Prelims
a) R1C34 = {59/68}
b) R12C5 = {39/48/57}, no 1,2,6
c) R1C67 = {17/26/35}, no 4,8,9
d) R23C3 = {19/28/37/46}, no 5
e) R23C4 = {29/38/47/56}, no 1
f) R4C34 = {14/23}
g) R46C5 = {29/38/47/56}, no 1
h) R4C67 = {29/38/47/56}, no 1
i) R56C4 = {49/58/67}, no 1,2,3
j) R57C5 = {18/27/36/45}, no 9
k) R56C6 = {18/27/36/45}, no 9
l) R6C23 = {18/27/36/45}, no 9
m) R89C3 = {19/28/37/46}, no 5
n) R8C45 = {19/28/37/46}, no 5
o) R9C45 = {17/26/35}, no 4,8,9
p) 7(3) cage at R2C6 = {124}
q) 19(3) cage at R7C1 = {289/379/469/478/568}, no 1
r) 26(4) cage at R1C8 = {2789/3689/4589/4679/5678}, no 1
1a. 45 rule on R89 1 outie R7C7 = 3, clean-up: no 5 in R1C6, no 8 in R4C6, no 6 in R5C5
1b. 45 rule on C89 2 outies R36C7 = 14 = {59/68}
1c. R36C7 = 14 -> R3456C8 = 14 = {1238/1247/1256/1346/2345} (cannot be {1256} which clashes with R36C7), no 9
1d. 45 rule on C6789 2 innies R5C7 + R7C6 = 13 = {49/58/67}, no 1,2
1e. 45 rule on N1 1 innie R1C3 = 1 outie R4C2 + 3, R1C3 = {5689} -> R4C2 = {2356}
1f. 45 rule on N1 3 innies R1C3 + R3C12 = 20 = {389/479/569/578}, no 1,2
1g. 45 rule on N4 3 innies R4C23 + R5C3 = 15, max R4C23 = 10 -> min R5C3 = 5
1h. 45 rule on C12 2 outies R67C3 = 7 = [16/25/34/52]
1i. Hidden killer quad 1,2,3,4 in R23C3, R4C3, R67C3 and R89C3, R23C3 and R89C3 each contain one of 1,2,3,4, R4C3 = {1234} -> R67C3 can only contain one of 1,2,3,4 = [16/25/52] -> R6C2 = {478}
1j. 19(3) cage at R7C1 = {289/469/568} (cannot be {478} because R7C3 only contains 2,5,6), no 7
1k. Max R7C3 = 6 -> min R7C12 = 13, no 2 in R7C12
1l. Naked triple {124} in 7(3) cage at R2C6, CPE no 2,4 in R2C45, clean-up: no 8 in R1C5, no 7,9 in R3C4
[16(4) cage at R8C1 omitted; it didn’t provide any eliminations and not specifically used later.]
2a. 3,7 in C3 only in R23C3 = {37} or R45C3 = [37] or R89C3 = {37} (locking cages)
2b. R4C23 + R5C3 = 15 (step 1g), R67C3 = 7 (step 1h)
2c. Consider permutations for R6C23 = [45/72/81]
R6C23 = [45] => no 5 in R4C2 => R45C3 cannot be [37]
or R6C23 = [72]
or R6C23 = [81] => R7C3 = 6, no 4 in R23C3 and R89C3 => R4C3 = 4 (hidden single in C3)
-> R45C3 cannot be [37]
2d. R23C3 = {37} or R89C3 = {37} (locking cages), locked for C3, clean-up: no 2 in R4C4
2e. 45 rule on C123 3 innies R145C3 = 18 = {189/459/468}, no 2, clean-up: no 3 in R4C4
2f. Naked pair {14} in R4C34, locked for R4, clean-up: no 7 in R4C67, no 7 in R6C5
3a. 45 rule on N5 3 innies R4C46 + R5C5 = 12 = {129/138/156/246/345} (cannot be {237} because R4C4 only contains 1,4, cannot be {147} because no 1,4,7 in R4C6), no 7, clean-up: no 2 in R7C5
3b. R4C4 = {14} -> no 1,4 in R5C5, clean-up: no 5,8 in R7C5
3c. 6,9 of {129/246} must be in R4C6 -> no 2 in R4C6, clean-up: no 9 in R4C7
3d. Consider placement for 1 in N5
R4C4 = 1 => R4C46 + R5C5 = {129/138/156}
or R56C6 = {18}, 8 locked for N5 => 3 in N5 only in R4C6 + R5C5 => R4C46 + R5C5 = {345}
-> R4C46 + R5C5 = {129/138/156/345}
3e. 1 in C5 only in R389C5 and R57C5 = [81]
3f. 45 rule on C5 3 innies R389C5 = 13 = {139/148/157/247/256/346} (cannot be {238} which clashes with R57C5 = [81], blocking cages)
3g. Consider combinations for R389C5
R389C5 = {139}, locked for C5 => R12C5 = [48]/{57} => R57C5 cannot be [54], clashes with R12C5
or R389C5 = {148/157} => R57C5 cannot be [54], clashes with R389C5
or R389C5 = {247/256/346} => R57C5 = [81]
-> R57C5 = [27/36/81], no 5 in R5C5, no 4 in R7C5
3h. R4C46 + R5C5 = {129/138/345} (cannot be {156} because 5,6 only in R4C6), no 6, clean-up: no 5 in R4C7
3i. Consider placement for 4 in C5
4 in R13C5 => R23C6 = {12}, 1 locked for C6 => R4C4 = 1 (hidden single in N4)
or R6C5 = 4 => R4C4 = 1
or R8C5 = 4 => R8C4 = 6, no 6 in R7C5 => no 3 in R5C5 => R4C46 + R5C5 = {129/138} (cannot be {345} because R5C5 only contains 2,8)
-> R4C46 + R5C5 = {129/138}, no 4,5, clean-up: no 6 in R4C7
3j. R4C4 = 1 -> R4C3 = 4, clean-up: no 6 in R23C3, no 8 in R56C6, no 5 in R6C3, no 2 in R7C3 (step 1i), no 6 in R89C3, no 9 in R8C5, no 7 in R9C5
3k. 19(3) cage at R7C1 (step 1j) = {469/568}, 6 locked for R7 and N7, clean-up: no 3 in R5C5, no 7 in R5C7 (step 1d)
[Continuing in N5]
3l. R46C5 = {29/38/56}/[74], R57C5 = [27/81] -> combined cage R4567C5 = {29}[81]/{38}[27]/{56}[27]/{56}[81]/[7481]
3m. R56C4 = {49/67} (cannot be {58} which clashes with R456C5), no 5,8
3n. R46C5 = {29/38/56} (cannot be [74] which clashes with R56C4), no 7 in R4C5, no 4 in R6C5
3o. Hidden killer pair 4,7 in R56C4 and R56C6 for N5, R56C4 contains one of 4,7 -> R56C6 must contain one of 4,7 = {27/45}, no 3,6
3p. 8 in N5 only in R456C5, locked for C5, clean-up: no 4 in R1C5, no 2 in R8C4
3q. R389C5 = {139/247/256/346} (cannot be {157} which clashes with R7C5)
3r. R389C5 = {247/256/346} (cannot be {139} because combined cage R12C5 = {57} + R389C5 = {139} which clashes with R7C5), no 1,9, clean-up: no 9 in R8C4, no 7 in R9C4
[Fairly straightforward from here.]
4a. R7C5 = 1 (hidden single in C5) -> R5C5 = 8, clean-up: no 3 in R46C5, no 5 in R7C6 (step 1d)
4b. R4C6 = 3 (hidden single in N5) -> R4C7 = 8, clean-up: no 6 in R1C3 (step 1e), no 8 in R1C4, no 5 in R1C7, no 6 in R36C7 (step 1b)
4c. Naked pair {59} in R36C7, locked for C7, 5 locked for 28(6) cage at R3C7, clean-up: no 4,8 in R7C6 (step 1d)
5a. R23C6 = {12/14} (cannot be {24} which clashes with R56C6), 1 locked for C6 and 7(3) cage at R2C7, clean-up: no 7 in R1C7
5b. Killer pair 2,4 in R23C6 and R56C6, locked for C6, clean-up: no 6 in R1C7
5c. 8 in C6 only in R89C6, locked for N8, clean-up: no 2 in R8C5
5c. Killer pair 3,6 in R8C45 and R9C45, locked for N8
5d. R1C6 = 6 (hidden single in C6) -> R1C7 = 2, R2C7 = 4, R5C7 = 6 -> R7C6 = 7 (step 1d), clean-up: no 8 in R1C3, no 2 in R56C6, no 7 in R6C4, no 3 in R8C45
5e. R5C4 = 7 (hidden single in N5) -> R6C4 = 6 -> R8C45 = [46], clean-up: no 2 in R9C45
5f. Naked pair {45} in R56C6, 5 locked for C6 and N5
5g. Naked pair {29} in R46C5, locked for C5, clean-up: no 3 in R12C5
5h. Naked pair {57} in R12C5, locked for N2, 5 locked for C5 -> R1C34 = [59], R12C5 = [75], R5C3 = 9, R9C45 = [53], R4C2 = 2 (step 1e), R6C3 = 1 -> R6C2 = 8, R7C34 = [62], R46C5 = [92], R4C89 = [75], clean-up: no 7 in R8C3
5i. Naked pair {38} in R1C89, locked for R1 and N3
5j. R1C89 = 11 -> R2C89 = 15 = {69}, locked for R2 and N3
5k. R36C7 = [59] = 14, R34C8 = [17] = 8 -> R56C8 = 6 = [24]
5l. R3456C9 = [7513] = 16 -> R7C89 = 13 = [94]
5m. Naked pair {14} in R1C12, locked for N1
5n. R1C12 = {14} = 5 -> R2C12 = 10 = {37} (cannot be {28} because 2,8 only in R2C1), locked for N1, 3 locked for R2
and the rest is naked singles.