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Although the cage pattern is extremely messy, 45s can be used on R4, R6 and C6 to get 3 innies in each case. There are also innie-outie differences for R4, R6 and C6 which I didn't find until I was working on The Messier, The Merrier 5; however I didn't find them useful so didn't use them. I've commented in the posts for individual puzzles where the software solvers used innie-outie differences or outies for R4, R6 and C6.

Prelims

Only 39 Prelims in this, compared with 40 for the Twosomes, but I’ve listed them all because the cage pattern is so confusing and have given both cells for the same reason.

a) R1C1 + R7C7 = {17/26/35}, no 4,8,9
b) R15C2 = {17/26/35}, no 4,8,9
c) R1C3 + R4C6 = {14/23}
d) R1C49 = {19/28/37/46}, no 5
e) R1C5 + R3C7 = {29/38/47/56}, no 1
f) R15C6 = {69/78}
g) R16C7 = {16/25/34}, no 7,8,9
h) R1C8 + R7C2 = {79}
i) R29C1 = {14/23}
j) R2C25 = {29/38/47/56}, no 1
k) R2C3 + R7C8 = {49/58/67}, no 1,2,3
l) R2C4 + R5C7 = {18/27/36/45}, no 9
m) R27C6 = {69/78}
n) R24C7 = {18/27/36/45}, no 9
o) R2C8 + R5C5 = {29/38/47/56}, no 1
p) R28C9 = {12}
q) R3C19 = {29/38/47/56}, no 1
r) R3C2 + R8C7 = {79}
s) R38C3 = {18/27/36/45}, no 9
t) R37C4 = {39/48/57}, no 1,2,6
u) R3C58 = {17/26/35}, no 4,8,9
v) R39C6 = {12}
w) R4C13 = {19/28/37/46}, no 5
x) R4C49 = {79}
y) R4C58 = {15/24}
z) R57C1 = {59/68}
aa) R5C3 + R8C6 = {59/68}
bb) R58C4 = {13}
cc) R59C8 = {17/26/35}, no 4,8,9
dd) R5C9 + R9C5 = {19/28/37/46}, no 5
ee) R6C18 = {39/48/57}, no 1,2,6
ff) R6C29 = {19/28/37/46}, no 5
gg) R6C35 = {49/58/67}, no 1,2,3
hh) R6C6 + R9C9 = {29/38/47/56}, no 1
ii) R7C39 = {16/25/34}, no 7,8,9
jj) R7C5 + R9C3 = {16/25/34}, no 7,8,9
kk) R8C18 = {14/23}
ll) R8C25 = {89}
mm) R9C24 = {18/27/36/45}, no 9

1. Naked pair {89} in R8C25, locked for R8 -> R8C7 = 7, R3C2 = 9, R7C2 = 7, R1C8 = 9, R8C25 = [89]
[I don’t know whether clean-up is necessary for this one, the easiest in the series, but I’ll do it anyway.]
Clean-up: no 1 in R1C1, no 1 in R15C2, no 1 in R1C49, no 2,4 in R1C5, no 2 in R2C2, no 4,6 in R2C3, no 2 in R2C4, no 2,3,4 in R2C5, no 6,8 in R2C6, no 2 in R24C7, no 2 in R2C8, no 2 in R3C19, no 1,2 in R3C3, no 3,5 in R3C4, no 2 in R3C7, no 6 in R5C1, no 5,6 in R5C3, no 2 in R5C5, no 6 in R5C6, no 1 in R5C8, no 1 in R5C9, no 3 in R6C1, no 4 in R6C3, no 4 in R6C6, no 1,2,3 in R6C9, no 3 in R7C4, no 1,2 in R9C4

2. Killer pair 6,7 in R15C6 and R27C6, locked for C6 -> R8C6 = 5 -> R5C3 = 9, clean-up: no 6 in R1C6, no 4 in R3C3, no 7 in R3C4, no 1 in R4C13, no 4 in R6C5, no 3 in R6C8, no 5 in R7C1, no 4 in R7C8, no 4 in R9C2, no 2 in R9C3, no 1 in R9C5, no 4,5,6 in R9C9

3. Naked pair {78} in R15C6, locked for C6 -> R2C6 = 9, R7C6 = 6, R7C1 = 9 -> R5C1 = 5, clean-up: no 2 in R1C1, no 3 in R1C2, no 7 in R2C3, no 4 in R2C4, no 6 in R2C8, no 6 in R3C9, no 4 in R5C9, no 8 in R6C5, no 7 in R6C8, no 1 in R7C39, no 3 in R7C7, no 3 in R9C2, no 1 in R9C3, no 2,3 in R9C9

4. Naked pair {12} in R28C9, locked for C9, clean-up: no 8 in R1C4, no 5 in R7C3, no 8 in R9C5

5. Naked pair {12} in R39C6, locked for C6 -> R6C6 = 3 -> R9C9 = 8, R4C6 = 4 -> R1C3 = 1, R58C4 = [13], R7C8 = 5, R2C3 = 8, clean-up: no 3 in R1C1, no 2 in R1C4, no 4 in R1C7, no 7 in R1C9, no 3 in R2C2, no 5 in R2C7, no 7 in R2C8, no 3 in R3C19, no 6 in R3C3, no 3 in R3C5, no 2,6 in R4C1, no 6 in R4C3, no 2 in R4C58, no 1 in R4C7, no 6 in R5C5, no 6,8 in R5C7, no 3 in R59C8, no 7 in R5C9, no 7 in R6C1, no 2 in R6C2, no 5 in R6C5, no 6 in R6C7, no 7 in R6C9, no 2 in R7C3, no 2 in R8C18, no 4 in R9C1, no 1,6 in R9C2, no 4 in R9C3, no 2 in R9C5

6. R4C58 = [51], R8C18 = [14], R28C9 = [12], R7C7 = 1 -> R1C1 = 7

and the rest is naked singles.

Prelims

Only 39 Prelims in this, compared with 40 for the Twosomes, but I’ve listed them all because the cage pattern is so confusing and have given both cells for the same reason.

a) R1C1 + R7C7 = {19/28/37/46}, no 5
b) R15C2 = {18/27/36/45}, no 9
c) R1C3 + R4C6 = {18/27/36/45}, no 9
d) R1C49 = {14/23}
e) R1C5 + R3C7 = {19/28/37/46}, no 5
f) R15C6 = {15/24}
g) R16C7 = {39/48/57}, no 1,2,6
h) R1C8 + R7C2 = {49/58/67}, no 1,2,3
i) R29C1 = {14/23}
j) R2C25 = {59/68}
k) R2C3 + R7C8 = {14/23}
l) R2C4 + R5C7 = {19/28/37/46}, no 5
m) R27C6 = {39/48/57}, no 1,2,6
n) R24C7 = {49/58/67}, no 1,2,3
o) R2C8 + R5C5 = {29/38/47/56}, no 1
p) R28C9 = {18/27/36/45}, no 9
q) R3C19 = {16/25/34}, no 7,8,9
r) R3C2 + R8C7 = {49/58/67}, no 1,2,3
s) R38C3 = {15/24}
t) R37C4 = {39/48/57}, no 1,2,6
u) R3C58 = {16/25/34}, no 7,8,9
v) R39C6 = {18/27/36/45}, no 9
w) R4C13 = {69/78}
x) R4C49 = {13}
y) R4C58 = {49/58/67}, no 1,2,3
z) R57C1 = {19/28/37/46}, no 5
aa) R5C3 + R8C6 = {89}
bb) R58C4 = {19/28/37/46}, no 5
cc) R59C8 = {59/68}
dd) R5C9 + R9C5 = {29/38/47/56}, no 1
ee) R6C18 = {17/26/35}, no 4,8,9
ff) R6C29 = {17/26/35}, no 4,8,9
gg) R6C35 = {18/27/36/45}, no 9
hh) R6C6 + R9C9 = {69/78}
ii) R7C39 = {19/28/37/46}, no 5
jj) R7C5 + R9C3 = {39/48/57}, no 1,2,6
kk) R8C18 = {59/68}
ll) R8C25 = {13}
mm) R9C24 = {17/26/35}, no 4,8,9

1. Naked pair {13} in R4C49, locked for R4, clean-up: no 6,8 in R1C3

2. Naked pair {13} in R8C25, locked for R8, clean-up: no 6,8 in R2C9, no 5 in R3C3, no 7,9 in R5C4

3. Killer pair 8,9 in R8C18 and R8C6, locked for R8, clean-up: no 1 in R2C9, no 4,5 in R3C2, no 1,2 in R5C4

4. Killer pair 8,9 in R4C13 and R5C3, locked for N4, clean-up: no 1 in R1C2, no 1 in R6C5, no 1,2 in R7C1

5. R4C58 = {49/58} (cannot be {67} which clashes with R4C13), no 6,7

6. Killer pair 8,9 in R4C13 and R4C58, locked for R4, clean-up: no 1 in R1C3, no 4,5 in R2C7

7. R39C6 = {18/27/36} (cannot be {45} which clash with R15C6), no 4,5

8. 9 in R6 only in R6C467
8a. 45 rule on R6 3 innies R6C467 = 20 = {389/479/569}, no 1,2

9. Killer triple 1,2,4 in R3C19, R3C3 and R3C58, locked for R3, clean-up: no 6,8,9 in R1C5, no 8 in R7C4, no 7,8 in R9C6

[I originally used combined cage R6C12 + R6C29 for the next step. I didn’t spot 45 rule on C6 until I was working on The Messier, The Merrier 3 but, having spotted it then, I’ve re-worked from this point because it’s a better and simpler way to continue. Maybe with some more re-work I could have avoided using the killer triples. However the killer triple in step 11 is so powerful that it feels the best way forward.]

10. 45 rule on C6 3 innies R468C6 = 18 = {279/468} (cannot be {459} because 4,5 only in R4C6, cannot be {567} because R8C6 only contains 8,9), no 5, clean-up: no 4 in R1C3
10a. 2,4 only in R4C6 -> R4C6 = {24}, clean-up: no 2,3 in R1C3
10b. R8C6 = {89} -> no 8,9 in R6C6, clean-up: no 6,7 in R9C9

11. Killer triple 5,6,7 in R6C18, R6C29 and R6C6, locked for C6, clean-up: no 5,7 in R1C7, no 2,3,4 in R6C35

12. R6C35 = [18], clean-up: no 8 in R1C2, no 4 in R1C7, no 6 in R2C2, no 3 in R2C8, no 5 in R4C8, no 3 in R5C9, no 7 in R6C18, no 7 in R6C29, no 9 in R7C1, no 4 in R7C8, no 9 in R7C9, no 5 in R8C3, no 2 in R8C4, no 4 in R9C3

13. Naked pair {24} in R38C3, locked for C3 -> R2C3 = 3 -> R7C8 = 2, clean-up: no 8 in R1C1, no 7 in R2C9, no 5 in R3C5, no 4 in R3C9, no 6 in R5C2, no 9 in R5C5, no 7 in R5C7, no 6 in R6C1, no 8 in R7C3, no 9 in R7C5, no 9 in R7C6, no 7 in R7C7, no 6,7,8 in R7C9, no 6 in R8C9, no 2 in R9C1

14. Naked quad {2356} in R6C1289, locked for R6 -> R6C6 = 7 -> R9C9 = 8, clean-up: no 2 in R1C1, no 9 in R1C7, no 5 in R27C6, no 4 in R2C8, no 6 in R5C8, no 4 in R7C5, no 6 in R8C1, no 3 in R9C5, no 2 in R9C6

15. R468C6 (step 10) = {279} (only remaining combination) -> R4C6 = 2 -> R1C3 = 7, R8C6 = 9, R5C3 = 8, clean-up: no 4 in R15C6, no 2 in R2C4, no 9 in R2C8, no 3 in R3C4, no 3 in R3C7, no 7 in R4C1, no 2 in R5C2, no 2 in R5C9, no 6 in R7C2, no 5 in R7C5, no 3 in R7C6, no 3 in R7C7, no 3 in R7C9, no 5 in R8C18, no 6 in R8C7, no 6 in R9C8

16. R8C18 = [86], clean-up: no 4 in R1C1, no 5 in R1C8, no 1 in R3C5, no 2 in R5C1, no 4 in R5C4, no 5 in R5C5, no 2 in R6C1, no 7 in R7C2

17. R9C3 = 5 (hidden single in C3) -> R7C5 = 7

and the rest is naked singles.

Prelims

Only 39 Prelims in this, compared with 40 for the Twosomes, but I’ve listed them all because the cage pattern is so confusing and have given both cells for the same reason.

a) R1C1 + R7C7 = {18/27/36/45}, no 9
b) R15C2 = {29/38/47/56}, no 1
c) R1C3 + R4C6 = {17/26/35}, no 4,8,9
d) R1C49 = {19/28/37/46}, no 5
e) R1C5 + R3C7 = {69/78}
f) R15C6 = {19/28/37/46}, no 5
g) R16C7 = {15/24}
h) R1C8 + R7C2 = {59/68}
i) R29C1 = {29/38/47/56}, no 1
j) R2C25 = {29/38/47/56}, no 1
k) R2C3 + R7C8 = {19/28/37/46}, no 5
l) R2C4 + R5C7 = {14/23}
m) R27C6 = {19/28/37/46}, no 5
n) R24C7 = {15/24}
o) R2C8 + R5C5 = {16/25/34}, no 7,8,9
p) R28C9 = {69/78}
q) R3C19 = {29/38/47/56}, no 1
r) R3C2 + R8C7 = {19/28/37/46}, no 5
s) R38C3 = {29/38/47/56}, no 1
t) R37C4 = {69/78}
u) R3C58 = {16/25/34}, no 7,8,9
v) R39C6 = {16/25/34}, no 7,8,9
w) R4C13 = {39/48/57}, no 1,2,6
x) R4C49 = {19/28/37/46}, no 5
y) R4C58 = {49/58/67}, no 1,2,3
z) R57C1 = {16/25/34}, no 7,8,9
aa) R5C3 + R8C6 = {17/26/35}, no 4,8,9
bb) R58C4 = {18/27/36/45}, no 9
cc) R59C8 = {19/28/37/46}, no 5
dd) R5C9 + R9C5 = {49/58/67}, no 1,2,3
ee) R6C18 = {19/28/37/46}, no 5
ff) R6C29 = {15/24}
gg) R6C35 = {69/78}
hh) R6C6 + R9C9 = {29/38/47/56}, no 1
ii) R7C39 = {16/25/34}, no 7,8,9
jj) R7C5 + R9C3 = {19/28/37/46}, no 5
kk) R8C18 = {16/25/34}, no 7,8,9
ll) R8C25 = {18/27/36/45}, no 9
mm) R9C24 = {69/78}

1. Naked quad {1245} in R1246C7, locked for C7 -> R5C7 = 3 -> R2C4 = 2, clean-up: R1C1 = {123}, no 8 in R1C2, no 7 in R1C6, no 8 in R1C9, no 9 in R2C25, no 4 in R2C8, no 6,7,8,9 in R3C2, no 5 in R3C8, no 7 in R4C4, no 4 in R4C7, no 8 in R4C9, no 7 in R5C4, no 5 in R5C5, no 8 in R5C6, no 7 in R6C1, no 4 in R7C1, no 8 in R7C6, no 8 in R7C8, no 6,7 in R8C4, no 5 in R8C6, no 9 in R9C1, no 5 in R9C6, no 7 in R9C8

2. Naked quad {6789} in R789C7 + R8C9, locked for N9, clean-up: no 1,3,4 in R2C3, no 1,2,4 in R5C8, no 2,3,4,5 in R6C6, no 1 in R7C3, no 1 in R8C1

3. 7 in C8 only in R456C8, locked for N6, clean-up: no 3 in R4C4, no 6 in R9C5

4. Killer quad 6,7,8,9 in R6C18, R6C35 and R6C6, locked for R6

5. Min R9C7 = 6 -> max R4C2 + R6C4 = 8, no 8,9 in R4C2

6. 3 in R6 only in R6C14, CPE no 3 in R4C2

7. 45 rule on R4 3 innies R4C267 = 10 = {127/136/145/235}
7a. 3 of {136} must be in R4C6 -> no 6 in R4C6, clean-up: no 2 in R1C3

8. 45 rule on C6 3 innies R468C6 = 18 = {279/369/378/567} (cannot be {189} because 8,9 only in R6C6), no 1, clean-up: no 7 in R1C3, no 7 in R5C3

9. R15C6 = {19/46}/[82] (cannot be [37] which clashes with R468C6), no 3,7

10. R27C6 = {19/46}/[82] (cannot be {37} which clashes with R468C6), no 3,7

11. Using steps 9 and 10 combined cage R1257C6 = {1946/1982/4682}
11a. 7 in C6 only in R468C6 (step 8) = {378/567} (cannot be {279} which clashes with R1257C6), no 2,9, clean-up: no 6 in R1C3, no 6 in R5C3, no 2 in R9C9
11b. 9 in C6 only in R1257C6 = {1946/1982}, 1 locked for C6, clean-up: no 6 in R39C6

12. 2 in N5 only in R5C56, locked for R5, clean-up: no 9 in R1C2, no 5 in R7C1, no 6 in R8C6

13. R468C6 (step 11a) = {378/567}
13a. 6,8 only in R6C6 -> R6C6 = {68}, clean-up: no 4 in R9C9

14. Killer pair 6,8 in R6C35 and R6C6, locked for R6, clean-up: no 2,4 in R6C18

15. 2 in N4 only in R46C2, locked for C2, clean-up: no 9 in R5C2, no 7 in R8C5, no 8 in R8C7

16. 9 in C2 only in R79C2, locked for N7, clean-up: no 2 in R3C3, no 1 in R7C5

17. 2 in N1 only in R13C1, locked for C1, clean-up: no 9 in R2C1, no 5 in R5C1, no 5 in R8C8

18. 5 in N9 only in R79C9, locked for C9, clean-up: no 6 in R3C1, no 1 in R6C2, no 8 in R9C5

19. 3 in N4 only in R4C13 + R6C1
19a. R4C13 = {39} or R6C1 = 3 -> no 9 in R6C1, clean-up: no 1 in R6C8

20. Killer pair 1,3 in R57C1 and R6C1, locked for C1 -> R1C1 = 2 -> R7C7 = 7, clean-up: no 8 in R1C4, no 8 in R1C5, no 8 in R29C1, no 8 in R2C9, no 3 in R3C2, no 8 in R3C4, no 8,9 in R3C9, no 9 in R4C3, no 4 in R6C7, no 4 in R8C8, no 3 in R9C3

21. 4 in C7 only in R12C7, locked for N3, clean-up: no 6 in R1C4, no 7 in R3C1, no 3 in R3C5

22. 2 in C7 only in R46C7, locked for N6, clean-up: no 8 in R4C4, no 4 in R6C2

23. R34C1 = {89} (hidden pair in C1), clean-up: no 6,7 in R3C9, no 5,7,8 in R4C3

24. R4C267 (step 7) = {127/136/235} (cannot be {145} which clashes with R4C49), no 4
24a. 7 of {127} must be in R4C6 -> no 7 in R4C2
24b. 3 of {235} must be in R4C6 -> no 5 in R4C6, clean-up: no 3 in R1C3

25. Naked pair {37} in R48C6, locked for C6, clean-up: no 4 in R39C6
25a. R48C6 = {37} -> R6C6 = 8 (step 11a) -> R9C9 = 3, R3C6 = 5, R9C6 = 2, R3C9 = 2 -> R3C1 = 9, R4C1 = 8 -> R4C3 = 4, clean-up: no 3,7 in R1C2, no 7 in R1C4, no 6 in R1C5, no 6 in R2C2, no 7 in R2C3, no 7 in R3C3, no 6 in R4C49, no 5,9 in R4C58, no 7,8 in R5C8, no 7 in R6C35, no 3 in R7C1, no 5 in R7C3, no 6 in R7C4, no 6,8 in R7C5, no 1 in R7C8, no 4 in R8C1, no 7 in R8C2, no 2,6,7 in R8C3, no 1 in R8C4, no 8 in R9C3

26. Naked pair {67} in R4C58, locked for R4 -> R4C6 = 3 -> R1C3 = 5, R8C6 = 7, R5C3 = 1, R5C1 = 6, R6C1 = 3 -> R6C8 = 7, R7C1 = 1, R8C1 = 5 -> R8C8 = 2, R4C58 = [76], R6C35 = [96], R7C8 = 4 -> R2C3 = 6, R1C2 = 4 -> R5C2 = 7, R29C1 = [74], R1C7 = 1 -> R6C7 = 5, R24C7 = [42], R4C2 = 5, R6C2 = 2 -> R6C9 = 4, R1C5 = 9 -> R3C7 = 6

and the rest is naked singles.

JSudoku used 45 rule on C6, 3 outies R15C3 + R9C9 = 9 and later a number of hidden killer pairs.

SS also used 3 outies for C6. It also used innie-outie difference for R4, innie-outie difference for R6 and innie-outie difference for C6.

Until I saw the solver logs I wasn't aware that there were useful innie-outie differences for this extremely messy cage pattern. With hindsight, having found innies for R4, R6 and C6 I ought to have looked for innie-outie differences.

Prelims

This is the only one in the series with 40 Prelims; it also has a Prelim for the 3-cell cage. I’ve listed them all because the cage pattern is so confusing and have given the cells for the same reason.

a) R1C1 + R7C7 = {15/24}
b) R15C2 = {15/24}
c) R1C3 + R4C6 = {39/48/57}, no 1,2,6
d) R1C49 = {29/38/47/56}, no 1
e) R1C5 + R3C7 = {16/25/34}, no 7,8,9
f) R15C6 = {19/28/37/46}, no 5
g) R16C7 = {59/68}
h) R1C8 + R7C2 = {19/28/37/46}, no 5
i) R29C1 = {49/58/67}, no 1,2,3
j) R2C25 = {29/38/47/56}, no 1
k) R2C3 + R7C8 = {69/78}
l) R2C4 + R5C7 = {19/28/37/46}, no 5
m) R27C6 = {29/38/47/56}, no 1
n) R24C7 = {29/38/47/56}, no 1
o) R2C8 + R5C5 = {16/25/34}, no 7,8,9
p) R28C9 = {15/24}
q) R3C19 = {59/68}
r) R3C2 + R8C7 = {49/58/67}, no 1,2,3
s) R38C3 = {12}
t) R37C4 = {19/28/37/46}, no 5
u) R3C58 = {39/48/57}, no 1,2,6
v) R39C6 = {17/26/35}, no 4,8,9
w) R4C13 = {18/27/36/45}, no 9
x) R4C49 = {15/24}
y) R4C58 = {39/48/57}, no 1,2,6
z) R57C1 = {59/68}
aa) R5C3 + R8C6 = {39/48/57}, no 1,2,6
bb) R58C4 = {18/27/36/45}, no 9
cc) R59C8 = {29/38/47/56}, no 1
dd) R5C9 + R9C5 = {89}
ee) R6C18 = {16/25/34}, no 7,8,9
ff) R6C29 = {59/68}
gg) R6C35 = {18/27/36/45}, no 9
hh) R6C6 + R9C9 = {14/23}
ii) R7C39 = {29/38/47/56}, no 1
jj) R7C5 + R9C3 = {16/25/34}, no 7,8,9
kk) R8C18 = {19/28/37/46}, no 5
ll) R8C25 = {19/28/37/46}, no 5
mm) R9C24 = {39/48/57}, no 1,2,6
nn) 10(3) cage R4C2 + R6C4 + R9C7 = {127/136/145/235}, no 8,9

1. Naked pair {12} in R38C3, locked for C3, clean-up: no 7,8 in R4C1, no 7,8 in R6C5, no 5,6 in R7C5, no 9 in R7C9

2. R4C13 = [18/27/36/63] (cannot be {45} which clashes with R4C49), no 4,5

3. R24C7 = {29/38/47} (cannot be {56} which clashes with R16C7), no 5,6

4. R29C1 = {49/67} (cannot be {58} which clashes with R57C1), no 5,8

5. Killer pair 6,9 in R29C1 and R57C1, locked for C1, clean-up: no 5,8 in R3C9, no 3 in R4C3, no 1 in R6C8, no 1,4 in R8C8

6. Killer pair 5,8 in R3C1 and R57C1, locked for C1,clean-up: no 2 in R6C8, no 1 in R7C7, no 2 in R8C8

7. 45 rule on R6 3 innies R6C467 = 15
7a. Max R6C67 = 13 -> min R6C4 = 2

8. 7 in C9 only in R17C9 -> either R1C4 = 4 or R7C3 = 4 (locking cages), CPE no 4 in R1C3 + R7C4, clean-up: no 6 in R3C4, no 8 in R4C6

9. 45 rule on R4 3 innies R4C267 = 18 = {189/279/369/468/567} (cannot be {378} which clashes with R4C13, cannot be {459} which clashes with R4C49)
9a. 6 of {369/468/567} must be in R4C2 -> no 3,4,5 in R4C2
9b. 8 of {468} must be in R4C7 -> no 4 in R4C7, clean-up: no 7 in R2C7

10. R6C467 (step 7) = {159/168/249/267/348/357} (cannot be {258/456} which clash with R6C29)
10a. Hidden killer triple 1,2,3 in R6C18, R6C35 and R6C467 for R6, R6C18 and R6C467 must each have one of 1,2,3 -> R6C35 must have one of 1,2,3 = {36}/[72/81] (cannot be {45} which doesn’t contain 1,2 or 3), no 4,5

11. Killer triple 4,5,6 in R6C18, R6C29 and R6C467, locked for R6, clean-up: no 3 in R6C35

12. R6C29 = {59/68}, R6C35 = [72/81] -> combined cage R6C2359 = {5972/5981/6872}
12a. R6C467 (step 10) = {159/168/267/348} (cannot be {249/357} which clash with R6C2359)
12b. 9 of {159} must be in R6C7 -> no 5 in R6C7, clean-up: no 9 in R1C7
12c. 7 of {267} must be in R6C4 -> no 2 in R6C4

13. R4C2 + R6C4 + R9C7 = {127/136/145/235}
13a. 7 in {127} must be in R6C4 -> no 7 in R4C2 + R9C7

14. 7 in N4 only in R456C3, locked for C3, clean-up: no 5 in R4C6, no 8 in R7C8, no 4 in R7C9

15. R4C267 (step 9) = {189/279/369/468}
15a. 2 of {279} must be in R4C2 -> no 2 in R4C7, clean-up: no 9 in R2C7

16. 6 in R4 only in R4C23, locked for R4, clean-up: no 8 in R6C9, no 8 in R7C1

17. R6C18 = [16/34/43] (cannot be {25} which clashes with R6C2359), no 2,5
17a. 2 in R6 only in R6C56, locked for N5, clean-up: no 8 in R1C6, no 5 in R2C8, no 4 in R4C9, no 7 in R8C4

18. 45 rule on C6 3 innies R468C6 = 16 = {178/259/349/457} (cannot be {358} because 5,8 only in R8C6)
18a. 5,8 of {178/457} must be in R8C6 -> no 7 in R8C6, clean-up: no 5 in R5C3

19. 8 in C1 only in R35C1 -> either R3C9 = 6 or R7C1 = 6 (locking cages), CPE no 6 in R7C9, clean-up: no 5 in R7C3

20. R6C2 cannot be 8, here’s how
20a. 8 in C1 only in R35C1
R3C1 = 8 => R3C9 = 6 => no 6 in R6C9 => no 8 in R6C2
R5C1 = 8 => no 8 in R8C2
20b. -> no 8 in R6C2, clean-up: no 6 in R6C9

21. Naked pair {59} in R6C29, locked for R6, clean-up: no 5 in R1C7

22. Naked pair {68} in R16C7, locked for C7, clean-up: no 1 in R1C5, no 2,4 in R2C4, no 3 in R24C7, no 5,7 in R3C2

23. R4C267 (step 15) = {279/369}, no 1,4, 9 locked for R4, clean-up: no 3 in R4C58

24. R4C2 + R6C4 + R9C7 = {127/136/235} (cannot be {145} because R4C2 only contains 2,6), no 4
24a. R4C2 = {26} -> no 6 in R6C4, no 2 in R9C7
24b. 1,5 only in R9C7 -> R9C7 = {15}

25. R6C467 (step 12a) = {267/348} (cannot be {168} because R6C4 only contains 3,7), no 1, clean-up: no 4 in R9C9
25a. R6C4 = {37} -> no 3 in R6C6, clean-up: no 2 in R9C9

26. 6 in R6 only in R6C78, locked for N6, clean-up: no 5 in R9C8

27. 6 in C9 only in R13C9, locked for N3 -> R16C7 = [86], clean-up: no 3 in R1C49, no 4 in R3C5, no 1 in R5C5, no 1 in R6C1, no 2,4 in R7C2

28. Naked pair {34} in R6C18, locked for R6 -> R6C6 = 2 -> R9C9 = 3, R6C5 = 1, R6C4 = 7, R6C3 = 8, clean-up: no 5 in R1C3, no 3,9 in R1C6, no 4 in R1C9, no 9 in R27C6, no 5,6 in R3C6, no 1 in R4C1, no 5 in R4C8, no 5 in R4C9, no 8 in R5C6, no 3 in R5C7, no 8 in R5C8, no 6 in R7C1, no 4 in R7C5, no 7 in R7C8, no 7 in R8C1, no 9 in R8C2, no 4 in R8C6, no 5 in R9C2, no 6 in R9C3, no 6 in R9C6

29. R5C2 = 1 (hidden single in N4) -> R1C2 = 5, R3C1 = 8 -> R3C9 = 6, R6C2 = 9, R6C9 = 5, R57C1 = [59], R7C8 = 6, R2C3 = 9, R1C3 = 3, R4C6 = 9, R4C7 = 7 -> R2C7 = 4, R3C2 = 4 -> R8C7 = 9, R5C7 = 2 -> R2C4 = 8, R7C7 = 5 -> R1C1 = 1, R38C3 = [21], R7C3 = 4 -> R7C9 = 7, R4C9 = 1 -> R4C4 = 5, R28C9 = [24], R1C9 = 9 -> R1C4 = 2, R9C7 = 1, R3C7 = 3, R2C8 = 1 -> R5C5 = 6

and the rest is naked singles.

My comments in The Messier, The Merrier 3 post also apply here. For The Messier, The Merrier 4 both JSudoku and SS used outies and innie-outie differences for R4, R6 and C6.

Prelims

Only 39 Prelims in this, compared with 40 for the Twosomes, but I’ve listed them all because the cage pattern is so confusing and have given both cells for the same reason.

a) R1C1 + R7C7 = {19/28/37/46}, no 5
b) R15C2 = {15/24}
c) R1C3 + R4C6 = {29/38/47/56}, no 1
d) R1C49 = {14/23}
e) R1C5 + R3C7 = {59/68}
f) R15C6 = {79}
g) R16C7 = {29/38/47/56}, no 1
h) R1C8 + R7C2 = {39/48/57}, no 1,2,6
i) R29C1 = {29/38/47/56}, no 1
j) R2C25 = {29/38/47/56}, no 1
k) R2C3 + R7C8 = {59/68}
l) R2C4 + R5C7 = {18/27/36/45}, no 9
m) R27C6 = {15/24}
n) R24C7 = {39/48/57}, no 1,2,6
o) R2C8 + R5C5 = {29/38/47/56}, no 1
p) R28C9 = {14/23}
q) R3C19 = {19/28/37/46}, no 5
r) R3C2 + R8C7 = {15/24}
s) R38C3 = {16/25/34}, no 7,8,9
t) R37C4 = {39/48/57}, no 1,2,6
u) R3C58 = {16/25/34}, no 7,8,9
v) R39C6 = {18/27/36/45}, no 9
w) R4C13 = {19/28/37/46}, no 5
x) R4C49 = {49/58/67}, no 1,2,3
y) R4C58 = {18/27/36/45}, no 9
z) R57C1 = {16/25/34}, no 7,8,9
aa) R5C3 + R8C6 = {18/27/36/45}, no 9
bb) R58C4 = {17/26/35}, no 4,8,9
cc) R59C8 = {49/58/67}, no 1,2,3
dd) R5C9 + R9C5 = {16/25/34}, no 7,8,9
ee) R6C18 = {18/27/36/45}, no 9
ff) R6C29 = {79}
gg) R6C35 = {16/25/34}, no 7,8,9
hh) R6C6 + R9C9 = {29/38/47/56}, no 1
ii) R7C39 = {19/28/37/46}, no 5
jj) R7C5 + R9C3 = {17/26/35}, no 4,8,9
kk) R8C18 = {59/68}
ll) R8C25 = {39/48/57}, no 1,2,6
mm) R9C24 = {39/48/57}, no 1,2,6

1. Naked pair {79} in R15C6, locked for C6, clean-up: no 2,4 in R1C3, no 2 in R39C6, no 2 in R5C3, no 2,4 in R9C9

2. Naked pair {79} in R6C29, locked for R6, clean-up: no 2,4 in R1C7, no 2 in R6C18

3. R39C6 = {18/36} (cannot be {45} which clashes with R27C6), no 4,5

4. 9 in R8 only in R8C1258
4a. R8C18 = {59/68}, R8C25 = {39/48/57} -> combined cage R8C1258 including 9 = {5948/6839}, no 7, 8 locked for R8, clean-up: no 1 in R5C3, no 5 in R8C25

5. R8C4 = 7 (hidden single in R8) -> R5C4 = 1, clean-up: no 5 in R1C2, no 4 in R1C9, no 8 in R2C4, no 5 in R37C4, no 8 in R4C8, no 6 in R4C9, no 2,8 in R5C7, no 6 in R6C3, no 6 in R7C1, no 5 in R9C2, no 1 in R9C3, no 6 in R9C5

6. 1 in C7 only in R789C7, locked for N9, clean-up: no 4 in R2C9, no 9 in R7C3

7. 1 in C9 only in R123C9, locked for N3, clean-up: no 6 in R3C5

8. R1C9 = {123}, R28C9 = 5 -> max R128C9 = 8 must contain 1 in R12C9, locked for N3, clean-up: no 9 in R3C1
[I first spotted the short chain between R1C9 = {123} and R28C9 = {14/23} but then saw the alternative way, which I think is cleaner.]

9. 1 in C8 only in R46C8 -> either R4C5 = 8 or R6C1 = 8 (locking cages), CPE no 8 in R4C123, clean-up: no 2 in R4C13

10. R1C2 = {124}, R1C49 = 5 = {14/23} -> variable combined cage R1C249 = 6,7,9 = {123/124/234}, 2 locked for R1, clean-up: no 8 in R7C7

11. 45 rule on R4 3 innies R4C267 = 13 = {139/148/157/238/247/256/346}
11a. 1 of {139} must be in R4C2 -> no 9 in R4C2
11b. 1 of {157} must be in R4C2, 5 of {256} must be in R4C7 -> no 5 in R4C2

12. 45 rule on R6 3 innies R6C467 = {238/256/346}
12a. R6C18 = {18/45} (cannot be {36} which clashes with R6C467), no 3,6

13. R4C13 = {19/37/46}, R4C49 = {49/58/67} -> combined cage R4C1349 = {1958/1967/3749/3758/4658}
13a. R4C267 (step 11) = {139/238/247/256/346} (cannot be {148/157} which clashes with R4C1349)
13b. 5 of {256} must be in R4C7 -> no 5 in R4C6, clean-up: no 6 in R1C3

14. 7 in R3 only in R3C19 = {37}, locked for R3; clean-up: no 4 in R3C58, no 9 in R7C4, no 4 in R8C3, no 6 in R9C6
[I ought to have spotted this earlier; it’s been available since step 5. I’d been looking for quite a long time for limited occurrences of a number in a row/column/nonet to see if I could find more locking cages moves.
Although it wasn’t immediately obvious how important it was, step 14 proved to be the key breakthrough because it eliminated 8,9 from R3C19 and 3 from R3C5.]

15. 9 in R3 only in R3C47, CPE no 9 in R1C5, clean-up: no 5 in R3C7

16. 8 in R3 only in R3C467, CPE no 8 in R1C5, clean-up: no 6 in R3C7

17. Hidden killer pair 8,9 in R3C46 and R3C7 for R3, R3C7 = {89} -> R3C46 must contain one of 8,9
17a. Hidden killer triple 7,8,9 in R1C6, R2C5 and R3C46 for N2, R1C6 = {79}, R3C46 contains one of 8,9 -> R2C5 must contain one of 7,8,9 -> R2C5 = {789}, clean-up: R2C2 = {234}

18. 8 in C2 only in R789C2, locked for N7, clean-up: no 3 in R2C1, no 2 in R7C9, no 6 in R8C8

19. 3 in N2 only in R12C4, locked for C4, clean-up: no 9 in R3C4, no 9 in R9C2

20. Naked pair {48} in R37C4, locked for C4, clean-up: no 1 in R1C9, no 5,9 in R4C9, no 5 in R5C7, no 4,8 in R9C2

21. Naked pair {23} in R1C49, locked for R1, clean-up: no 8 in R4C6, no 4 in R5C2, no 8 in R6C7, no 9 in R7C2, no 7 in R7C7

22. R3C7 = 9 (hidden single in R3) -> R1C5 = 5, clean-up: no 1 in R1C1, no 3 in R24C7, no 6 in R2C8, no 2 in R3C8, no 6 in R4C6, no 4 in R4C8, no 2 in R5C5, no 4 in R5C7, no 2 in R5C9, no 2 in R6C3, no 2,6 in R6C7, no 3,7 in R7C2, no 1 in R7C6, no 3 in R9C3

23. R1C2 = 1 (hidden single in R1) -> R5C2 = 5, clean-up: no 7 in R1C8, no 2 in R6C5, no 4 in R6C8, no 2 in R7C1, no 6 in R8C3, no 4 in R8C6, no 1,5 in R8C7, no 2 in R9C5, no 8 in R9C8

24. R2C9 = 1 (hidden single in C9) -> R8C9 = 4, R8C7 = 2, R3C2 = 4, R37C4 = [84], R7C2 = 8, R1C8 = 4, clean-up: no 6,8 in R1C1, no 3 in R2C2, no 6 in R2C3, no 2 in R2C6, no 5 in R3C3, no 8 in R4C7, no 3 in R5C1, no 7 in R5C3, no 7 in R5C5, no 9 in R59C8, no 3 in R5C9, no 2,6 in R7C3, no 5 in R7C6, no 6 in R7C7, no 6 in R7C9, no 3 in R8C3, no 8 in R8C5, no 7 in R9C1, no 3 in R9C5, no 1 in R9C6

25. R2C6 = 4, R7C6 = 2, R4C6 = 3 -> R1C3 = 8, R9C6 = 8 -> R3C6 = 1, R3C5 = 2, R3C3 = 6 -> R8C3 = 1, R3C8 = 5, R5C9 = 6, R9C5 = 1, R2C2 = 2 -> R2C5 = 9

and the rest is naked singles.

I'm not really sure why the software solvers, in particular SS, found this puzzle so hard. I don't think I used any particularly difficult steps so I'm not sure what they missed.

The SS log included steps such as Cage Blockers Extended, Cage Blockers Complex, Cage Placement Insane and Cage Blockers Insane. JSudoku's solving path seemed to be simpler but my impression, without going through it in detail, was that it was still harder than mine.

I don't think either solver found step 4 which led to two early placements. Maybe they aren't set up to do combined cages which must contain a particular number. I don't think they have any problem with combined cages which don't have that limitation.