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HATMAN
 Post subject: Rainbow Killer 5
PostPosted: Wed Oct 18, 2023 5:10 am 
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Joined: Wed Apr 30, 2008 9:45 pm
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Rainbow Killer 5

The same rules as for Rainbow Killer 4
Rainbow as in "All the numbers in the rainbow"
The numbers are from 0-9
There is a zero in every nonet, row and column. Zero is chosen as it increases the killer combinations.
Each other number is missing from one and only one nonet, row and column.

I tried to do 5.5 and failed miserably, so I simplified it and still failed. So I simplified it further. Once I succeeded on 5.1, I slowly built up and eventually solved 5.5.
You may follow my increasing difficulty path, or you may wish to go straight to 5.5.

Rain K5.1
Image

Rain K5.2
Image

Rain K5.3
Image

Rain K5.4
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Rain K5.5
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Andrew
 Post subject: Re: Rainbow Killer 5
PostPosted: Sun Oct 22, 2023 9:26 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
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Thanks HATMAN for a nice starting puzzle. It's a few weeks since I solved Rainbow Killer 4 but my recollection is that was easier than Rainbow Killer 5.1 so I recommend anyone who hasn't previously tried a Rainbow Killer to try Rainbow Killer 4 first.

Here's how I solved Rainbow Killer 5.1:
Zero is in every row, column and nonet. One of 1-9 is missing in each row, column and nonet.

Prelims

a) R12C6 = {04/13}
b) R23C7 = {19/28/37/46}, no 0,5
c) R2C89 = {19/28/37/46}, no 0,5
d) R3C12 = {69/78}
e) R56C6 = {07/16/25/34}, no 8,9
f) R67C3 = {29/38/47/56}, no 0,1
g) R78C6 = {79}
h) R9C12 = {49/58/67}, no 0,1,2,3
i) 5(3) cage at R1C1 = {014/023}
j) 22(3) cage at R1C4 = {589/679}
k) 4(3) cage at R5C8 = {013}
l) 20(3) cage at R7C1 = {389/479/569/578}, no 0,1,2
m) 6(3) cage at R7C4 = {015/024/123}, no 6,7,8,9
n) 5(3) cage at R8C3 = {014/023}
o) 10(4) cage at R4C1 = {0127/0136/0145/0235/1234}, no 8,9
p) 28(4) cage at R4C7 = {4789/5689}, no 0,1,2,3

1a. Naked pair {79} in R78C6, locked for C6 and N8, clean-up: no 0,2 in R3C4, no 0 in R56C6
1b. R56C6 = {16/25} (cannot be {34} which clashes with R12C6), no 3,4
1c. 45 rule on N3, no 5 in N3
1d. 45 rule on N9, no 9 in N9
1e. 5(3) cage at R1C1 = {014/023}, 0 locked for N1
1f. 22(3) cage at R1C4 = {589/679}, 9 locked for N2, clean-up: no 0 in R3C6
1g. 4(3) cage at R5C8 = {013}, locked for N6
1h. 28(4) cage at R4C7 = {4789/5689}, 8,9 locked for N6

2a. Combined 20(3) cage at R7C1 + R9C12 form 33(5) cage = {36789/45789}, 7,8,9 locked for N7, clean-up: no 2,3,4 in R6C3
2b. 0 in N2 either in R12C6 = {04} or 17(3) cage at R2C4 = {089} -> no 4 in R23C4
2c. 0 in N3 either in 16(3) cage at R1C7 = {079} or in R3C89 = {04} -> no 4 in 16(3) cage
2d. 10(4) cage at R4C1 = {0127/0136/0145/0235} or 10(4) cage = {1234} => R45C3 = {09} -> 0 in R45C123, locked for N4
2e. 45 rule on N12 sum of missing numbers = 3 -> 1,2 missing from N12
2f. 45 rule on N1 R3C3 + missing number = 10, one of 1,2 missing -> R3C3 = {89}
2g. Killer pair 8,9 in R3C12 and R3C3, locked for R3 and N1, clean-up: no 1 in R3C56, no 1,2 in R2C7
2h. N1 must contain 5 only in 15(3) cage at R1C3 = {357/456}, no 1,2
2i. 45 rule on N8 R9C4 + missing number = 12, R78C6 = {79} -> R9C4 = 4, 8 missing from N8, clean-up: no 9 in R9C12
2j. No 9 in R9 -> all other rows must contain 9
2k. R9C4 = 4 -> R89C3 = {01}, locked for C3 and N7, clean-up: no 8,9 in R45C3
2l. 6(3) cage at R7C4 = {015/123}, 1 locked for N8
2m. 20(3) cage at R7C1 = {389/479} (cannot be {569/578} which clash with R9C12), no 5,6
2n. 45 rule on N5 R5C5 + missing number = 13, N5 must contain 5,8,9 -> missing number must be one of 4,6,7, R5C5 = {679}

3a. Consider combinations for 22(3) cage at R1C4 = {589/679}
22(3) cage = {589}, 8 locked for N2 => 17(3) cage at R2C4 cannot be [890] => 0 in N2 only in R12C6 = {04}
or 22(3) cage at {679} => 16(3) cage at R1C7 cannot be {079} (ALS clash) => 0 in N3 only in R3C89 = {04}, locked for R3
-> no 0 in R3C4, also no 0 in R2C4 because {089} can only be [890]
3b. 0 in N2 only in R12C6 = {04}, locked for C6, 4 locked for N2, clean-up: no 5 in R3C56
3c. 0 in R3 only in R3C89 = {04}, locked for N3, clean-up: no 6 in R23C7, no 6 in R2C89
3d. Killer pair 6,7 in R3C12 and R3C56, locked for R3, clean-up: no 3 in R2C7
3e. 5 is missing from N3 -> 6 in N3 only in 16(3) cage, locked for R1
3f. 22(3) cage = {589/679}
3g. 6 of {679} must be in R2C5 -> no 7 in R2C5
3h. 17(3) cage at R2C4 = {179/278/359/368} (cannot be {269} which clashes with R3C56)
3i. 1,2 of {179/278} must be in R3C4 -> no 1,2 in R2C4
3j. R3C3 = {89} -> no 8 in R2C4
3k. 8 missing from N8 -> N2 must contain 8 which is only in 22(3) cage = {589}, 5 locked for N2
3l. 17(3) cage = {179/278/368}
3m. 6,7 only in R2C4 -> R2C4 = {67}
3n. Consider combinations for R3C12 = {69/78}
R3C12 = {69}, 6 locked for R3 => R3C56 = [72]
or R3C12 = {78} => R3C3 = 9 => 17(3) cage = {179}
-> 17(3) cage = {179/368}, no 2
3o. Combined cage 17(3) + R3C56 = {179}{36}/{368}[72]
3o. No 5 in R3 -> all other rows must contain 5
3p. Hidden killer pair 1,2 in R3C46 and R3C7 for R3, R3C46 contain one of 1,2 -> R3C7 = {12}, R2C7 = {89}
3q. Consider combinations for 17(3) cage
17(3) cage = {179} = [791] => 6,7 in N3 only in 16(3) cage at R1C7 = {367} => R1C45 = {89} (hidden pair in N2, which is missing one of 1,2), locked for N2
or 17(3) cage = {368} => R3C56 = [72] => R3C7 = 1, R2C7 = 9
-> no 9 in R2C5
3r. 9 in R2 only in R2C789, locked for N3 -> 16(3) cage = {268/367}, no 1

4a. 8,9 in N4 only in R6C123, locked for R6
4b. 0 in N4 only in 10(4) cage at R4C1 = {0127/0136/0145} (cannot be {0235} which clashes with R45C3), 1 locked for N4
4c. 17(3) cage at R6C1 = {269/278/359/368/458} (cannot be {467} because R6C123 contains both of 8,9)
4d. Hidden killer pair 17(3) and R6C3 for R6, 17(3) cage only contains one of 8,9 -> R6C3 = {89}, R7C3 = {23}
4e. 1,2 missing from N12 -> N7 must contain 2 in R7C23, locked for R7
4f. 1,2 missing from N12 -> N6 must contain 2 in R6C79, locked for R6, clean-up: no 5 in R5C6
4g. 1,2 missing from N12 -> N4 must contain 2 in 10(4) cage at R4C1 = {0127} or R45C3 = {27} -> 7 in R45C123 (locking cages), locked for N4
4h. 11(3) cage at R5C4 = {038/137/146/236/245} (cannot be {029/128} because 2,8,9 only in R5C4, cannot be {056} which clashes with R56C6), no 9
4i. 8 of {038} must be in R5C4 -> no 0 in R5C4
4j. N5 must contain 0,8 in either 14(3) cage = {068} or 11(3) cage = {038} -> 14(3) cage cannot be {059}, locking-out cages
4k. 14(3) cage = {068/149/167/257} (cannot be {158/239/248/347/356} which clash with 11(3) cage = {038}), no 3
4l. 4 of {149} must be in R4C4 -> no 9 in R4C5
4m. 11(3) cage at R5C4 = {038/137/245} (cannot be {146/236} which clash with 14(3) cage = {068}), no 6
4n. The missing number in N5 must be one of {467} (step 2n), 3 in N5 only in 11(3) cage = {038/137}, no 2,4,5
4o. 7 of {137} must be in R56C4 (R56C4 cannot be {13} which clashes with R3C4), no 7 in R6C5
4p. Caged X-Wing for 3 in 11(3) cage and 4(3) cage at R5C8 for R56, no other 3 in R56, clean-up: no 6 in R4C3
4q. 17(3) cage = {269/368/458}, (cannot be {359} which clashes with R67C3 = [92])
4r. 2,3 of {269/368} must be in R7C2 -> no 6 in R7C2
4s. Consider combinations for 17(3) cage
17(3) cage = {269/368}, 6 locked for N4
or 17(3) cage = {458} => R7C3 = 2 (hidden single in N7), R45C3 = [36] (cannot be {45} which clashes with 17(3) cage)
-> 10(4) cage at N4 = {0127/0145}, no 3,6

5a. 11(3) cage at R5C4 (step 4n) = {038/137}
5b. Consider placements for R2C4 = {67}
R2C4 = 6 => R3C56 = [72], N5 must contain 2 => 14(3) cage = {257}, 0 must be in 11(3) cage = {038}
or R2C4 = 7 => 11(3) cage = {038}
-> 11(3) cage = {038}, R5C4 = 8, R6C45 = {03}, locked for R6, 0 locked for N5, R6C8 = 1, R5C89 = {03}, 0 locked for R5, clean-up: no 9 in R2C9, no 6 in R5C6
5c. 17(3) cage at R6C1 (step 4q) = {269/368/458}
5d. N4 must contain 5 in 10(4) cage = {0145} or R45C3 = {45} or 17(3) cage = {458} which must then be {58}4 -> no 4 in R6C23, no 5 in R7C2
5e. N7 must contain 5 in R9C12 = {58}, locked for R9, 8 locked for N7
5f. 20(3) cage at R7C1 = {479} (only remaining combination), 4 locked for N7
5g. Naked pair {23} in R7C23, 3 locked for R7
5h. 17(3) cage = {269/368}, no 5, 6 locked for R6 and N4, clean-up: no 3 in R4C3
5i. No 3 in N4, no 6 in N7
5j. R6C6 = 5 -> R5C6 = 2, clean-up: no 7 in R3C5, no 7 in R4C3
5k. Naked pair {36} in R3C56, locked for N2, 6 locked for R3
5l. R23C4 = [71] -> R3C3 = 9 (cage sum), R6C3 = 8, R7C3 = 3, R7C2 = 2, R3C7 = 2 -> R2C7 = 8
5m. Naked triple {367} in 16(3) cage at R1C7, locked for C1, 3 locked for N3 -> R2C89 = [91]
5n. N12 are missing 1,2 (step 2f), no 2 in N2 -> no 1 in N1 -> 5(3) cage at R1C1 = {023} = [203], R12C6 = [40], R1C3 = 5, 22(3) cage at R1C4 = [985], clean-up: no 4 in R45C3
5o. R4C1 = 0 (hidden single in N4)
5p. R45C3 = [27]
5q. 14(3) cage at R4C4 = {167} = [671] -> R5C5 = 9
5r. 4 missing from N5 -> 7 missing from N6 (the only remaining missing number not fixed) -> R6C79 = [42]
5s. Naked triple {589} in R4C789, 5 locked for R4 and N6 -> R4C2 = 4, R5C7 = 6, R2C23 = [64], R6C12 = [69], R8C2 = 7, R3C12 = [78], R9C12 = [85], R78C6 = [79], R78C1 = [94]
5t. 6(3) cage at R7C4 = {015} (only remaining combination, cannot be {123} because 2,3 only in R8C4) -> R7C5 = 1, R78C4 = {05}, 0 locked for C4 -> R6C45 = [30]

[Now, at last, to return to N9, which is the least helpful nonet.]
6a. 12(3) cage at R7C7 = {048} (only remaining combination) -> R7C7 = 0, R7C89 = {48}, 8 locked for N9, R78C4 = [50], R89C3 = [10]
6b. R9C7 = 1 (hidden single in N9) -> R8C7 + R9C8 = 9 = [36]

and the rest is naked singles.

Solution:
2 0 5 9 8 4 7 3 6
3 6 4 7 5 0 8 9 1
7 8 9 1 3 6 2 4 0
0 4 2 6 7 1 9 5 8
5 1 7 8 9 2 6 0 3
6 9 8 3 0 5 4 1 2
9 2 3 5 1 7 0 8 4
4 7 1 0 6 9 3 2 5
8 5 0 4 2 3 1 6 7

It's probably just coincidence that the missing numbers for the nonets happen to be the same as for the nonets of Rainbow Killer 1, while the missing numbers for the rows are 1 to 9, in that order.
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Andrew
 Post subject: Re: Rainbow Killer 5
PostPosted: Tue Oct 24, 2023 2:49 am 
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Then I did Rainbow Killer 5.3 next, since the expansion from a 3-cell cage to a 4-cell cage looked interesting.

I've improved my analysis of step 4 (but have not changed by WT for version 5.1), then found a nice forcing chain which I hadn't spotted while solving the earlier version.

Here's how I solved Rainbow Killer 5.3:
Zero is in every row, column and nonet. One of 1-9 is missing in each row, column and nonet.

Prelims

a) R12C6 = {04/13}
b) R23C7 = {19/28/37/46}, no 0,9
c) R2C89 = {19/28/37/46}, no 0,9
d) R3C12 = {69/78}
e) R56C6 = {07/16/25/34}, no 8,9
f) R67C3 = {29/38/47/56}, no 0,1
g) R78C6 = {79}
h) 5(3) cage at R1C1 = {014/023}
i) 22(3) cage at R1C4 = {589/679}
j) 4(3) cage at R5C8 = {013}
k) 20(3) cage at R7C1 = {389/479/569/578}, no 0,1,2
l) 6(3) cage at R7C4 = {015/024/123}, no 6,7,8,9
m) 10(4) cage at R4C1 = {0127/0136/0145/0235/1234}, no 8,9
n) 28(4) cage at R4C7 = {4789/5689}, no 0,1,2,3
o) 10(4) cage at R8C3 = {0127/0136/0145/0235/1234}, no 8,9

1a. Naked pair {79} in R78C6, locked for C6 and N8, clean-up: no 0,2 in R3C4, no 0 in R56C6
1b. R56C6 = {16/25} (cannot be {34} which clashes with R12C6), no 3,4
1c. 45 rule on N3, no 5 in N3
1d. 45 rule on N9, no 9 in N9
1e. 5(3) cage at R1C1 = {014/023}, 0 locked for N1
1f. 22(3) cage at R1C4 = {589/679}, 9 locked for N2, clean-up: no 0 in R3C6
1g. 4(3) cage at R5C8 = {013}, locked for N6
1h. 28(4) cage at R4C7 = {4789/5689}, 8,9 locked for N6

2a. 0 in N2 either in R12C6 = {04} or 17(3) cage at R2C4 = {089} -> no 4 in R23C4
2b. 0 in N3 either in 16(3) cage at R1C7 = {079} or in R3C89 = {04} -> no 4 in 16(3) cage
2c. 10(4) cage at R4C1 = {0127/0136/0145/0235} or 10(4) cage = {1234} => R45C3 = {09} -> 0 in R45C123, locked for N4
2d. 45 rule on N12 sum of missing numbers = 3 -> 1,2 missing from N12
2e. 45 rule on N1 R3C3 + missing number = 10, one of 1,2 missing -> R3C3 = {89}
2f. Killer pair 8,9 in R3C12 and R3C3, locked for R3 and N1, clean-up: no 1,2 in R2C7, no 1 in R3C56
2g. N1 must contain 5 only in 15(3) cage at R1C3 = {357/456}, no 1,2
[Alternatively N1 is missing one of 1,2, which must be in 5(3) cage at R1C1]
2h. 45 rule on N8 R9C4 + missing number = 12, R78C6 = {79} -> R9C4 = 4, 8 missing from N8
2i. R9C4 = 4 -> R8C3 + R9C23 = {015/123}, no 6,7, 1 locked for N7
2j. 6(3) cage at R7C4 = {015/123}, 1 locked for N8
2k. 6 in N8 only in 11(3) cage at R8C5 = {056/236}
2l. 45 rule on N5 R5C5 + missing number = 13, N5 must contain 5,8,9 -> missing number must be one of 4,6,7, R5C5 = {679}

3a. Consider combinations for 22(3) cage at R1C4 = {589/679}
22(3) cage = {589}, 8 locked for N2 => 17(3) cage at R2C4 cannot be [890] => 0 in N2 only in R12C6 = {04}
or 22(3) cage at {679} => 16(3) cage at R1C7 cannot be {079} (ALS clash) => 0 in N3 only in R3C89 = {04}, locked for R3
-> no 0 in R3C4, also no 0 in R2C4 because {089} can only be [890]
3b. 0 in N2 only in R12C6 = {04}, locked for C6, 4 locked for N2, clean-up: no 5 in R3C56
3c. 0 in R3 only in R3C89 = {04}, locked for N3, clean-up: no 6 in R23C7, no 6 in R2C89
3d. Killer pair 6,7 in R3C12 and R3C56, locked for R3, clean-up: no 3 in R2C7
3e. 5 is missing from N3 -> 6 in N3 only in 16(3) cage, locked for R1
3f. 22(3) cage = {589/679}
3g. 6 of {679} must be in R2C5 -> no 7 in R2C5
3h. 17(3) cage at R2C4 = {179/278/359/368} (cannot be {269} which clashes with R3C56)
3i. 1,2 of {179/278} must be in R3C4 -> no 1,2 in R2C4
3j. R3C3 = {89} -> no 8 in R2C4
3k. 8 missing from N8 -> N2 must contain 8 which is only in 22(3) cage = {589}, 5 locked for N2
3l. 17(3) cage = {179/278/368}
3m. 6,7 only in R2C4 -> R2C4 = {67}
3n. Consider combinations for R3C12 = {69/78}
R3C12 = {69}, 6 locked for R3 => R3C56 = [72]
or R3C12 = {78} => R3C3 = 9 => 17(3) cage = {179}
-> 17(3) cage = {179/368}, no 2
3o. Combined cage 17(3) + R3C56 = {179}{36}/{368}[72]
3o. No 5 in R3 -> must contain all the other numbers
3p. Hidden killer pair 1,2 in R3C46 and R3C7 for R3, R3C46 contain one of 1,2 -> R3C7 = {12}, R2C7 = {89}
3q. Consider combinations for 17(3) cage
17(3) cage = {179} = [791] => 6,7 in N3 only in 16(3) cage at R1C7 = {367} => R1C45 = {89} (hidden pair in N2, which is missing one of 1,2), locked for N2
or 17(3) cage = {368} => R3C56 = [72] => R3C7 = 1, R2C7 = 9
-> no 9 in R2C5
3r. 9 in R2 only in R2C789, locked for N3 -> 16(3) cage = {268/367}, no 1

4a. 1,2 missing from N12 -> N6 must contain 2 in R6C79, locked for R6, clean-up: no 5 in R5C6, no 9 in R7C3
4b. The missing number in N5 must be one of {467} (step 2l)
4c. 14(3) cage at R4C4 and 11(3) cage at R5C4 must together contain all of 0,3,8 for N5, 14(3) cage cannot be {158}, 11(3) cage cannot be {128}, both of which clash with R56C6 -> either 11(3) cage = {038} or 14(3) cage = {068} with 11(3) cage = {137} -> 11(3) cage = {038/137}, 3 locked for N5
4d. Consider combinations for 17(3) cage at R2C4 (step 3n) = {179/368}
17(3) cage = {179} = [791] => 11(3) cage must be {038} (cannot be {137} because 1,7 then only in R6C5)
or 17(3) cage = {368} = [638] => R3C56 = [72] => R56C6 = {16}, 1 locked for N5
-> 11(3) cage = {038}, 0,8 locked for N5, no 8 in C6 -> all other columns must contain 8
4e. Caged X-Wing for 0,3 in 11(3) cage and 4(3) cage at R5C8 for R56, no other 0,3 in R56, clean-up: no 6,9 in R4C3, no 8 in R7C3
4f. 14(3) cage = {149/167/257}
4g. 4 of {149} must be in R4C5 -> no 9 in R4C5
4h. 11(3) cage at R8C5 (step 2k) = {056/236}
4i. Consider placements for 2 in N5
R4C4 = 2 => R4C56 = [75] => R3C56 = {36}
or R4C5 = 2, no 2 in R89C5 => no 3 in R9C6 => R3C6 = 3 (hidden single in C6)
or 2 in R45C6 => R3C56 = {36}
-> R3C56 = {36}, locked for N2, 6 locked for R3, R23C4 = [71], R3C3 = 9, R3C7 = 2 -> R2C7 = 8, R2C5 = 5, clean-up: no 3 in R2C89, no 0 in R4C3, no 2 in R7C3
4j. Naked triple {367} in 16(3) cage at R1C7, 3,7 locked for R1
4k. 15(3) cage at R1C3 = {456} (only remaining combination) -> R1C3 = 5, R2C23 = {46}, 4 locked for R2 and N1 -> R12C6 = [40]
4l. 5(3) cage at R1C1 = {023} -> R1C12 = {02}, R2C1 = 3, clean-up: no 4 in R45C3, no 6 in R67C3
4m. R6C9 = 2 (hidden single in N6), R7C5 = 1 (hidden single in N8)
4n. 28(4) cage at R4C7 = {4789/5689}, 8 locked for R4, clean-up: no 1 in R5C3
4o. 8 in C3 only in R56C3, locked for N4

5a. N4 must contain 9, only in R6C12, locked for 17(3) cage, no 9 in R7C2
5b. 17(3) cage at R6C1 = {179/269/359} (cannot be {089} because 0,8 only in R7C2), no 0,4,8
5c. 1,9 of {179} must be in R6C12 -> no 7 in R6C12
5d. 2,3,7 only in R7C2 -> R7C2 = {237}
5e. 0 in N4 only in R4C12 -> 10(4) cage at R4C1 = {0127/0136/0145/0235}
5f. Consider combinations for R45C3 = [18]/{27}/[36]
R45C3 = [18]
or R45C3 = {27} => 10(4) cage = {0136/0145}, 1 locked for N4
or R45C3 = [36] => 10(4) cage = {0127/0145}, 1 locked for N4
-> no 1 in R6C12
5g. 17(3) cage = {269/359}, no 7
5h. R8C3 + R9C23 (step 2i) = {015} (cannot be {123} which clashes with R7C2} -> R9C2 = 5, R89C3 = {01}, 1 locked for C3, 0 locked for N7, clean-up: no 8 in R5C3
[Cracked, the rest is fairly straightforward]
5i. R6C3 = 8 (hidden single in N4) -> R7C3 = 3, R7C2 = 2, R6C12 = {69}, 6 locked for R6 and N4, clean-up: no 1 in R5C6
5j. Naked pair {27} in R45C3, locked for N4
5k. 10(4) cage = {0145}, 3 missing from N4
5l. 20(3) cage at R7C1 = {479} (only remaining combination)
5m. N7 must contain 8 -> R9C1 = 8, R3C12 = [78], R78C1 = {49}, locked for C1 and N7, R6C1 = [69], R8C2 = 7, R79C6 = [79]
5n. Naked pair {03} in R6C45 -> R5C4 = 8, R6C8 = 1, R2C89 = [91], R6C6 = 5 -> R5C6 = 2, R45C3 = [27]
5o. R7C5 = 1 -> R78C4 = 5 = {05} (cannot be {23} because 2,3 only in R8C4), locked for C4 and N8, R6C45 = [30]
5p. R1C4 = 9, R4C4 = 6, R4C6 = 1 -> R4C5 = 7 (cage sum)

6a. 9 missing from N9, 7 in seven other nonets -> no 7 in N6
6b. R6C7 = 4, R5C7 = 6 (hidden single in N6), naked triple {589} in R4C789, 5 locked for R4
6c. 12(3) cage at R7C7 = {048} (only remaining combination) -> R7C7 = 0, R7C89 = {48}, locked for N9, 4 locked for R7 -> R78C4 = [50], R89C3 = [10]
6d. R9C7 = 1 (hidden single in N9) -> R8C7 + R9C8 = 9 = [36]

and the rest is naked singles.
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Andrew
 Post subject: Re: Rainbow Killer 5
PostPosted: Tue Oct 24, 2023 4:39 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Finally I moved on to Rainbow Killer 5.5 (I didn't do 5.2 and 5.4). I was able to use many of my previous steps so it didn't seem any harder than Rainbow Killer 5.3. I also found a useful one at the end of step 2, something I hadn't used or needed before.

Here's how I solved Rainbow Killer 5.4:
Zero is in every row, column and nonet. One of 1-9 is missing in each row, column and nonet.

Prelims

a) R12C6 = {04/13}
b) R23C7 = {19/28/37/46}, no 0,9
c) R2C89 = {19/28/37/46}, no 0,9
d) R3C12 = {69/78}
e) R56C6 = {07/16/25/34}, no 8,9
f) R67C3 = {29/38/47/56}, no 0,1
g) R78C6 = {79}
h) 5(3) cage at R1C1 = {014/023}
i) 22(3) cage at R1C4 = {589/679}
j) 4(3) cage at R5C8 = {013}
k) 20(3) cage at R7C1 = {389/479/569/578}, no 0,1,2
l) 6(3) cage at R7C4 = {015/024/123}, no 6,7,8,9
m) 5(3) cage at R8C3 = = {014/023}
n) 28(4) cage at R4C7 = {4789/5689}, no 0,1,2,3

1a. Naked pair {79} in R78C6, locked for C6 and N8, clean-up: no 0,2 in R3C4, no 0 in R56C6
1b. R56C6 = {16/25} (cannot be {34} which clashes with R12C6), no 3,4
1c. 45 rule on N3, no 5 in N3
1d. 45 rule on N9, no 9 in N9
1e. 5(3) cage at R1C1 = {014/023}, 0 locked for N1
1f. 22(3) cage at R1C4 = {589/679}, 9 locked for N2, clean-up: no 0 in R3C6
1g. 4(3) cage at R5C8 = {013}, locked for N6
1h. 28(4) cage at R4C7 = {4789/5689}, 8,9 locked for N6

2a. 0 in N2 either in R12C6 = {04} or 17(3) cage at R2C4 = {089} -> no 4 in R23C4
2b. 0 in N3 either in 16(3) cage at R1C7 = {079} or in R3C89 = {04} -> no 4 in 16(3) cage
2c. 45 rule on N12 sum of missing numbers = 3 -> 1,2 missing from N12
2d. 45 rule on N1 R3C3 + missing number = 10, one of 1,2 missing -> R3C3 = {89}
2e. Killer pair 8,9 in R3C12 and R3C3, locked for R3 and N1, clean-up: no 1,2 in R2C7, no 1 in R3C56
2f. N1 must contain 5 only in 15(3) cage at R1C3 = {357/456}, no 1,2
[Alternatively N1 is missing one of 1,2, which must be in 5(3) cage at R1C1]
2g. 45 rule on N8 R9C4 + missing number = 12, R78C6 = {79} -> R9C4 = 4, 8 missing from N8
2h. R9C4 = 4 -> R89C3 = {01}, locked for C3 and N7, clean-up: no 8,9 in R45C3
2i. 6(3) cage at R7C4 = {015/123}, 1 locked for N8
2j. 6 in N8 only in 11(3) cage at R8C5 = {056/236}
2k. 45 rule on N5 R5C5 + missing number = 13, N5 must contain 5,8,9 -> missing number must be one of 4,6,7, R5C5 = {679}
2l. N147 must all contain both of 8,9 -> C123 must all contain both of 8,9
2m. Hidden killer pair 8,9 in R3C3 and R67C3 for C3, R3C3 = {89} -> R67C3 must contain one of 8,9 = {29/38}

3a. Consider combinations for 22(3) cage at R1C4 = {589/679}
22(3) cage = {589}, 8 locked for N2 => 17(3) cage at R2C4 cannot be [890] => 0 in N2 only in R12C6 = {04}
or 22(3) cage at {679} => 16(3) cage at R1C7 cannot be {079} (ALS clash) => 0 in N3 only in R3C89 = {04}, locked for R3
-> no 0 in R3C4, also no 0 in R2C4 because {089} can only be [890]
3b. 0 in N2 only in R12C6 = {04}, locked for C6, 4 locked for N2, clean-up: no 5 in R3C56
3c. 0 in R3 only in R3C89 = {04}, locked for N3, clean-up: no 6 in R23C7, no 6 in R2C89
3d. Killer pair 6,7 in R3C12 and R3C56, locked for R3, clean-up: no 3 in R2C7
3e. 5 is missing from N3 -> 6 in N3 only in 16(3) cage, locked for R1
3f. 22(3) cage = {589/679}
3g. 6 of {679} must be in R2C5 -> no 7 in R2C5
3h. 17(3) cage at R2C4 = {179/278/359/368} (cannot be {269} which clashes with R3C56)
3i. 1,2 of {179/278} must be in R3C4 -> no 1,2 in R2C4
3j. R3C3 = {89} -> no 8 in R2C4
3k. 8 missing from N8 -> N2 must contain 8 which is only in 22(3) cage = {589}, 5 locked for N2
3l. 17(3) cage = {179/278/368}
3m. 6,7 only in R2C4 -> R2C4 = {67}
3n. Consider combinations for R3C12 = {69/78}
R3C12 = {69}, 6 locked for R3 => R3C56 = [72]
or R3C12 = {78} => R3C3 = 9 => 17(3) cage = {179}
-> 17(3) cage = {179/368}, no 2
3o. 3o. Combined cage 17(3) + R3C56 = [683][72] }/[791]{36}
3o. No 5 in R3 -> must contain all the other numbers
3p. Hidden killer pair 1,2 in R3C46 and R3C7 for R3, R3C46 contain one of 1,2 -> R3C7 = {12}, R2C7 = {89}
3q. Consider combinations for 17(3) cage
17(3) cage = {179} = [791] => 6,7 in N3 only in 16(3) cage at R1C7 = {367} => R1C45 = {89} (hidden pair in N2, which is missing one of 1,2), locked for N2
or 17(3) cage = {368} => R3C56 = [72] => R3C7 = 1, R2C7 = 9
-> no 9 in R2C5
3r. 9 in R2 only in R2C789, locked for N3 -> 16(3) cage = {268/367}, no 1

4a. 1,2 missing from N12 -> N6 must contain 2 in R6C79, locked for R6, clean-up: no 5 in R5C6, no 9 in R7C3
4b. The missing number in N5 must be one of {467} (step 2k)
4c. 14(3) cage at R4C4 and 11(3) cage at R5C4 must together contain all of 0,3,8 for N5, 14(3) cage cannot be {158}, 11(3) cage cannot be {128}, both of which clash with R56C6 -> either 11(3) cage = {038} or 14(3) cage = {068} with 11(3) cage = {137} -> 11(3) cage = {038/137}, 3 locked for N5
4d. Consider combinations for 17(3) cage at R2C4 (step 3n) = {179/368}
17(3) cage = {179} = [791] => 11(3) cage must be {038} (cannot be {137} because 1,7 then only in R6C5)
or 17(3) cage = {368} = [638] => R3C56 = [72] => R56C6 = {16}, 1 locked for N5
-> 11(3) cage = {038}, 0,8 locked for N5, no 8 in C6 -> all other columns must contain 8
4e. Caged X-Wing for 0,3 in 11(3) cage and 4(3) cage at R5C8 for R56, no other 0,3 in R56, clean-up: no 6 in R4C3, no 8 in R7C3
4f. 14(3) cage = {149/167/257}
4g. 4 of {149} must be in R4C5 -> no 9 in R4C5
4h. 11(3) cage at R8C5 (step 2j) = {056/236}
4i. Consider placements for 2 in N5
R4C4 = 2 => R4C56 = [75] => R3C56 = {36}
or R4C5 = 2, no 2 in R89C5 => no 3 in R9C6 => R3C6 = 3 (hidden single in C6)
or 2 in R45C6 => R3C56 = {36}
-> R3C56 = {36}, locked for N2, 6 locked for R3, R23C4 = [71], R3C3 = 9, R3C7 = 2 -> R2C7 = 8, R2C5 = 5, R6C3 = 8 -> R7C3 = 3, clean-up: no 3 in R2C89, no 6 in R5C3
4j. Naked triple {367} in 16(3) cage at R1C7, 3,7 locked for R1
4k. 15(3) cage at R1C3 = {456} (only remaining combination) -> R1C3 = 5, R2C23 = {46}, 4 locked for R2 and N1 -> R12C6 = [40]
4l. 5(3) cage at R1C1 = {023} -> R1C12 = {02}, R2C1 = 3, clean-up: no 4 in R45C3
4m. R6C9 = 2 (hidden single in N6), R7C5 = 1 (hidden single in N8)
4n. Naked pair {27} in R45C3, locked for N4
4o. Naked pair {03} in R6C45, locked for R6 and N5 -> R5C4 = 8, R6C8 = 1, R1C45 = [98], R2C89 = [91], clean-up: no 6 in R5C6
4p. 14(3) cage = {167} (only remaining combination, cannot be {257} which clashes with R4C3) = [671], R45C3 = [27], R5C5 = 9, R56C6 = [25]

5a. 10(3) cage at R4C2 = {136/145} (cannot be {019} because 0,9 only in R4C2, cannot be {046} which clashes with R6C12, ALS block), no 0,9
5b. R4C1 = 0 (hidden single in N4) -> R1C12 = [20]
5c. N4 must contain 9 only in R6C12 -> 17(3) cage at R6C1 = {269} (only remaining combination) -> R6C12 = {69}, 6 locked for R6 and N4, R7C2 = 2, clean-up: no 3 in 10(3) cage
5d. R7C5 = 1 -> R78C4 = {05} (cannot be {23} because 2,3 only in R8C4), locked for N8, 0 locked for C4
5e. 12(3) cage at R7C7 = {048} (only possible combination), cannot be {057} which clashes with R7C4), locked for R7 and N9 -> R78C4 = [50], R89C3 = [10]
5f. R7C7 = 0 (hidden single in C7)
5g. R9C7 = 1 (hidden single in N9) -> R8C7 + R9C8 = 9 = {36}/[72]
5h. Naked triple {236} in R9C568, locked for R9
5i. 28(4) cage at R4C7 = {5689} (only remaining combination) -> R5C7 = 6, R4C789 = {589}, 5 locked for R4 -> R4C2 = 4, R2C23 = [64], R6C12 = [69], clean-up: no 3 in R9C8
5j. Naked pair {37} in R18C7, locked for C7 -> R6C7 = 4
5k. 20(3) cage = {479} (only remaining combination, cannot be {578} which clashes with R9C2) = [947], R8C7 = 3 -> R9C8 = 6

and the rest is naked singles.
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