Prelims
a) 11(2) cage at R1C1 = {29/38/47/56}, no 1
b) R12C5 = {89}
c) 11(2) cage at R1C9 = {29/38/47/56}, no 1
d) 6(2) cage in N1 = {15/24}
e) 4(2) cage in N3 = {13}
f) R3C34 = {29/38/47/56}, no 1
g) R3C56 = {29/38/47/56}, no 1
h) R34C7 = {29/38/47/56}, no 1
i) R45C3 = = {29/38/47/56}, no 1
j) R56C7 = {29/38/47/56}, no 1
k) R67C3 = {29/38/47/56}, no 1
l) R7C45 = {29/38/47/56}, no 1
m) R7C67 = {29/38/47/56}, no 1
n) 8(2) cage in N7 = {17/26/35}, no 4,8,9
o) 16(2) cage in N9 = {79}
p) 11(2) cage at R8C2 = {29/38/47/56}, no 1
q) 11(2) cage at R8C8 = {29/38/47/56}, no 1
r) 11(3) cage at R4C4 = {128/137/146/236/245}, no 9
s) 11(3) cage at R4C6 = {128/137/146/236/245}, no 9
Steps resulting from Prelims
1a. Naked pair {89} in R12C5, locked for C5 and N2, clean-up: no 2,3 in R3C3, no 2,3 in R3C56, no 2,3 in R7C4
1b. Naked pair {13} in 4(2) cage, locked for N3, clean-up: no 8 in 11(2) cage at R1C9, no 8 in R4C7
1c. Naked pair {79} in 16(2) cage, locked for N9, clean-up: no 2,4 in R7C6, no 2,4 in 11(2) cage at R8C8
1d. 9 in D\ only in 11(2) cage at R1C1 and R3C3, locked for N1
1e. 9 in N5 only in R5C46, locked for R5, clean-up: no 2 in R4C3, no 2 in R6C7
2. 1 in D\ only in 11(3) cage at R4C4, locked for N5
2a. R5C5 = 1 (hidden single in D/)
2b. 11(3) cage at R4C4 = {128/137/146}, no 5
2c. 11(3) cage at R4C6 = {128/137/146}, no 5
3. 45 rule on D\ 2 innies R3C3 + R7C7 = 12 = [48/75/84/93], no 2,6, no 5 in R3C3, clean-up: no 5,6 in R3C4, no 5,9 in R7C6
4. 45 rule on D/ 2 innies R3C7 + R7C3 = 12 = {48/57}/[93], no 2,6, no 9 in R7C3, clean-up: no 5,9 in R4C7, no 2,5,9 in R6C3
5. 45 rule on D\ 4 innies R3C3 + R4C4 + R6C6 + R7C7 = 22 = {2938/2947/2956/3847/3856/4756} (other combinations clash with one or both of the 11(2) cages on D\)
5a. R3C3 + R4C4 + R6C6 + R7C7 = {2938/3847/4756} (cannot be {2947/2956/3856} which clash with combinations for R3C3 + R7C7)
5b. 2 of {2938} must be in R6C6 (R3C3 + R4C4 cannot be [92] because of CCC with R3C34), no 2 in R4C4, clean-up: no 8 in R6C6 (step 2b)
5c. 4 of {3847} must be in R3C3 + R7C7, 4 of {4756} must be in R6C6 (R3C3 + R4C4 cannot be [74] because of CCC with R3C34), no 4 in R4C4, clean-up: no 6 in R6C6 (step 2b)
6. Killer quad 3,5,6,8 in R3C3 + R4C4 + R6C6 + R7C7 and 11(2) cage at R8C8, locked for D\
7. 6(2) cage in N1 = {15} (cannot be {24} which clashes with 11(2) cage at R1C1), locked for N1
7a. 1 in R1 only in R1C46, locked for N2
7b. 5 in D\ only in R7C7 + R8C8 + R9C9, locked for N9
8. 45 rule on D/ 4 innies R3C7 + R4C6 + R6C4 + R7C3 = 22 = {2938/2947/2956/3847/3856/4756} (other combinations clash with one or both of the 11(2) cages on D/)
8a. R3C7 + R4C6 + R6C4 + R7C3 = {2938/3847/4756} (cannot be {2947/2956/3856} which clash with combinations for R3C7 + R7C3)
8b. 2 of {2938} must be in R6C4 (R3C7 + R4C6 cannot be [92] because of CCC with R34C7), no 2 in R4C6, clean-up: no 8 in R6C4 (step 2c)
9. 9 in N1 only in 11(2) cage at R1C1 and R3C3 -> 11(2) cage at R1C1 = {29} or R3C34 = [92] (locking cages), CPE no 2 in R3C1
9a. 2 in R3 only in R3C4 and R3C9 -> R3C34 = [92] or R3C79 = [92] (locking cages), no 9 in R3C9
10. 9 in R3 only in R3C37 -> R3C3 + R4C4 + R6C6 + R7C7 = [9823] or R3C7 + R4C6 + R6C4 + R7C3 = [9823] (locking cages, permutations of {2938} follow from steps 3 and 4 which place the 3s and therefore the 2,8s) -> 8 locked in R4C46, locked for R4 and N5, 2 locked in R6C46, locked for R6 and N5, 3 locked in R7C37, locked for R7, clean-up: no 3 in R5C3, no 8 in R7C4, no 8 in R7C7, no 5 in R8C3, no 4 in R3C3 (step 3), no 7 in R3C4
10a. 2 in C5 only in R789C5, locked for N8
11. R3C3 + R4C4 + R6C6 + R7C7 (step 5a) = {2938/3847/4756} = [7645/8734/9823] (cannot be [8347] which clashes with R3C34], no 3 in R4C4, no 7 in R6C6
12. R3C3 = 9 or R7C3 = 3 (from locking cages in step 10) => R6C3 = 8 -> no 8 in R3C3, clean-up: no 7 in R4C4, no 3 in R6C6, no 4 in R7C7 (all step 11), no 3 in R3C4, no 7 in R7C6
12a. Killer pair 7,9 in 11(2) cage at R1C1 and R3C3, locked for N1
12b. Killer pair 3,5 in R7C7 and 11(2) cage at R8C8, locked for N9
13. R3C7 = 9 or R7C7 = 3 (from locking cages in step 10) => no 3 in R4C7 -> no 8 in R3C7, clean-up: no 3 in R4C7, no 4 in R7C3 (step 4), no 7 in R6C3
14. R3C7 + R4C6 + R6C4 + R7C3 (step 8a) = {2938/3847/4756} = [4378/5467/7645/9823] (cannot be [4783/5647/7465] which clash with R67C3]), no 7 in R4C6, no 3 in R6C4
15. 3 in R7 only in R7C37, R7C3 = 3 or R7C7 = 3 => R7C6 = 8 -> no 8 in R7C3, clean-up: no 4 in R3C7, no 3 in R4C6, no 7 in R6C4 (all step 14), no 7 in R4C7, no 3 in R6C3
16. Killer quad 3,5,7,9 in R7C3, R7C45, R7C7 and R7C8, locked for R7, clean-up: no 1,3 in R8C3
17. Killer quad 3,5,7,9 in R3C7, R56C7, R7C7 and R8C7, locked for C7 -> R2C7 = 1, R3C8 = 3, R2C3 = 5, R3C2 = 1, clean-up: no 7 in R3C7, no 6 in R4C6, no 4 in R6C4 (all step 14), no 6 in R1C9, no 6 in R45C3, no 4 in R4C7, no 6 in R6C3, no 7 in R8C3, no 8 in R9C9
17a. Naked quad {2468} in R46C46, 4,6 locked for N5
17b. 4 in N5 only in R46C6, locked for C6, clean-up: no 7 in R3C5
17c. 6 in N5 only in R46C4, locked for C4, clean-up: no 5 in R7C5
17d. 8 on D\ only in R4C4 + R8C8, CPE no 8 in R8C4
18. Naked pair {26} in 8(2) cage, locked for N7, clean-up: no 5,9 in 11(2) cage at R8C2
18a. Killer pair {37} in R7C3 and 11(2) cage at R8C2, locked for N7 and D/, clean-up: no 4 in 11(2) cage at R1C9
18b. Killer pair 5,9 in 11(2) cage at R1C9 and R3C7, locked for N3
19. R9C3 = 1 (hidden single in C3)
19a. R7C9 = 1 (hidden single in R7)
19b. Killer pair 4,8 in R7C1 and 11(2) cage at R8C2, locked for N7
20. 45 rule on R3 3 remaining innies R3C179 = 19 = {289/568} (cannot be {469} which clashes with R3C56, cannot be {478} because R3C7 only contains 5,9), no 4,7
20a. 4 in R3 only in R3C456, locked for N2
21. 6 in C3 only in R18C3
21a. 45 rule on C3 3 remaining innies R138C3 = 17 = {269/467} (cannot be {368} because R3C3 only contains 7,9), no 3,8
21b. 8 in C3 only in R56C3, locked for N4
22. R3C3 + R4C4 + R6C6 + R7C7 (step 11) = [7645/9823]
22a. R3C3 = 7 or R3C3 = 9 => R7C7 = 3 => R7C3 = 7 -> 7 locked in R37C3, locked for C3, clean-up: no 4 in R45C3
23. R3C7 + R4C6 + R6C4 + R7C3 (step 14) = [5467/9823]
23a. R3C7 = 5 or R3C7 = 9 => R7C3 = 3 => R7C7 = 5 -> 5 locked in R37C7, locked for C7, clean-up: no 6 in R56C7
24. 5 in R7 only in R7C47, R7C45 = [56] or R7C67 = [65] (locking cages), 6 locked in R7C56, locked for R7 and N8 -> R7C2 = 2, R8C3 = 6, clean-up: no 9 in R1C1, no 9 in R7C4, no 5 in R9C9
24a. Killer pair 2,4 in 11(2) cage at R1C1 and R1C3, locked for N1
24b. 2 in N1 only in R1C13, locked for R1, clean-up: no 9 in R2C2
24c. 6 on D\ only in R4C4 + R9C9, CPE no 6 in R4C9
24d. Naked pair {26} in R2C8 + R6C4, CPE no 2 in R2C4, no 6 in R6C8
25. 9 in D/ only in R1C9 + R3C7 -> R1C9 + R2C8 = [92] or R3C79 (step 9a) = [92] (locking cages), CPE no 2 in R2C9
[While checking my walkthrough I noticed that I’d missed another CPE using these 9s; however step 27 gives the same elimination and more.]
26. R7C8 = 9 (hidden single in R7), R8C7 = 7, clean-up: no 4 in R56C7, no 4 in R9C1
26a. 4 in D/ only in R4C6 + R8C2, CPE no 4 in R4C2
27. 9 in C7 only in R36C7 -> R34C7 = [92] or R56C7 = [29] (locking cages), 2 locked in R45C7, locked for C7 and N6
28. R3C3 + R4C4 + R6C6 + R7C7 (step 22) = [7645/9823], R3C34 = [74/92] -> R34C4 = [46/28]
28a. R3C3 + R4C4 + R6C6 + R7C7 = [7645/9823], R7C67 = [83/65] -> R67C6 = [46/28]
28b. Killer pair 4,8 in R4C6 + R67C6, locked for C6
29. R3C7 + R4C6 + R6C4 + R7C3 (step 23) = [5467/9823], R34C7 = [56/92] -> R4C67 = [46/82]
29a. Killer pair 6,8 in R4C4 and R4C67, locked for R4
29b. R3C7 + R4C6 + R6C4 + R7C3 = [5467/9823], R67C3 = [47/83] -> R6C34 = [46/82]
29c. Killer pair 2,4 in R6C34 and R6C6, locked for R6
30. 6 in C5 only in R37C5 -> R3C56 = [65] or R7C45 = [56] (locking cages), CPE no 5 in R1C4 + R89C6
31. 5 in R1 only in R1C69 -> R12C6 = [56] (no 6 in R3C56 when R1C6 = 5) or R1C9 + R2C8 = [56] (locking cages) -> 6 locked in R2C68, locked for R2
31a. R2C68 = {26} (hidden pair in R2)
31b. Naked quad {2468} in R2467C6, locked for C6, clean-up: no 5 in R3C5
31c. 5 in N2 only in R13C6, locked for C6
32. 5 in D/ only in R1C9 + R3C7 -> R1C9 + R2C8 = [56] or R34C7 = [56] (locking cages), CPE no 6 in R1C7
[At this stage I could see a contradiction chain which reduces the innies on each diagonal to one combination. However having tried to avoid such steps I decided to continue without using such a step.]
33. 3 in R7 only in R7C37, R7C3 = 3 => R6C3 = 8 or R7C7 = 3 => no 3 in R5C7 => no 8 in R6C7
-> no 8 in R6C7, clean-up: no 3 in R5C7
33a. Naked pair {28} in R5C37, locked for R5
34. 7 in C3 only in R37C3, R3C3 = 7 => R3C4 = 4 or R7C3 = 7 => no 4 in R7C45
-> no 4 in R7C4, clean-up: no 7 in R7C5
34a. Naked pair {46} in R37C5, locked for C5
35. R37C5 = {46} = 10, R3C56 = 11, R7C45 = 11 -> R3C6 + R7C4 = 12
[This establishes that R3C6 and R7C4 cannot both contain the same number.]
35a. Naked pair {57} in R3C6 + R7C4, CPE no 7 in R12C4 + R9C6
35b. R12C4 = [13]
and the rest is naked singles.