Prelims
a) R1C34 = {39/48/57}, no 1,2,6
b) R1C67 = {19/28/37/46}, no 5
c) R34C1 = {59/68}
d) 10(2) cage at R3C3 = {19/28/37/46}, no 5
e) 10(2) cage at R3C4 = {19/28/37/46}, no 5
f) 8(2) cage at R3C6 = {17/26/35}, no 4,8,9
g) 12(2) cage at R3C7 = {39/48/57}, no 1,2,6
h) R34C9 = {19/28/37/46}, no 5
i) R67C1 = {29/38/47/56}, no 1
j) 10(2) cage at R6C3 = {19/28/37/46}, no 5
k) 10(2) cage at R6C4 = {19/28/37/46}, no 5
l) 10(2) cage at R6C6 = {19/28/37/46}, no 5
m) 10(2) cage at R6C7 = {19/28/37/46}, no 5
n) R67C9 = {29/38/47/56}, no 1
o) R9C34 = {19/28/37/46}, no 5
p) R9C67 = {16/25/34}, no 7,8,9
q) 23(3) cage at R1C5 = {689}
r) 6(3) cage at R1C8 = {123}
s) 9(3) cage at R5C1 = {126/135/234}, no 7,8,9
t) 24(3) cage at R5C7 = {789}
u) 19(3) cage at R8C9 = {289/379/469/478/568}, no 1
v) 15(5) cage at R4C5 = {12345}
Steps resulting from Prelims and Initial Placements
1a. Naked triple {689} in 23(3) cage at R1C5, locked for C5 and N2, clean-up: no 3,4 in R1C3, no 1,2,4 in R1C7, no 1,2,4 in R4C3, no 2 in R4C7
1b. Naked triple {123} in 6(3) cage at R1C8, locked for N3, clean-up: no 7 in R1C6, no 9 in R4C6, no 7,8,9 in R4C9
1c. Naked triple {789} in 24(3) cage at R5C7, locked for R5 and N6, clean-up: no 1 in R3C6, no 1,2,3 in R7C6, no 2,3,4 in R7C9
1d. 45 rule on R1234 1 innie R4C5 = 1
1e. 45 rule on R6789 1 innie R6C5 = 2
1f. 45 rule on C1234 1 innie R5C4 = 4
1g. 45 rule on C6789 1 innies R5C6 = 3 -> R5C5 = 5
1h. Naked triple {126} in 9(3) cage at R5C1, locked for N4
1i. Naked triple {347} in 14(3) cage at R7C5, locked for N8
Clean-ups: no 8 in R1C3, no 7 in R1C7, no 8 in R3C1, no 6,7,8,9 in R3C3, no 7 in R3C6, no 7,8,9 in R3C7, no 9 in R3C9, no 5 in R4C7, no 3,7 in R6C3, no 3,6 in R6C7, no 5,9 in R7C1, no 6,7,8,9 in R7C3, no 8,9 in R7C4, no 8 in R7C6, no 6,7,8,9 in R7C7, no 9 in R7C9, no 3,6,7 in R9C3, no 3,4 in R9C7
2a. R67C1 = {38/47}/[92] (cannot be [56] which clashes with R34C1), no 5,6
2b. 2 in C7 only in R789C7, locked for N9
2c. 1 in C9 only in R12C9, locked for N3
2d. 1 in C8 only in R678C8, locked for 25(5) cage at R6C8, no 1 in R8C67
3. 45 rule on R1 1 innie R1C5 = 2 outies R2C19 + 3
3a. Max R2C19 = 6 -> max R2C1 = 5
4. 45 rule on C1 2 outies R19C2 = 1 innie R5C1 + 10, min R19C2 = 11, no 1 in R19C2
5. 45 rule on C9 2 outies R19C8 = 1 innie R5C9 + 1
5a. Min R5C9 = 7 -> min R19C8 = 8 -> min R9C8 = 5
6. 45 rule on N3 5(2+3) outies R12C6 + R4C689 = 18
[The first key step.]
6a. Min R4C689 = 12 -> max R12C6 = 6, no 7 in R2C6
[I continued working on these outies until I spotted step 6c which made that work unnecessary.]
6b. 7 in N2 only in R123C4, locked for C4, clean-up: no 3 in R3C3, no 3 in R7C3
6c. Consider permutations for 12(2) cage at R3C7 = [48/57]
12(2) cage = [48] => R6C6 = 7 (hidden single in N5) => R7C7 = 3 => R4C7 = 6, R3C6 = 2
or 12(2) cage = [57] => R3C6 = 2 => R4C7 = 6
-> 8(2) cage at R3C6 = [26], clean-up: no 4 in R1C6, no 8 in R1C7, no 4 in R3C3, no 4 in R3C9, no 8 in R4C3, no 8 in R4C4, no 5 in R7C9, no 1 in R9C6, no 5 in R9C7
[Cracked. The rest is straightforward.]
[At this stage I spotted R9C67 = [52] (cannot be [61] which clashes with 10(2) cage at R6C7, but it wasn’t necessary after working through the many placements in the rest of step 6.]
6d. R1C67 = [19], clean-up: no 3 in R1C4
6e. Naked pair {57} in R1C34, locked for R1
6f. 10(2) cage at R3C3 = [19], clean-up: no 5 in R3C1, no 1 in R7C7
6g. 6 in N5 only in R6C46 –> 4 in R7C37, locked for R7, clean-up: no 7 in R6C1
6h. R2C6 = 4 (hidden single in C6)
6i. R3C7 = 4 (hidden single in N3) -> R4C6 = 8, R6C4 = 6 -> R7C3 = 4, R6C6 = 7 -> R7C7 = 3, R6C7 = 1 -> R7C6 = 9, R9C7 = 2 -> R9C6 = 5, R4C1 = 5 -> R3C1 = 9, R7C5 = 7, R8C6 = 6, clean-up: no 4,8 in R6C1, no 4 in R6C9, no 2 in R7C1, no 8 in R9C34
6j. R9C34 = [91], 10(2) cage at R6C3 = [82], R67C1 = [38], 10(2) cage at R3C4 = [37], R1C34 = [57], R46C2 = [49], R67C9 = [56], R6C8 = 4, R8C4 = 8
7. R2C1 = 2 -> R1C12 = 12 = [48], 23(3) cage at R1C5 = [698]
7a. R3C9 = 7 -> R4C9 = 3
7b. 4 in N9 only in 19(3) cage at R8C9 = {478} (only remaining combination) -> R8C9 = 4
and the rest is naked singles.
