 1 locked for N3
9g. 0,3 in N3 only in R4C7 -> one of 0,3 missing in N3 -> 7,9 paired with 0,3, 1 paired with 2
9h. 11(3) cage at R4C4 corresponds with 15(3) cage at R7C6
9i. 11(3) cage = {146} -> 15(3) cage (step 8k) = {258}, locked for N6
9j. 13(3) cage at R8C6 (step 8l) = {346}
9k. 24(3) cage at R2C6 corresponds with 9(3) cage at R9C4 -> 9(3) cage at R9C4 = {036}, 8(3) cage at R10C4 = {125}, 1 locked for N8
9l. 24(3) cage at R9C5 corresponds with 9(3) cage at R2C5 -> 9(3) cage at R2C5 = {036}, 7(3) cage at R1C6 = {124}

10a. 20(3) cage at R3C4 = {578} corresponds with 13(3) cage at R8C6 = {346}, 4 paired with 5, 6 paired with 8 -> 7 paired with 3 -> 0 paired with 9
10b. R6C5 corresponds with R7C7, R7C7 = {5689} -> R6C5 = {4689}, R6C4 = {1246}
10c. R11C5 = 2 (hidden single in C5)
10d. R2C7 corresponds with R11C5 -> R2C7 = {12}
10e. R10C7 = 4 (hidden single in C7)
10f. R3C5 corresponds with R10C7 -> R3C5 = 5
10g. 20(3) cage at R3C4 = {578} -> R5C6 = 5
10h. 13(3) cage at R8C6 = {346} -> R8C6 = 4
10i. R12C6 = 1 -> R10C4 = 5, R1C6 = 2 -> R2C7 = 1, R3C6 = 4, R9C6 = 0 -> R8C5 = 1
10j. 11(3) cage at R4C4 = {146} -> R4C4 = 1, clean-up: no 9 in R6C5
10k. R4C4 corresponds with R9C8 -> R9C8 = 2
10l. R4C6 = 9 -> R5C7 = 2
10m. 9(3) cage at R9C4 = {036} -> R10C5 = 0
10n. R2C5 = 3 (hidden single in C5) -> R3C6 = 6, R4C7 = 0, R11C6 = 3 -> R9C4 = 6, R9C7 = 3, R10C8 = 6, clean-up: no 4 in R6C6, no 9 in R6C8
10o. R10C8 = 6 corresponds with R3C4 -> R3C4 = 8, R2C6 = 7, R3C7 = 9, R4C58 = [78], R9C5 = 9, R10C6 = 8, R11C7 = 7

11a. R5C5 = 4 (hidden single in C5) -> R6C6 = 6, this is the repeated number in R6
11b. R6C6 corresponds with R7C6 -> R7C6 = 8, this is the repeated number in R7, 6,8 are repeated in C6, R8C7 = 5
11c. 7(2) cage at R5C8 = {016} (only remaining combination), locked for R5 and N4 -> R6C57 = [68], R6C5 = 4, R6C8 = 7, R7C57 = [86], clean-up: no 0,2 in R6C2, no 2 in R6C3, no 0,2 in R7C2, no 1,4,5 in R7C10
11d. Naked pair {15} in 6(2) cage at R6C2, locked for R6, 5 locked for N2, clean-up: no 0 in R7C10

12a. R67C2 corresponds to 5(2) cage at R6C10 = {23} -> R67C2 = [17] (cannot be [13] which would place 5 in both of R67C3), R67C3 = [51], 5(2) cage = [32]
12b. Naked pair {29} in R6C911, locked for R6, 2 locked for N4
12c. R6C1 = 0 -> R4C2 + R5C3 = 12 = [39]
12d. R9C3 = 4 -> R7C1 + R8C2 = 9 = [36/90]
12e. R4C3 corresponds with R9C9 -> R9C9 = 7
12f. 5 missing from N5 -> 4 missing from N4 -> R4C9 = 5
12g. R7C3 corresponds with R6C9 -> R6C911 = [29]
12h. R6C1 corresponds with R7C11 -> R7C11 = 0
12i. R6C8 corresponds with R7C4 -> R7C14 = [93], R8C2 = 0
12j. R7C911 = [40], R9C9 = 7 -> R8C10 = 9 (cage sum)
12k. 7(3) cage at R5C8 corresponds with R8C234, R8C234 = [062] -> 7(3) cage = [160]
12l. R5C34 = [87], R7C8 = 5, R8C89 = [38]

I wonder whether:
any other solvers forgot that 45s can be used on some of the Triankle Killers. In one of the earlier ones 45 immediately gave the missing number in one triangle.

Solution:
Attachment:
Triankle Killer 7.jpg
Triankle Killer 7.jpg [ 66.3 KiB | Viewed 429 times ]


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