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For such a multiple of 45, I had to use at least one 45-cage! This killer might be tricky,
but I think there are several things that can be attempted fruitfully.

ASSASSIN 180



3x3::k:1792:3585:1538:1538:3332:3589:3589:3847:3847:1792:3585:3585:4108:3332:8206:3589:1808:2321:6674:6674:6674:4108:8206:8206:8206:1808:2321:6674:2332:2332:4108:8206:11552:8206:2338:2338:2596:2596:2598:2598:8206:11552:11552:3371:3371:2605:2605:4143:4143:11552:11552:2355:4148:3381:3638:1335:4143:4153:11552:11552:2355:4148:3381:3638:1335:4153:4153:11552:2372:3397:2118:3381:2120:2120:3402:3402:11552:2372:3397:2118:2118:

solution



Here is however an easier version (I guess the solving path is less narrow than the first one)


SMOOTH ASSASSIN 180



3x3::k:1792:3585:1538:1538:3332:3589:3589:3847:3847:1792:3585:3585:4108:3332:3598:3589:1808:2321:6674:6674:6674:4108:3598:3598:2840:1808:2321:6674:2332:2332:4108:1823:11552:2840:2338:2338:2596:2596:2598:2598:1823:11552:11552:3371:3371:2605:2605:4143:4143:11552:11552:2355:4148:3381:3638:1335:4143:4153:11552:11552:2355:4148:3381:3638:1335:4153:4153:11552:2372:3397:2118:3381:2120:2120:3402:3402:11552:2372:3397:2118:2118:

Solution :

Hello ?!
I feel sorry there is no activity with this killer (or no hint asked by PM for those who want it)

Thanks for the puzzle, Manu. Actually there was quite a lot of activity before I got the solution. But nothing elegant about my method to report.

I did enjoy making inferences about the 32- and 45- cages, about which cells could see which others and which cells and groups were copies; but it didn't lead me directly to the solution.

cheers

_________________
Joe

Thanks for this Assassin, manu! University kept me busy, so I didn't have time to solve your Killer.

It's fairly difficult (at least the way I solved it), so this might be another reason why no one has written a walkthrough yet.

A180 Walkthrough

1. C789
a) 16(2) = {79} locked for C8
b) 8(3) = 1{25/34} -> 1 locked for N9
c) Outies N9 = 14(3): R6C79 <> 7,8,9 since R6C8 >= 7
d) Innies N69 = 9(2) <> 9
e) 15(2): R1C9 <> 6,8
f) 13(2): R5C9 <> 4,6

2. R1234
a) Outies N1 = 8(1+1) -> R4C1 = (3467)
b) 9 locked in R4C456 @ R4 for N5
c) Innies N3 = 14(3) <> 6 because {167} blocked by Killer pair (67) of 15(2) and {356} is a Killer triple of 7(2)
d) Innies+Outies N3: R1C6 = R3C7 <> 6

3. C789 !
a) ! Innies C9 = 23(4): R4C9 <> 1 since [9185] clashes with 9(2) @ R4C8 = [18]
b) 9(2) @ R4C8: R4C8 <> 8; R4C9 <> 2
c) ! 8 locked in R15C8 @ C8 -> 15(2) + 13(2) = 28(2+2) = [8749/8758/6985]
-> R5C8 <> 6; R5C9 <> 7
d) 9(2) @ R4 <> {45} since it's a Killer pair of 13(2) @ N6

4. R456
a) Innies N69 = 9(2) <> {45} since it's a Killer pair of 13(2) @ N6
b) Hidden Killer pair (45) in Outies N9 for N6 since 13(2) can only have one of (45)
-> Outies N9 = 14(3) = {149/257/347} <> 6 because R6C8 = (79)
c) R4C6 <> 6 since it sees all 6 of N6
d) 10(2) @ R5C3: R5C3 <> 1
e) Innies+Outies R5: 3 = R4C6 - R5C5: R4C6 <> 1,2,3; R5C5 <> 3,7,8
f) Hidden Killer pair (59) locked in Outies N9 for N6 since 13(2) can't have both of them
-> Outies N9 = 14(3) = {149/257} <> 3
g) 9(2) @ C7: R7C7 <> 2,3,6
h) Hidden Killer pair (23) in 13(3) @ N9 since 8(3) can only have one of (23) -> 13(3) <> 1{48/57}

5. C789 !
a) ! 6 locked in Innies N69 + 13(2) @ C6 -> Either Innies N69 = 9(2) = {36} xor 13(2) = {67} -> Innies N69 <> {27}
b) Hidden Killer pair (79) in each of Innies N3 + R789C7 for C7 since Innies N3 can only have one of them
and two of (79) in R789C7 blocked by R7C8 = (79)
-> Innies N3 = 14(3) <> 8
c) Killer pair (79) locked in R789C7 + R7C8 for N9
d) 13(3) <> 1
e) 9 locked in R15C9 @ C9 = 9[5/7] (see step 3c)
f) 13(2) @ N6: R5C8 <> 5
g) ! 9 locked in Innies C9 = 23(4) = 9{167/257/347/356} <> 8 because [9851] causes a clash with 13(2) @ N6 = [85] and R15C9 = (579)
h) 9(2) @ R4C8 <> 1
i) 1 locked in R456C7 @ N6 for C7
j) Innies+Outies N3: R1C4 = R3C7 <> 1,8

6. C789
a) Killer triple (478) in 9(2) + 13(2) @ C7 blocks Innies N3 = {347}
b) Innies N3 = 2{39/57} -> 2 locked for C7+N3
c) 7(2) <> 5
d) 5 locked in 8(3) @ C8 = {125} locked for N9
e) 3 locked in 13(3) @ N9 for C9 -> 13(3) = 3{28/46}
f) 9(2) @ C7 = [18/54]
g) 9(2) @ R4C8: R4C8 <> 6
h) 9(2) @ R2C9 <> 6,7
i) 13(2) <> 8
j) Innies+Outies N3: R1C4 = R3C7 <> 4

7. R789+N5
a) Killer triple (125) locked in 8(2) + R9C89 for R9
b) 13(2) <> 8
c) 8 locked in R9C56 @ R9 for N8
d) R456C6 <> 8 since they see all 8 of N8
e) Innies+Outies R5: 3 = R4C6 - R5C5: R5C5 <> 5

8. R456 !
a) Innies R5 = 12(3) = {156/237/246} <> 8 since other combos blocked by Killer quad (1348) of both 10(2) + R5C8
b) 45(9) must have 8 -> 8 locked for C5
c) ! Outies R6789 = 15(3) = 5{19/37/46} <> 2 because {267} blocked by R4C9 = (67)
-> 5 locked for C6+N5+45(9)
d) Outies R6789 = 15(3): R5C6 <> 1,3,6 since R5C7 = (136)
e) Innies R5 = 12(3): R5C5 <> 4 since R5C6 = (457)
f) Innies+Outies R5: 3 = R4C6 - R5C5: R4C6 <> 7

9. R123
a) 13(2) <> 5
b) Hidden Single: R3C5 = 5 @ C5
c) 5 locked in 16(3) @ N8 for 16(3) -> 16(3) <> 1,6
d) 26(4) = 9{278/368/467} -> 9 locked for R3+N1
e) Innies+Outies N3: R1C6 = R3C7 <> 9
f) Outies N1 = 8(1+1) <> 3
g) 6(2): R1C3 <> 1

10. R456 !
a) 5 locked in 9(2) + R6C3 for N4
b) ! Innies N4 = 16(3) + 9(2) = 25(5) = 45{169/178/268/367} since R4C1 = (467), {13579} has no combo for 9(2),
235{69/78} unplaceable since 9(2) can only be {27/36} which leaves no candidate for R4C1
-> 4 locked for N4; 9(2) <> 2,7
c) 10(2)s @ C1 <> 6
d) ! Innies R6 = 21(4): R6C3 <> 1 since 5,9 only possible there
e) 5(2) + 8(2) = 13(4) = 1{237/246/345} -> 1 locked for N7

11. C123
a) Innies N14 = 14(3) <> 9 since 2{39} blocked by Killer triple (239) of both 10(2) @ C1
b) 9 locked in both 10(2) @ C1 @ N4 -> 1 locked for N4
c) Hidden Single: R2C3 = 1 @ C3, R1C4 = 1 @ R1 -> R1C3 = 5, R4C2 = 5 @ N4 -> R4C3 = 4
d) 14(3) = {167} -> 6,7 locked for C2+N1
e) 7(2) = {34} locked for C1+N1
f) 26(4) = {2789} -> R4C1 = 7; 2,8 locked for R3

12. N34
a) Innies+Outies N3: R1C4 = R3C7 <> 2
b) Innies N4 = 9(2) = {36} locked for C3+N4

13. Rest is singles.

Thanks for your answers guys !
I would not want to make too difficult assassins ; I am not a good solver, since many assassins posted by others seem very hard for me whereas they are quite straightforward for the others, and I miss many obvious things . I realize that a puzzle maker is not always able to consider pertinently the difficulty of his work, particularly if the solving path is narrow (like this one in fact), and if his puzzle was made with a particular intention.

I haven't even got around to trying A180 yet, but I did finish the Smooth version. It was a fun puzzle so thanks manu!

I enjoyed it mostly because I was able to use a wide variety of different steps.


Here's my wt, please advise of errors or clarifications necessary:

Like Ronnie I started with the Smooth version. However my solving path was very different. I was struggling after step 9, then I had another look at "clones" and found a bit more from them; after that it was straightforward.


Here is my walkthrough for Smooth A180.

Prelims

a) R12C1 = {16/25/34}, no 7,8,9
b) R1C34 = {15/24}
c) R12C5 = {49/58/67}, no 1,2,3
d) R1C89 = {69/78}
e) R23C8 = {16/25/34}, no 7,8,9
f) R23C9 = {18/27/36/45}, no 9
g) R34C7 = {29/38/47/56}, no 1
h) R4C23 = {18/27/36/45}, no 9
i) R45C5 = {16/25/34}, no 7,8,9
j) R4C89 = {18/27/36/45}, no 9
k) R5C12 = {19/28/37/46}, no 5
l) R5C34 = {19/28/37/46}, no 5
m) R5C89 = {49/58/67}, no 1,2,3
n) R6C12 = {19/28/37/46}, no 5
o) R67C7 = {18/27/36/45}, no 9
p) R67C8 = {79}
q) R78C1 = {59/68}
r) R78C2 = {14/23}
s) R89C6 = {18/27/36/45}, no 9
t) R89C7 = {49/58/67}, no 1,2,3
u) R9C12 = {17/26/35}, no 4,8,9
v) R9C34 = {49/58/67}, no 1,2,3
w) 8(3) cage in N9 = {125/134}
x) 26(4) cage at R3C1 = {2789/3689/4589/4679/5678}, no 1

Steps resulting from Prelims
1a. Naked pair {79} in R67C8, locked for C8, clean-up: no 6,8 in R1C9, no 2 in R4C9, no 4,6 in R5C9
1b. 8(3) cage in N9 = {125/134}, 1 locked for N9, clean-up: no 8 in R6C7

[There’s a fairly obvious “clone” R5C5 = R7C6 because R5C5 “sees” all cells of the 45(9) cage except for R7C6. However it’s not clear how it can be useful so I’ll leave it for now.]

2. 45 rule on R1234 2 innies R4C56 = 10 = [19/28/37/46/64], no 5, no 1,2,3 in R4C6, clean-up: no 2 in R5C5

3. 45 rule on C7 1 outie R1C6 = 1 innie R5C7 + 2, no 1,2 in R1C6, no 8,9 in R5C7

4. 45 rule on N9 2 innies R7C78 = 1 outie R6C9 + 11
4a. Max R7C78 = 17 -> max R6C9 = 6
4b. Min R7C78 = 12, no 2 in R7C7, clean-up: no 7 in R6C7

5. 45 rule on N1 1 outie R4C1 = 1 innie R1C3 + 2 -> R4C1 = {3467}

6. 45 rule on N2 2 innies R1C46 = 1 outie R4C4 + 2
6a. Min R1C46 = 4 -> min R4C4 = 2

7. 45 rule on N7 3 innies R789C3 = 18 = {279/378/459/468} (cannot be {189/369/567} which clash with R78C1), no 1

8. 45 rule on N3 1 outie R1C6 = 1 innie R3C7, no 2 in R3C7, clean-up: no 9 in R4C7
8a. 9 in R4 locked in R4C46, locked for N5, clean-up: no 1 in R5C3

9. 45 rule on C7 3 innies R125C7 = 12
9a. 45 rule on N3 3 innies R123C7 = 14 = {149/158/239/248/257/347} (cannot be {167} which clashes with R1C89, cannot be {356} which clashes with R23C8), no 6, clean-up: no 6 in R1C6 (step 8), no 5 in R4C7, no 4 in R5C7 (step 3)
9b. 7 of {257} must be in R12C7 (R12C7 cannot be {25} because R125C7 cannot be {25}5)
7 of {347} must be in R12C7 (R12C7 cannot be {34} which clashes with R34C7 = [74])
-> no 7 in R3C7, clean-up: no 7 in R1C6 (step 8), no 4 in R4C7, no 5 in R5C7 (step 3)

[Maybe I’m missing something simpler but I’ve now realised how I can use the “clone” that I mentioned earlier or, more accurately, another “clone” which follows from that one.]
10. R5C5 “sees” all cells of the 45(9) cage except for R7C6 -> R5C5 = R7C6
10a. R4C5 “sees” all cells of the 45(9) cage except for R5C7 and R7C6, but R5C5 = R7C6 -> R4C5 = R5C7
10b. R4C5 = R5C7 -> R45C5 = R5C57 = 7
10c. 45 rule on R5 1 remaining innie R5C6 = 5
10d. R5C57 = 7 = [16/43/61], no 3 in R5C5, no 2,7 in R5C7
10e. 5 in N8 locked in R789C4, locked for C4
10f. Now I’ll do clean-up for step 10: no 1 in R1C3, no 4,9 in R1C6 (step 3), no 4,5,9 in R3C7 (step 8), no 2,6,7 in R4C7, no 3 in R5C7 (step 3), no 4 in R5C5 (step 10d), no 2,3,4 in R4C5, no 3 in R4C1 (step 5), no 8 in R5C89, no 4 in R89C6

11. Naked pair {16} in R5C57, locked for R5 -> R5C8 = 4, R5C9 = 9, R1C9 = 7, R1C8 = 8, R1C6 = 3, R34C7 = [38], R5C7 = 1 (step 3), R45C5 = [16], R4C6 = 9 (step 2), R67C8 = [79], clean-up: no 4 in R2C1, no 5,7 in R2C5, no 1,2,6 in R23C9, no 2,5 in R4C89, no 3 in R6C12, no 6 in R6C7, no 5,6 in R7C7, no 5 in R8C1, no 6 in R89C6, no 4,5 in R89C7
11a. Naked pair {36} in R4C89, locked for R4 and N6, clean-up: no 4 in R1C3 (step 5), no 2 in R1C4
11b. Naked pair {67} in R89C7, locked for N9 -> R7C7 = 4, R6C7 = 5, R6C9 = 2, clean-up: no 8 in R6C12, no 1 in R8C2
11c. Naked triple {348} in R6C456, locked for R6 and N5, clean-up: no 6 in R6C12
11d. Naked pair {27} in R45C4, locked for C4
11e. Naked pair {45} in R23C9, locked for C9 and N3, clean-up: no 2 in R23C8
11f. Naked pair {38} in R78C9, locked for C9 and N9 -> R4C89 = [36], R9C9 = 1, clean-up: no 8 in R8C6, no 7 in R9C12
11g. R6C3 = 6 (hidden single in R6)

12. 45 rule on N12 2 outies R4C14 = 1 innie R1C6 + 6
12a. R1C6 = 3 -> R4C14 = 9 = [72], R1C3 = 5 (step 5), R1C4 = 1, R4C23 = [54], R5C4 = 7, R5C3 = 3, clean-up: no 2 in R1C1, no 2,6 in R2C1, no 8 in R2C5, no 3 in R9C1, no 8,9 in R9C4
12b. R3C5 = 5 (hidden single in C5), R23C9 = [54]

13. X-Wing for 9 in R12C5 and R12C7, no other 9 in R12
13a. Naked pair {49} in R12C5, locked for C5 and N2
13b. Naked pair {68} in R23C4, locked for C4 and N2, clean-up: no 7 in R9C3
13c. R8C4 = 9 (hidden single in C4), clean-up: no 5 in R7C1
13d. Naked pair {68} in R78C1, locked for C1 and N7 -> R5C12 = [28], R9C1 = 5, R9C2 = 3

and the rest is naked singles.

To test my "udosuk Style Killer Cages" Excel program's ability to correctly identify, draw and colour non-diagonal connecting tubes between remote parts of a cage, manu's "SMOOTH ASSASSIN 180" seemed a good starting point.
So after having made three cages having remote parts, I had JSudoku generate some puzzles. Here are two of them.
One very easy with a SS Score of 0.66, and one more difficult with a SS Score of 1.78.
Both puzzles have the same cage layout, but different solutions.

Assassin 180 v0.10                                                         Assassin 180 v1.5
                    
SS Score: 0.66                                                             SS Score: 1.78
3x3::k:1280:5121:3074:3074:10571:5381:5381:2823:2823:1280:5121:5121:2828:10571:2574:5381:4170:2065:6418:6418:6418:2828:2574:2574:2840:4170:2065:6418:2076:2076:2828:10571:11552:2840:7756:7756:3620:3620:1318:1318:10571:11552:11552:3657:3657:7756:7756:4911:4911:11552:11552:819:10571:2357:3894:10571:4911:3385:11552:11552:819:10571:2357:3894:10571:3385:3385:11552:4164:4170:4166:2357:1864:1864:7756:7756:11552:4164:4170:4166:4166:          3x3::k:1536:4097:2050:2050:10827:4357:4357:2311:2311:1536:4097:4097:2572:10827:2574:4357:6474:1297:6162:6162:6162:2572:2574:2574:3864:6474:1297:6162:1820:1820:2572:10827:11552:3864:9036:9036:2596:2596:2086:2086:10827:11552:11552:2377:2377:9036:9036:5423:5423:11552:11552:819:10827:2613:3126:10827:5423:4921:11552:11552:819:10827:2613:3126:10827:4921:4921:11552:2116:6474:4934:2613:3144:3144:9036:9036:11552:2116:6474:4934:4934:

And here scribble images for Assassin 180 and Smooth Assassin 180:

_________________
Quis custodiet ipsos custodes?

Normal: [D  Y-m-d,  G:i]     PM->email: [D, d M Y H:i:s]

As someone who doesn't post very often (OK not for a long time now) but who solves on pencil and paper, I would like to thank Borge for the nice images.

Also everyone else for the good puzzles.

Thanks manu for another challenging puzzle!

Congratulations of solving A180 without using "clones".

Having used them for Smooth A180 I was very pleased to have decided to use them again for A180, once I'd spotted the extra step needed for them to be useful. I also used Ronnie's cage blocker (step 1b of her Smooth A180 walkthrough). Thanks Ronnie for posting your walkthrough! I'm not sure that I'd have been able to solve A180 without that very useful step. Maybe I might eventually have found it using permutation analysis as in Afmob's walkthrough.


Here is my walkthrough for A180. Thanks Afmob for pointing out that step 22 and a couple of earlier details were unnecessary; I've deleted them.

Prelims

a) R12C1 = {16/25/34}, no 7,8,9
b) R1C34 = {15/24}
c) R12C5 = {49/58/67}, no 1,2,3
d) R1C89 = {69/78}
e) R23C8 = {16/25/34}, no 7,8,9
f) R23C9 = {18/27/36/45}, no 9
g) R4C23 = {18/27/36/45}, no 9
h) R4C89 = {18/27/36/45}, no 9
i) R5C12 = {19/28/37/46}, no 5
j) R5C34 = {19/28/37/46}, no 5
k) R5C89 = {49/58/67}, no 1,2,3
l) R6C12 = {19/28/37/46}, no 5
m) R67C7 = {18/27/36/45}, no 9
n) R67C8 = {79}
o) R78C1 = {59/68}
p) R78C2 = {14/23}
q) R89C6 = {18/27/36/45}, no 9
r) R89C7 = {49/58/67}, no 1,2,3
s) R9C12 = {17/26/35}, no 4,8,9
t) R9C34 = {49/58/67}, no 1,2,3
u) 8(3) cage in N9 = {125/134}
v) 26(4) cage at R3C1 = {2789/3689/4589/4679/5678}, no 1

Steps resulting from Prelims
1a. Naked pair {79} in R67C8, locked for C8, clean-up: no 6,8 in R1C9, no 2 in R4C9, no 4,6 in R5C9
1b. 8(3) cage in N9 = {125/134}, 1 locked for N9, clean-up: no 8 in R6C7

2. 45 rule on N1 1 outie R4C1 = 1 innie R1C3 + 2
2a. R1C3 = {1245} -> R4C1 = {3467}

3. 45 rule on N69 2 innies R45C7 = 9 = {18/27/36/45}, no 9
3a. 9 in R4 locked in R4C456, locked for N5, clean-up: no 1 in R5C3

4. 45 rule on R1234 1 innie R4C6 = 1 outie R5C5 + 3, no 1,2,3 in R4C6, no 7,8 in R5C5

5. R5C5 “sees” all cells of the 45(9) cage except for R7C6 -> R5C5 = R7C6, no 7,8,9 in R7C6
5a. R4C5 “sees” all cells of the 45(9) cage except for R5C7 and R7C6, but R5C5 = R7C6 -> R4C5 = R5C7, no 9 in R4C5

6. R45C7 = 9 (step 3), R4C5 = R5C7 (step 5a) -> R4C57 = 9
6a. 9 in R4 locked in R4C46
6b. 45 rule on R4 3 remaining innies R4C146 = 18 = {279/369/459} (cannot be {189} because R4C1 only contains 3,4,6,7), no 1,8, clean-up: no 5 in R5C5 (step 4), no 5 in R7C6 (step 5)
6c. 4,7 of {279/459} must be in R4C1, no 4,7 in R4C46, clean-up: no 1,4 in R5C5 (step 4), no 1,4 in R7C6 (step 5)

7. 45 rule on N3 3 innies R123C7 = 14 = {149/158/239/248/257/347} (cannot be {167} which clashes with R1C89, cannot be {356} which clashes with R23C8), no 6
7a. 45 rule on N3 1 outie R1C6 = 1 innie R3C7, no 6 in R1C6

[Now step 1b of Ronnie’s Smooth A180 walkthrough, expressed slightly differently.]
8. R1C89 = [69/87]
8a. R5C89 = [49/58/85] (cannot be [67] which clashes with the permutations for R1C89 in either C8 or C9), no 6,7
8b. R4C89 = {18/36}/[27] (cannot be {45} which clashes with R5C89), no 4,5
8c. R45C7 (step 3) = {18/27/36} (cannot be {45} which clashes with R5C89), no 4,5, clean-up: no 4,5 in R4C5 (step 5a)
8d. 4 in R4 locked in R4C123, locked for N4, clean-up: no 6 in R5C12, no 6 in R5C4, no 6 in R6C12
8e. 4 in C1 locked in R1234C1, CPE no 4 in R3C23

9. 45 rule on R5 3 innies R5C567 = 12 = {156/237/246} (cannot be {138} which clashes with R5C12, cannot be {147} because R5C5 only contains 2,3,6, cannot be {345} which clashes with R5C89), no 8, clean-up: no 8 in R4C5 (step 5a), no 1 in R4C7 (step 3)
9a. 4,5 of {156/246} must be in R5C6 -> no 1,6 in R5C6

10. Hidden killer pair 4,5 in R5C89 and R6C79 for N6, R5C89 contains one of 4,5 -> R6C79 must contain one of 4,5
10a. 45 rule of N9 3 outies R6C789 = 14 = {149/257/347} (cannot be {158/248/356} because R6C8 only contains 7,9, cannot be {167/239} which don’t contain 4 or 5), R6C79 = {12345}, clean-up: no 2,3 in R7C7
10b. 6 in N6 locked in R45C7 or R4C89 -> 3 locked in R45C7 or R4C89 (9(2) locking cages), locked for N6, clean-up: no 6 in R7C7

11. 6 in C7 locked in R4589C7
11a. R45C7 = {18/27/36}, R89C7 = {49/58/67} -> combined cage R4589C7 = {1867/3649/3658} (other combinations don’t contain 6) -> no 2,7 in R45C7, clean-up: no 2,7 in R4C5 (step 5a)

12. R5C567 (step 9) = {156/237/246}
12a. 3 of {237} must be R5C7 -> no 3 in R5C56, clean-up: no 3 in R7C6 (step 5)
12b. 4,7 of {237/246} must be in R5C6 -> no 2 in R5C6

13. 45 rule on N7 3 innies R789C3 = 18 = {279/378/459/468} (cannot be {189/369/567} which clash with R78C1), no 1

14. 10(2) cages in N4 = {19/28/37} -> combined cage R56C12 = {1928/1937/2837}
14a. 45 rule on N4 3 innies R4C1 + R56C3 = 16 = {169/268/367/457} (cannot be {178/259/349/358} which clash with R56C12)
14b. 1 of {169} must be in R6C3 -> no 9 in R6C3
14c. R4C23 = {36/45} (cannot be {18/27} which clash with R56C12)

15. 8 in R4 locked in R4C789, locked for N6, clean-up: no 5 in R5C89
15a. R5C89 = [49], R1C9 = 7, R1C8 = 8, R67C8 = [79], clean-up: no 5,6 in R2C5, no 3 in R23C8, no 1,2 in R23C9, no 2 in R4C8, no 1 in R4C9, no 1 in R5C12, no 6 in R5C3, no 1 in R5C4, no 3 in R6C12, no 5 in R7C7, no 5 in R8C1, no 4 in R89C7
15b. 9 in C7 locked in R123C7 -> R123C7 (step 7) = {149/239}, no 5, clean-up: no 5 in R1C6 (step 7a)

16. Naked quad {2378} in R5C1234, locked for R5 -> R5C567 = [651], R4C6 = 9, R4C5 = 1 (step 5a), R7C6 = 6 (step 5), clean-up: no 4 in R123C7 (step 15b), no 1,4 in R1C6 (step 7a), no 7 in R2C5, no 9 in R3C7 (step 7a), no 8 in R4C9, no 8 in R7C7, no 8 in R8C1, no 3,4 in R89C6, no 7 in R9C3

17. Naked pair {36} in R4C89, locked for R4 -> R4C4 = 2, R4C1 = 7 (step 6b), R1C3 = 5 (step 2), R1C4 = 1, R4C23 = [54], R4C7 = 8, clean-up: no 2 in R1C1, no 2,6 in R2C1, no 8 in R2C5, no 3 in R5C12, no 5 in R89C7, no 3 in R9C1, no 1 in R9C2, no 8,9 in R9C4

18. Naked triple {239} in R123C7, locked for C7 and N3 -> R6C7 = 5, R7C7 = 4, R6C9 = 2, clean-up: no 5 in R23C8, no 6 in R23C9, no 1 in R8C2
18a. Naked pair {28} in R5C12, locked for R5 and N4 -> R5C34 = [37]
18b. R6C3 = 6 (hidden single in R6)

[I’ll detour slightly to look at larger cages and it proves to be very helpful.]
19. R4C1 = 7 -> 26(4) cage at R3C1 = {2789/4679}, no 3, 9 locked for R3 and N1

20. 32(7) cage at R2C6 = {1235678} (only remaining combination) -> R3C5 = 5, 7 locked for C6, clean-up: no 2 in R89C6
20a. R23C9 = [54], clean-up: no 6 in R3C12 (step 19)
20b. Naked triple {289} in R3C123, locked for R3 and N1 -> R3C67 = [73], R2C6 = 2, R1C6 = 3, R12C7 = [29], R12C5 = [94], R23C4 = [86], R23C8 = [61], R4C89 = [36]

21. R9C9 = 1 (hidden single in C9), R89C6 = [18], R6C6 = 4, R6C4 = 3, R7C3 = 7 (cage sum)

and the rest is naked singles.