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HATMAN
 Post subject: Rainbow Killer 6
PostPosted: Sun Nov 19, 2023 8:24 am 
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Rainbow Killer 6
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.

Three similar puzzles with increasing difficulty, on the assassin scale I think about 0.8, 1.0 and 1.5.

Rainbow Killer 6-1

Image

Rainbow Killer 6-4

Image

Rainbow Killer 6-5
The four green cells are a disjoint killer cage so no repeats.

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Andrew
 Post subject: Re: Rainbow Killer 6
PostPosted: Wed Nov 22, 2023 2:47 am 
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Thanks HATMAN for another nice puzzle. The first one was fairly easy the way I solved it.

Here's how I solved Rainbow Killer 6-1:
Zero is in every row, column and nonet. One of 1-9 is missing in each row, column and nonet.

Prelims

a) R23C9 = {39/48/57}, no 0,1,2,6
b) R3C12 = {59/68}
c) R34C4 = {08/17/26/35}, no 4,9
d) R3C56 = {04/13}
e) R45C3 = {79}
f) R4C67 = {49/58/67}, no 0,1,2,3
g) 6(2) cage at R5C6 = {06/15/24}, no 3,7,8,9
h) R56C7 = {29/38/47/56}, no 0,1
i) R6C34 = {04/13}
j) R67C6 = {29/38/47/56}, no 0,1
k) R7C45 = {05/14/23}, no 6,7,8,9
l) R89C2 = {19/28/37/46}, no 0,5
m) R9C89 = {08/17/26/35}, no 4,9
n) 20(3) cage at R1C4 = {389/479/569/578}, no 0,1,2
o) 19(3) cage at R1C6 = {289/379/469/478/568}, no 0,1
p) 9(3) cage at R5C8 = {018/027/036/045/126/135/234}, no 9

1a. Naked pair {79} in R45C3, locked for C3 and N4
1b. 20(3) cage at R1C4 and 19(3) cage at R1C6 form combined 39(6) cage = {456789}, locked for N2, clean-up: no 0 in R3C56
1c. Naked pair {13} in R3C56, locked for R3, clean-up: no 9 in R2C9
1d. R3C4 = 0 (hidden single 0 in N2) -> R4C4 = 8, clean-up: no 5 in R4C67, no 4 in R6C3, no 5 in R7C5, no 3 in R7C6
1e. 2 missing from N2, must be in all other nonets
1f. Killer pair 7,9 in R4C3 and R4C67, locked for R4
1g. 0 in N1 must be in one of the 14(3) cages = {059/068} -> naked quint {05689} for that 14(3) cage and R3C12, locked for N1 -> R3C3 = {24}
1h. The other 14(3) cage in N1 cannot contain any of 0,5,6,8,9 = {347} (only possible combination), locked for N1 -> R3C3 = 2 (hidden single 2 in N1)
1i. 1 missing from N1, must be in all other nonets
1j. 15(3) cage at R5C1 = {168/258/348/456}, no 0
1k. 2 must be in N4 either in 10(3) cage = {028/235} or 15(3) cage = {258} -> 15(3) cage = {168/258/456} (cannot be {348}, locking-out cages), no 3
1l. 0,1,2 must all be in N4, the only way two can be in the same cage is 10(3) cage at R4C1 = {028} when 15(3) cage = {456} and R6C6 = 1, otherwise 10(3) cage, 15(3) cage and R6C6 must each contains one of 0,1,2 -> R6C6 = {01}, clean-up: no 1 in R6C4
1m. N5 must contain both of 0,1 only in 15(3) cage at R4C5 and 6(2) cage at R5C6 -> 15(3) cage = {069/159}, 9 locked for R5 and N5, 6(2) cage = {06/15}, combined cages = {01569}, 5,6,9 locked for N5, R45C3 = [97], R4C67 = [76] (only remaining permutation), clean-up: no 5 in R5C7, no 2,4,5 in R6C7
1n. N5 must contain 2 -> R6C6 = 2, R7C6 = 9

[Applying 45 rule to nonets which must total at least 36 but cannot total 45 because that doesn’t provide a space for 0]
2a. 45 rule on N3 cages total 34 -> min R1C9 = 2
2b. 45 rule on N7 cages total 34 -> min R7C3 = 2
2c. 45 rule on N8, with innie placed, total 37 -> 8 missing from N8, must be in all other nonets
2d. 45 rule on N9 cages total 28 -> min R9C7 = 8
2e. 45 rule on N9, without innie placed, total 28 -> R9C7 + missing number = 17 = {89} but 8 already missing from N8 -> R9C7 = 8, 9 missing from N9, must be in all other nonets, clean-up: no 3 in R56C7, no 2 in R8C2, no 0 in R9C89
2f. 45 rule on N6, with innie placed, total 38 -> 7 missing from N6, must be in all other nonets, also missing from R6 so must be in all other rows, clean-up: no 4 in R5C7
2g. R56C7 = [29]
2h. 45 rule on N4 total 41 -> R6C3 = {01} + missing number = 4 -> missing number = {34}
2i. 45 rule on N5 total 37 -> R6C4 = {34} + missing number = 7 -> missing number = {34} -> the missing numbers in N45 are 3,4 -> must be in all other nonets and must be missing from R45 -> must be in all other rows
2j. R3 must contain 4 in R3C789, locked for N3, clean-up: no 8 in R3C9
2k. 11(3) cage at R1C7 = {047/137), no 5, 7 locked for C7 and N3, clean-up: no 5 in R23C9
2m. R3 must contain 7 -> R3C7 = 7 -> R12C7 = 4 = {13}, locked for C7 and N3, R2C9 = 8 -> R3C9 = 4
2n. 0 in N3 only in R12C8, locked for C8
2o. 0 in C7 only in R78C7, locked for N9
2p. 12(3) cage at R4C8 = {345} (only remaining combination) -> R4C8 = 4, R45C9 = {35}, locked for C9 and N6, clean-up: no 3,5 in R9C8
2q. R6C9 = 0 (hidden single 0 in N6), R6C3 = 1 -> R6C4 = 3, R56C8 = [18], clean-up: no 2 in R7C5
2r. 6(2) cage = [06] (only remaining combination), R6C12 = {45}, locked for N4, R5C1 = 6 (cage sum), R5C45 = {59}, 5 locked for R5 and N5 -> R4C5 = 1, R3C45 = [31], R45C9 = [53], R5C2 = 8 -> R4C12 = 2 = {02}, clean-up: no 2,4 in R7C4, no 2 in R9C2
2r. 3 missing from R4 and N4, 4 missing from R5 and N5, must be in all other rows and nonets

3a. 10(3) cage at R8C5 = {235} (only remaining combination, cannot be {037} because 0,7 only in R8C5, cannot be {046} which clashes with R7C5) -> R8C5 = 2, R89C6 = {35}, 5 locked for C6 and N8, R7C4 = 1 -> R7C5 = 4
3b. Naked pair {67} in R89C4, locked for C4, 7 locked for N8 -> R9C5 = 0
3c. Killer pair {67} in R9C4 and R9C89, locked for R9, clean-up: no 3,4 in R8C2
3d. 12(3) cage at R7C3 = {048/345}, no 6, 4 locked for C3 and N7, clean-up: no 6 in R8C2
3e. 10(3) cage at R7C7 = {037/235} -> R7C8 = 3, R7C9 = {27}
3f. Killer pair 2,7 in R7C9 and R9C89, locked for N9
3g. 10(3) cage at R8C7 = {145} (only remaining combination, cannot be {046} because 0,4 only in R8C7) = [451] -> 10(3) cage at R7C7 = {037} = [037], R89C6 = [35], clean-up: no 9 in R9C2
3h. 11(3) cage at R1C8 = {029} (only remaining combination) -> R3C8 = 9, R12C8 = {02}, 2 locked for C8 and N3 -> R9C89 = [62], clean-up: no 5 in R3C12
3i. R3C12 = [86]
3j. Naked quad R1278C3 = {0358}, locked for C3 -> R9C3 = 4, R78C3 = 8 = [80]
3k. Naked pair {35} in R12C3, locked for N1

4a. 45 rule on N7 total 34, 6 missing from N7 -> R7C2 = 5, R7C1 = 2, R6C12 = [54], R4C12 = [02]
4b. Naked triple {459} in R125C4, 4 locked for N2, R12C6 = [86], R2C5 = 5 (cage sum), R5C45 = [59], R1C5 = 7
4c. R2C3 = 3 -> R2C12 = 11 = [47], R8C2 = 9 -> R9C2 = 1

and the rest is naked singles.

Solution:
9 0 5 4 7 8 3 2 6
4 7 3 9 5 6 1 0 8
8 6 2 0 3 1 7 9 4
0 2 9 8 1 7 6 4 5
6 8 7 5 9 0 2 1 3
5 4 1 3 6 2 9 8 0
2 5 8 1 4 9 0 3 7
7 9 0 6 2 3 4 5 1
3 1 4 7 0 5 8 6 2
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Andrew
 Post subject: Re: Rainbow Killer 6
PostPosted: Thu Nov 23, 2023 9:37 pm 
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The middle puzzle of the three is definitely quite a bit harder, with three pairs of two-cell cages having been combined (must be why it's 6-4 with 6-2 and 6-3 not having made enough of a difference to be worth posting). Some of the steps I used this time could have been used for 6-1 but I was aiming to solve it with simple steps as much as possible. Since the only changes from 6-1 have been combining pairs of cages, 6-4 has the same solution.

Here's how I solved Rainbow Killer 6-4:
Zero is in every row, column and nonet. One of 1-9 is missing in each row, column and nonet.

Prelims

a) R23C9 = {39/48/57}, no 0,1,2,6
b) R3C12 = {59/68}
c) 6(2) cage at R5C6 = {06/15/24}, no 3,7,8,9
d) R67C6 = {29/38/47/56}, no 0,1
e) R7C45 = {05/14/23}, no 6,7,8,9
f) R89C2 = {19/28/37/46}, no 0,5
g) R9C89 = {08/17/26/35}, no 4,9
h) 20(3) cage at R1C4 = {389/479/569/578}, no 0,1,2
i) 19(3) cage at R1C6 = {289/379/469/478/568}, no 0,1
j) 9(3) cage at R5C8 = {018/027/036/045/126/135/234}, no 9

1a. 20(3) cage at R1C4 and 19(3) cage at R1C6 form combined 39(6) cage = {456789}, locked for N2
1b. 0 in N2 only in R3C456, locked for R3
1c. 0 in N1 must be in one of the 14(3) cages = {059/068} -> naked quint {05689} for that 14(3) cage and R3C12, locked for N1
1d. The other 14(3) cage in N1 cannot contain any of 0,5,6,8,9 = {347} (only possible combination), locked for N1, no 1,2 in either of the 14(3) cages
1e. Caged X-Wing, one of the 14(3) cages contains 4,7 in R12, combined 39(6) cage contains 4,7 in R12 -> no 4,7 in R12C789, clean-up: no 5,8 in R3C9
1f. Naked quad {0123} in R3C3456, locked for R3, clean-up: no 9 in R2C9
1g. R3C456 = {013/023} = 4,5 -> R4C4 = {78}
1h. N1 and N2 must each be missing one of 1,2 -> 1,2 must be in all other nonets

2a. 45 rule on N3 cages total 34 -> min R1C9 = 2
2b. 45 rule on N7 cages total 34 -> min R7C2 = 2
2c. 45 rule on N8 complete cages total 28 -> R7C6 = {89} and missing number must be one of 8,9 -> no other 8,9 in N8, R6C6 = {23}
2d. 45 rule on N9 cages total 28 -> R9C7 = {89} and missing number must be one of 8,9 -> no other 8,9 in N9, clean-up: no 0 in R9C89
-> N8 and N9 must each be missing one of 8,9 -> 8,9 must be in all other nonets
2e. 45 rule on N4 cages total 45 -> R6C4 must equal whichever number is missing from N4, which must contain 0,1,2,8,9 -> R6C4 = {34567}
2f. 45 rule on N6 cages total 45 -> R4C6 must equal whichever number is missing from N6, which must contain 0,1,2,8,9 -> R4C6 = {34567}
2g. N5 must contain all of 0,1,2, 15(3) cage at R4C6 and 6(2) cage at R5C6 can each only contain one of 0,1,2 -> R6C6 = 2 (only other position for one of 0,1,2), R7C6 = 9 -> R9C7 = 8, clean-up: no 4 in 6(2) cage, no 2 in R8C2
2h. 15(3) cage = {069/159} (cannot be {078} which clashes with R4C4, cannot be {168} which clashes with 6(2) cage) -> 15(3) cage and 6(2) cage form combined cage {01569}, locked for N5
2i. N5 must contain 8 -> R4C4 = 8, R3C456 = 4 = {013}, locked for R3 -> R3C3 = 2
2j. 45 rule of N456, including values in R4C4 and R6C6, total 121 -> missing three numbers total 14 which cannot include 1,2 = {347/356} -> all other nonets must include 3
[Note that the missing three numbers may, at this stage, be 3,5,6 because 3 may be the missing number in N5]
2k. 45 rule on N3 cages total 34, R1C9 + missing number total 11 but N3 must contain 2,3,8,9 -> no 2,3,8,9 in R1C9
2l. 360 rule on whole grid (rather than the Sudoku regular 405 because each of 1-9 missing), cages total 339 -> the four zero cells total 21, R3C3 + R9C7 = [82] = 10 -> R1C9 + R7C2 = 11 = {56}, no other 5,6 in N37, clean-up: no 7 in R3C9, no 4 in R89C2
2m. Naked pair {56} in R1C9 + R7C2, CPE no 5,6 in R1C2 + R7C9
2n. N37 are missing 5,6 -> N456 must be missing 3,4,7
2o. One of the 11(3) cages in N3 must contain 0 = {029} (only possibly combination, cannot be {038} which clashes with R2C9, cannot be {047} because 4,7 only in R3C78), 9 locked for N3 -> R3C9 = 4, R2C9 = 8
2p. Naked pair {79}, 9 locked for R3 and N3 -> R3C12 = {68}, locked for N1
2q. 5 missing from R3 and N12 are missing 1,2 -> 5 must be missing from N3, R1C9 = 6 -> R7C2 = 5, clean-up: no 0 in R7C45, no 2 in R9C8
2r. 24(4) cage at R4C6 doesn’t contain 8 = {2679/3579/4569}, no 0,1, 9 locked for C7 and N6 -> R3C7 = 7, R12C7 = 4 = {13}, locked for C7 and N3, R3C8 = 9, R12C8 = {02}, locked for C8
2s. 0 in C7 only in R78C7, locked for N9
2t. 24(4) cage = {2679/4569} (cannot be {3579} because 3,7 only in R4C6), no 3, 6 locked for C7 and N6
2u. R4C6 = {47} -> no 4 in R456C7

3a. One of the 10(3) cages in N9 must be {037} (cannot be {046} because no 0,4,6 in R78C9), locked for N9, clean-up: no 1,5 in R9C89
3b. R9C89 = [62], clean-up: no 8 in R8C2
3c. 5 in N9 only in 10(3) cage at R8C7 = {145}, locked for R8, 1,4 locked for N9, clean-up: no 9 in R9C2
3d. R7C7 = 0, R7C89 = {37}, locked for R7, clean-up: no 2 in R7C45
3e. Naked pair {14} in R7C45, locked for R7 and N8
3f. R7C3 = 8 -> R89C3 = 4 = [04] (cannot be [31] which clashes with R89C2), locked for C3 and N7
3g. R7C1 = 2 -> R89C1 = 10 = {37}/[91]
3h. 1 in C3 only in R456C3 (at this stage 6 may be missing from C3 since it’s missing from N7), locked for N4
3i. 20(4) cage at R4C3 = {1379/1469}, no 5, 9 locked for C3 and N4
3j. 5 in C3 only in R12C3, locked for N1
3k. 10(3) cage at R8C5 = {037/235}, no 6, 3 locked for N8
3l. 13(3) cage at R8C4 = {067} (only remaining combination, cannot be {256} because 2,6 only in R8C4) -> R8C4 = 6, R9C45 = {07}, locked for N8, 7 locked for R9, R8C56 = [23], R9C6 = 5, clean-up: no 1 in R6C5
3m. 20(3) cage at R1C4 = {479/578}, 7 locked for N2
3n. R35C6 = {01} (hidden pair in C6), clean-up: no 0 in R6C5

4a. N6 must contain 8 only in 9(3) cage at R5C8 = {018} -> R6C9 = 0, R56C8 = {18}, 1 locked for C8 and N6
4b. 12(3) cage at R4C8 = {345} (only remaining combination) -> R4C8 = 4, R45C9 = {35}, locked for C9, 5 locked for N6, R4C6 = 7
4c. 6 in N5 only in R456C5, locked for C5
4d. N2 must contain 6 -> R2C6 = 6, R1C6 = {48} -> R2C5 = {59}
4e. 14(3) cage at R2C1 (steps 1c and 1d) = {347} (cannot be {059} which clashes with R2C5), locked for R2 and N1
4f. R1C3 = 5, R1C12 = {09}, locked for R1
4g. 20(4) cage at R4C3 (step 3i) = {1379/1469}, R6C4 = {34} -> no 3 in R456C3

5a. 2 in N4 only in 10(3) cage at R4C1 = {028/235} = [028]/5{23}, no 3 in R4C1, no 0 in R45C2
5b. 15(3) cage at R5C1 = {078/357/456}
5c. 0 of {078} must be in R5C1 -> no 8 in R5C1
5d. 0 in N4 only in R45C1, locked for C1 -> R1C1 = 9, R8C1 = 7 -> R9C1 = 3, R2C1 = 4
5e. 15(3) cage at R5C1 = {078/456} (cannot be {357} because 3,7) only in R6C2), no 3
5f. Combined cage 10(3) and 15(3) = [028]{56}4 (cannot be 5{23}[087] which clashes with R2C2) -> 10(3) cage = [028], 15(3) cage = {56}4, 6 locked for C1 and N4
5g. R56C8 = [18] -> R5C6 = 0, R6C5 = 6, R56C1 = [65], R3C6 = 1
5h. Naked pair {59} in R5C45, locked for R5 and N5 -> R4C5 = 1, R7C45 = [14], R6C4 = 3, R3C4 = 0, R9C4 = 7, R1C456 = [478], R2C4 = 9 (cage sum)

and the rest is naked singles.
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Andrew
 Post subject: Re: Rainbow Killer 6
PostPosted: Sun Nov 26, 2023 2:57 am 
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Rainbow Killer 6-5 was a lot harder than 6-4, definitely in the hard Assassin category. Combining that further pair of two-cell cages made a big difference, taking away some of my previous steps. The disjoint cage, which is presumably essential to solve it, was also helpful fairly early, before my hardest step.

Here's how I solved Rainbow Killer 6-5:
Zero is in every row, column and nonet. One of 1-9 is missing in each row, column and nonet.

The four zero cells form a disjoint cage.

Prelims

a) R23C9 = {39/48/57}, no 0,1,2,6
b) R3C12 = {59/68}
c) 6(2) cage at R5C6 = {06/15/24}, no 3,7,8,9
d) R89C2 = {19/28/37/46}, no 0,5
e) R9C89 = {08/17/26/35}, no 4,9
f) 20(3) cage at R1C4 = {389/479/569/578}, no 0,1,2
g) 19(3) cage at R1C6 = {289/379/469/478/568}, no 0,1
h) 9(3) cage at R5C8 = {018/027/036/045/126/135/234}, no 9

1a. 20(3) cage at R1C4 and 19(3) cage at R1C6 form combined 39(6) cage = {456789}, locked for N2
1b. 0 in N2 only in R3C456, locked for R3
1c. 0 in N1 must be in one of the 14(3) cages = {059/068} -> naked quint {05689} for that 14(3) cage and R3C12, locked for N1
1d. The other 14(3) cage in N1 cannot contain any of 0,5,6,8,9 = {347} (only possible combination), locked for N1, no 1,2 in either of the 14(3) cages
1e. Caged X-Wing, one of the 14(3) cages contains 4,7 in R12, combined 39(6) cage contains 4,7 in R12 -> no 4,7 in R12C789, clean-up: no 5,8 in R3C9
1f. Naked quad {0123} in R3C3456, locked for R3, clean-up: no 9 in R2C9
1g. R3C456 = {013/023} = 4,5 -> R4C4 = {78}
1h. N1 and N2 must each be missing one of 1,2 -> 1,2 must be in all other nonets

2a. 45 rule on N3 cages total 34 -> min R1C9 = 2
2b. 45 rule on N7 cages total 34 -> min R7C2 = 2
2c. 45 rule on N9 cages total 28 -> R9C7 = {89} and missing number must be one of 8,9 -> no other 8,9 in N9, clean-up: no 0 in R9C89
2d. 45 rule on N8 total of cages at R8C4 and R8C5 = 23 -> R7C456 must total at least 13 -> max R6C6 = 3
2e. 45 rule on N4 cages total 45 -> R6C4 must equal whichever number is missing from N4, which must contain 0,1,2 -> R6C4 = {3456789}
2f. 45 rule on N6 cages total 45 -> R4C6 must equal whichever number is missing from N6, which must contain 0,1,2 -> R4C6 = {3456789}
2g. N5 must contain all of 0,1,2, 15(3) cage at R4C6 and 6(2) cage at R5C6 can each only contain one of 0,1,2 -> R6C6 = {012} (only other position for one of 0,1,2)

3a. One of the 14(3) cages in N1 = {059/068} and the other {347} (steps 1c and 1d), combined 39(6) cage in N2 (step 1a) = {456789}
3b. Consider permutations for R23C9 = [39/57/84]
R23C9 = [39] => R3C12 = {68}
or R23C9 = [57] => one 11(3) cage in N3 must be {029/038} and the other {146} (cannot be {128/236} which clash with the first cage), Caged X-Wing for 6 in combined 39(6) cage and 11(3) cage = {146}, no other 6 in R12 => 6 in N1 only in R3C12 = {68}
or R23C9 = [84] => 8 in N2 only in R1C456, locked for R1 => 8 in N1 only in R3C12 = {68}
-> R3C12 = {68}, locked for R3 and N1
3c. 14(3) cages in N1 are now {059} and {347}, Caged X-Wing for 5,9 in that 14(3) cage and combined 39(6) cage, no other 5,9 in R12, clean-up: no 7 in R3C9
3d. 11(3) cages in N3 now {029/056} (cannot be {038} which clashes with R2C9, cannot be {047} because 4,7 only in R3) and {137} (cannot be {128} because no 1,2,8 in R3, cannot be {146} because combined with {029} clashes with R3C9, cannot be {236} which clashes with the other cage, cannot be {245} because 4,5 only in R3), 3,7 locked for N3 -> R23C9 = [84]
3e. 360 rule on whole grid (rather than the Sudoku regular 405 because each of 1-9 missing), cages total 339 -> the four zero cells total 21 and form a disjoint cage, cannot total 21 with R1C9 = 2 and R3C3 = 1 -> R1C9 = 6, clean-up: no 5 in 11(3) cages, no 6 in R7C2 (disjoint cage), no 2 in R9C8
3f. 5 missing from N3, must be in all other nonets
3g. R1C9 = 6 -> R3C3 + R7C2 + R9C7 = 15 = [159/249/258] -> R7C2 = {45}
3h. 45 rule on N7 cages total 34 -> R7C2 + missing number total 11 = [4 + missing 7]/[5 + missing 6]
3i. R89C2 = {19/28/37} (cannot be {46} which clashes with R7C2 + missing number), no 4,6
3j. 12(3) cages in N7 = {039/048/129/147/156/237} (cannot be {057/246} which clash with R7C2 + missing number, cannot be {138} which clashes with R89C2, cannot be {345} which clashes with R7C2), also 12(3) cage at R7C3 cannot be {129} which clashes with R3C3
3m. Possible groupings which must include 0,1,2,5, starting with R89C2, then the two 12(3) cages and R7C2, {19}{048}{237}5, {28}{039}{147/156}{45}, {37}{048}{129}5
3n. One of 6,7 missing from N7

4a. 11(3) cage at R1C7 = {029/137}
4b. 24(4) cage at R4C6 cannot contain 0 because 7{089} clashes with R9C7, 8{079} clashes with 11(3) cage and 9{078} clashes with 11(3) cage + R9C7 (ALS block), no 0 in R456C7
4c. 24(4) cage cannot contain 1 because 6{189} clashes with R9C7, 8{169} clashes with 11(3) cage and 9{168} clashes with 11(3) cage + R9C7 (ALS block), no 1 in R456C7
4d. C7 must contain both of 0,1, one of the 10(3) cages in N9 must be {037/046} with the latter only {46}0
4e. Consider combinations for 11(3) cage at R1C7
11(3) cage = {029}, locked for C7 => 1 in R78C7, locked for N9
or 11(3) cage = {137}, locked for C7 => 0 in R78C7 => one of the 10(3) cages must be 0{37}, locked for N9
-> R9C89 = [35/53/62], no 1,7
4f. The 10(3) cages must contain both of 0 and 1 -> {037}/{46}0 and {127/145} (cannot be {136} which clashes with R9C89), {145} can only be in R8 because R7C2 = {45}
4g. Consider combinations for 11(3) cage at R1C7
11(3) cage = {029}, locked for C7
or 11(3) cage = {137}, 0 in R78C7 => one of the 10(3) cages = 0{37} => the other 10(3) cage = {145}
-> no 2 in R78C7
4h. C7 must contain 2
4i. 11(3) cage = {029}, locked for C7
or 11(3) cage = {137}, locked for C7, 0 in R78C7 and one of 4,5 in R78C7 (step 4f) => 2 in 24(4) cage = 7{269}/8{259} (cannot be 5{289} which clashes with R9C7, cannot be 9{258} which leaves 6 missing from R456C7 and thus N6 but whichever number is missing from N6 must be in R4C6, step 2f), 9 locked for C7
-> R9C7 = 8, 9 missing from N9, must be in all other nonets, clean-up: no 2 in R8C2
4j. N4 must contain 5,9 -> no 5,9 in R6C4 (step 2e)
4k. N6 must contain 5,9 -> no 5,9 in R4C6 (step 2f)
4l. 9 in N5 only in 15(3) cage = {069/159/249}, no 3,7,8
4m. 45 rule on N8 contains 9 -> cells in N8 must total at least 37, total of 16(3), 13(3) and 10(3) cells is 39 -> R6C6 = {12}, missing number in N8 one of 7,8
4n. 0 in N5 only in 15(3) cage = {069} or 6(2) cage = {06}, 6 locked for N5 (locking cages)
4o. R3C3 + R7C2 + R9C7 (step 3g) = [258] (only remaining combination) -> R3C3 = 2. R7C2 = 5, 1 missing from N1, 2 missing from N2, 6 missing from N7
4p. R3C456 = {013} = 4 -> R4C4 = 8 (cage sum)

[It gets a bit easier now but still resists to the end, as did Rainbow Killer 6-4.]
5a. 24(4) cage at R4C6 = {2679/3579/4569}, 9 locked for R7 and N6 -> R3C7 = 7, R12C7 = 4 = {13}, locked for C7 and N3, R12C8 = {02}, locked for C8, R3C8 = 9
5b. 24(4) cage = {2679/4569} (cannot be {3579} because 3,7 only in R4C6), no 3, 6 locked for C7 and N6
5c. R4C6 = {47} -> no 4 in R456C7
5d. 0 in C7 only in R78C7 -> one of the 10(3) cages in N7 must be {037} (cannot be {046} = {46}0) -> the other 10(3) cage = {145} must be in R8 -> 10(3) cage at R8C7 = {145}, locked for R8 and N9, clean-up: no 9 in R9C2, no 3 in R9C89
5e. R9C89 = [62] -> R7C7 = 0, R7C89 = {37}, locked for R7, clean-up: no 8 in R8C2
5f. 2 in C7 only in 24(4) cage = {2679} -> R4C6 = 7, R456C7 = {269}
5g. 7 missing from N6, 7 must be in all the other nonets -> 8 missing from N8 (step 4m)
5h. Cells in N8 total 37, 16(3), 13(3) and 10(3) cages in N8 total 39 -> R6C6 = 2, clean-up: no 4 no 4 in 6(2) cage at R5C6
5i. 15(3) cage at R4C5 = {069/159}, no 4
5j. R6C6 = 2 -> R7C456 = {149}, locked for R7 and N8 -> R7C3 = 8, R89C3 = 4 = [04] (cannot be {13} which clashes with R89C2)
5k. R7C1 = 2 -> R89C1 = 10 = {37}/[91]
5l. 1 in C3 only in R456C3, locked for N4
5m. 20(4) at R4C3 = {1379/1469}, no 5, 9 locked for C3 and N4
5n. R6C4 = {34} -> no 3 in R456C3
5o. 5 in C3 only in R12C3, locked for C3

6a. 8 in N6 only in 9(3) cage at R5C8 = {018} -> R6C9 = 0, R56C8 = {18}, 1 locked for C8 and N6, clean-up: no 6 in R5C6
6b. Naked pair {35}, locked for C9 and N6 -> R4C8 = 4
6c. 10(3) cage at R8C5 = {235} (cannot be {037} = [730] which clashes with R8C12, ALS block) = [235], naked pair {07} in R9C45, 7 locked for R9 and N8 -> R8C4 = 6, naked pair {01} in R35C6, locked for C6, clean-up: no 1 in R6C5
6d. R2C6 = 6 (hidden single in C6) -> R1C6 + R2C5 = 13 = {49}/[85], no 7
6e. 2 in N4 only in 10(3) cage at R4C1 = {028/235} = [028]/[523] (cannot be [532] which clashes with R4C9) -> R4C1 = {05}, R4C2 = 2, R5C2 = [38]
6f. 0 in N4 only in R45C1, locked for C1
6g. 14(3) cages in N1 (steps 1c and 1d) = {347} and {059} = [905], 9 in R12C1, locked for C1 and N1
6h. R8C1 = 7 -> R9C1 = 3, R89C2 = [91], naked pair {49} in R12C1, 4 locked for C1 and N1
6i. Combined cage 10(3) and 15(3) = [028]{56}4 (cannot be [523][087] which clashes with R12C2, ALS block) -> 10(3) cage = [028], 15(3) cage = {56}4, 6 locked for C1 and N4
6j. R5C8 = 1, R5C6 = 0 -> R6C5 = 6, R3C6 = 1
6k. Naked pair {59} in R5C45, locked for R5 and N5 -> R4C5 = 1
6l. R6C4 = 3 -> R3C45 = [03], R9C4 = 7
6m. R1C5 = 7 (hidden single in N2) -> R12C4 = 13 = {49}, locked for C4 and N2, R2C5 = 5
6n. R1C3 = 5 (hidden single in N1) -> R1C12 = 9 = [90]
6o. Naked triple {179} in R456C3, locked for C3 -> R2C3 = 3

and the rest is naked singles.
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