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HATMAN
PostPosted: Sat Feb 24, 2024 3:54 pm 
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Joined: Wed Apr 30, 2008 9:45 pm
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Location: Saudi Arabia
I will put all my Rainbow Killers here.
Other Rainbows will be posted seperately in Other Rainbow Puzzles.

The Rainbow Killer puzzles may be X, Windoku, Semi-Symmetric or just plain killers.

Rainbow:
The numbers are from 0-9.
There is a zero in every nonet, row and column. Zero was chosen as it increases the killer combinations, so I am sticking to it.
Each other number is missing from one and only one nonet, row and column.
It is proven that the nonet is at the crossover of the row and column.

SemiSymmetric: there are five unknown pairs of numbers.
Opposite each cell number: the opposing number is itself or its partner.
At least one cell pair is required to be self-symmetric.

From this: r5c5 = 0 zero and its partner is missing in n5, r5 and c5.
Explanation of r5c5 = 0:
Consider R5 SS so must all be anti-symmetric hence r5c5 paired with missing number R5m
Consider C5 SS so must all be anti-symmetric hence r5c5 paired with missing number C5m
Must be same missing number R5m = C5m -> r5c5 paired with missing number for N5 N5m
Four pairs must be missing from opposite nonets and one pair is r5c5 missing N5 N5m
Zero is present in every nonet so cannot be one of the four pairs, so its partner cannot be one of the four pairs, so its partner is missing from N5 = N5m so r5c5 = 0

Self-symmetric is harder to find than anti symmetric. With X I have found that the only pair that is self-symmetric or split symmetry is zero and its partner (not proven).
I have been trying to find a solution where all the pairs are split symmetry - unsuccessful so far. In my solutions I try and ensure that at least one pair is split symmetry - to avoid short-cutting.


Rainbow Killer SS 12

Image

Very easy, about 0.6.


Last edited by HATMAN on Thu Mar 07, 2024 6:57 am, edited 3 times in total.

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Andrew
PostPosted: Sun Feb 25, 2024 5:45 am 
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Location: Lethbridge, Alberta, Canada
Yes, very easy, so I won't post a walkthrough

Just a few comments, but spoiler warning:
The 45s are very useful for finding which numbers are missing from R5, C5 and then pairs of numbers missing from R12, R34, R67 and R89 after some early placements.
I usually find semi-symmetry hard but it's extremely helpful for this puzzle to get a quicker solution.

Solution:
0 1 3 9 4 6 2 7 8
2 4 7 1 8 0 5 6 3
6 8 9 5 3 2 1 0 4
4 7 8 3 2 5 0 9 1
3 5 2 8 0 9 6 4 7
1 9 0 4 6 7 8 3 5
5 0 1 6 7 4 9 8 2
7 2 4 0 9 1 3 5 6
9 3 6 2 5 8 7 1 0
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HATMAN
PostPosted: Tue Feb 27, 2024 7:55 am 
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Joined: Wed Apr 30, 2008 9:45 pm
Posts: 694
Location: Saudi Arabia
Rainbow Killer SS 12 H

Easy 12 was intended as a basis for this and a KiMo (which has not happened yet). In any case I do like to put up easier ones occasionally. I agree that anything below 1.0 is not worth a walkthrough. If there is something tricky or interesting - only that bit merits posting.

About 1.25 - one major step and a few tricky bits.

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Andrew
PostPosted: Sat Mar 02, 2024 1:37 am 
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Location: Lethbridge, Alberta, Canada
Thanks HATMAN for a challenging puzzle with some interesting steps. These days I use forcing chains where necessary; some of those steps may be quicker if looked at as contradictions. Also my solving path is how I came across the steps, I don't optimise my WTs; maybe there's something which could have been done earlier.

A very different puzzle from the easier version which has cages in all cells; my comments for that in hidden text don't apply for this one with so much "empty space"

Here's how I solved Rainbow Killer SS 12 H:
Prelims

a) R1C12 = {01}
b) R12C8 = {49/58/67}, no 0,1,2,3
c) R12C9 = {29/38/47/56}, no 0,1
d) R2C12 = {06/15/24}, no 3,7,8,9
e) R34C5 = {05/14/23}, no 6,7,8,9
f) R34C6 = {07/16/25/34}, no 8,9
g) R34C7 = {01}
h) R6C34 = {04/13}
i) R67C5 = {49/58/67}, no 0,1,2,3
j) R7C34 = {07/16/25/34}, no 8,9
k) R89C1 = {79}
l) R89C2 = {05/14/23}, no 6,7,8,9
m) R89C8 = {06/15/24}, no 3,7,8,9
n) R89C9 = {06/15/24}, no 3,7,8,9
o) 25(4) cage at R3C1 = {1789/2689/3589/3679/4579/4678}, no 0

1a. Naked pair {01} in R1C12, locked for R1 and N1, clean-up: no 5,6 in R2C12
1b. Naked pair {24} in R2C12, locked for R2 and N1, clean-up: no 9 in R1C8, no 7,9 in R1C9
1c. Naked pair {01} in R34C7, locked for C7
1d. Naked pair {79} in R89C1, locked for C1 and N7, clean-up: no 0 in R7C4
1e. 0 in N3 only in R3C789, locked for R3, clean-up: no 5 in R4C5, no 7 in R4C6

2a. 45 rule on R5 total = 44 -> no 1 in R5
2b. Using the property of Rainbow SS puzzles which HATMAN states, R5C5 = 0, which is paired with 1, the missing number in R5, so no 1 in C5 and N5, 1 in every other row, column and nonet, clean-up: no 4,5 in R3C5, no 6,7 in R3C6, no 4 in R4C5, no 3,4 in R6C3
2c. R5C5 = 0 -> R5C46 = 17 = {89}, locked for R5 and N5; 8 and 9 must be paired, clean-up: no 4,5 in R7C5
2d. Naked pair {23} in R34C5, locked for C5
2e. 10(3) cage at R5C1 = {235} (only remaining combination), locked for R5 and N4
2f. Naked triple {467} in 17(3) cage at R5C7, locked for N6

3a. R1C12 = {01} correspond with R9C89 and 0,1 paired -> R9C89 = {01}, locked for R9 and N9, clean-up: no 4,5 in R8C2
3b. R9C89 = {01} -> R8C89 = {56}, locked for R8 and N9
3c. R2C12 = {24} correspond with R8C89 = {56} -> 2,4 paired with 5,6 -> 3 and 7 must be paired
3d. R34C5 = {23} correspond with R67C5 -> R67C5 = {67}, locked for C5, 2 paired with 6 -> 4 paired with 5
3e. R34C7 correspond with R67C3 -> R67C3 = {01}, locked for C3, R7C4 = {67}
3f. Naked pair {67} in R7C45, locked for R7 and N8
3g. R12C9 = [29/38/47/83] (cannot be {56} which clashes with R8C9), no 5,6
3h. 0 in R4 only in R4C789, locked for N6
3i. R2C46 = {01} (hidden pair in R2), 1 locked for N2, clean-up: no 6 in R4C6
3j. 12(4) cage at R8C3 = {0246/0345/1236/1245} (cannot be {0129/0138} because 0,1 only in R8C4), no 8,9
3k. 0,1 only in R8C4 -> R8C4 = {01}
3l. R4C6 corresponds to R6C4, no 7 in either -> R4C6 = 5, R6C4 = 4, R3C6 = 2, R34C5 = [32], R6C3 = 0, R7C34 = [16], R67C5 = [67], clean-up: no 4 in R9C2
3m. R12C9 correspond with R89C1, R89C1 = {79} -> R12C9 = {38}, locked for C9 and N3, clean-up: no 5 in R12C8
3n. R12C7 correspond with R89C3, no 8,9 in R89C3 -> no 9 in R12C7
3o. R2C4 corresponds with R8C6, R2C4 = {01} -> R8C6 = {01}
3p. Naked pair {01) in R8C46, 0 locked for R8 and N8, clean-up: no 5 in R9C2
3q. Naked pair {23} in R89C2, locked for C2 and N7 -> R2C12 = [24], R8C3 = 4
3r. 10(3) cage at R5C1 = [352] -> 17(3) cage at R5C7 = [647], clean-up: no 9 in R2C8
3s. Naked pair {67} in R12C8, locked for N3, 6 locked for C8 -> R2C7 = 5, R8C8 = 5, R9C8 = 1, R89C9 = [60]
3t. 12(4) cage at R8C3 = {0246/0345/1236/1245}
3u. 2,3 only in R9C4 -> R9C4 = {23}
3v. R9C4 = {23} corresponds with R1C6 -> R1C6 = {67}
3w. Naked pair {67} in R1C68, locked for R1
3x. Naked pair {89} in R28C5, locked for C5
3y. Naked pair {23} in R9C24, locked for R9

4a. R4C2 corresponds with R6C8, no 0,1 in R6C8 -> no 1 in R4C2
4b. 25(4) cage at R3C1 = {4579/4678} (cannot be {1789} which clashes with R6C1) -> R4C1 = 4, 7 locked for C2
4c. R4C1 corresponds with R6C9 -> R6C9 = 5
4d. 20(4) cage at R1C3 = {0389/0569/0578/1379/1568}
4e. Consider placements for R9C3 = {56}
R9C3 = 5 => R19C5 = [54] => 20(4) cage = {0389/1379}
or R9C3 = 6 => R8C3 => R89C4 = 2 = [02] => 20(4) cage = {1379}
-> 20(4) cage = {0389/1379}, no 5,6
4f. Naked pair {89} in R1C4 + R2C5, locked for N2
4g. R1C4 = {89} corresponds with R9C6 -> R9C6 = {89}
4h. Naked pair {89} in R8C5 + R9C6, locked for N8
4i. Consider placements for R7C6 = {34}
R7C6 = 3 => R9C4 = 2
or R7C6 = 4, R9C5 = 5, R9C3 = 6, R8C3 = 4 => R89C4 = 2 = [02]
-> R9C4 = 2, R89C2 = [23]
4j. R1C6 corresponds with R9C4, R9C4 = 2 -> R1C6 = 6, R12C8 = [76]
4k. R3C8 corresponds with R7C2, R4C9 corresponds with R6C1
4l. Consider combinations for 25(4) cage = {4579/4678}
25(4) cage = {4579} => R3C8 = 0 and/or R4C9 = 1 => R34C7 = [10]
or 25(4) cage = {4678} => R6C1 = 1 and/or R7C2 = 0 => R3C8 = 0 and/or R4C9 = 1 => R34C7 = [10]
-> R34C7 = [10]
4m. R3C8 = 0 (hidden single in R3) -> R7C2 = 0, R1C12 = [01], R6C1 = 1 (hidden single in R6), R4C9 = 1 (hidden single in R4)
4n. R4C8 corresponds with R6C2 = {89} -> R4C8 = {89}
4o. R3C9 corresponds with R7C1
4p. Consider placements for R7C1 = {58}
R7C1 = 5 => 25(4) cage = {4678}
or R7C1 = 8, R3C9 = 9 => 25(4) cage = {4678} (cannot be {4579} = [5749] which clashes with R3C4)
-> 25(4) cage = {4678}, no 5,9

[Took me a while to spot this key step]
5a. Consider permutations for R3C4 = {57}
R3C4 = 5 => R19C5 = [45], R9C3 = 6
or R3C4 = 7, no 5 in C4 => all other columns must contain 5, R7C1 = 5 (then hidden single in C1) => R9C3 = 6
-> R9C3 = 6, R8C3 = 4, R9C4 = 2 -> R8C4 = 0 (cage sum)
[Cracked, straightforward from here]
5b. R2C4 = 1 -> 20(4) cage at R1C3 (step 4d) = {1379} -> R1C34 = [39], R2C3 = 7
5c. R1C3 = 3 corresponds with R9C7 -> R9C7 = 7
5d. R2C3 = 7 corresponds with R8C7 -> R8C7 = 3
5e. Naked pair {68} in R3C12, locked for 25(4) cage at R3C1, 8 locked for R3
5f, R4C2 = 7 corresponds with R6C8 -> R6C8 = 3
5g. All other columns contain 7 -> no 7 in C4 -> R3C4 = 5, R3C3 = 9
5h. R3C4 = 5 corresponds with R7C6 -> R7C6 = 4, R9C5 = 5
5i. R1C57 = [42], R4C38 = [89], R6C27 = [98], R7C789 = [982]
5j. R7C9 = 2 corresponds with R3C1 -> R3C1 = 6

and the rest is naked singles.
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