Prelims
a) R1C12 = {19/28/37/46}, no 0,5
b) R12C3 = {07/16/25/34}, no 8,9
c) R12C4 = {06/15/24}, no 3,7,8,9
d) R1C56 = {07/16/25/34}, no 8,9
e) R1C89 = {06/15/24}, no 3,7,8,9
f) R23C1 = {19/28/37/46} no 0,5
g) R23C6 = {49/58/67}, no 0,1,2,3
h) R23C9 = {19/28/37/46} no 0,5
i) R3C23 = {08/17/26/35}, no 4,9
j) R3C78 = {19/28/37/46} no 0,5
k) R4C12 = {08/17/26/35}, no 4,9
l) R45C3 = {29/38/47/56}, no 0,1
m) R4C45 = {39/48/57}, no 0,1,2,6
n) R45C6 = {03/12}
o) R4C78 = {08/17/26/35}, no 4,9
p) R45C9 = {19/28/37/46} no 0,5
q) R56C1 = {07/16/25/34}, no 8,9
r) R56C4 = {69/78}
s) R7C12 = {19/28/37/46} no 0,5
t) R78C3 = {19/28/37/46} no 0,5
u) R7C45 = {04/13}
v) R78C6 = {06/15/24}, no 3,7,8,9
w) R78C7 = {59/68}
x) R7C89 = {19/28/37/46} no 0,5
y) R89C1 = {05/14/23}, no 6,7,8,9
z) R89C4 = {19/28/37/46} no 0,5
aa) R89C9 = {05/14/23}, no 6,7,8,9
ab) R9C23 = {08/17/26/35}, no 4,9
ac) R9C56 = {69/78}
ad) R9C78 = {03/12}
45 rule on the whole grid shows that R258C258 total 45 but that’s no help as some numbers in cells which don’t “see” each other may be repeated.
Each row column and nonet must contain 0 and be missing one of 1-9 -> totals in each must be between 36 and 44 with each row, each column and each nonet having a different total.
1a. 45 rule on N1 cages total 35 -> no 0 in R2C2
1b. 45 rule on N2 cages total 35 -> no 0 in R2C5
1c. 45 rule on N3 cages total 35 -> no 0 in R2C8
1d. 45 rule on N4 cages total 35 -> no 0 in R5C2
1e. 45 rule on N5 cages total 39 -> max R5C5 = 5
1f. 45 rule on N7 cages total 33 -> min R8C2 = 3
1g. 45 rule on N8 cages total 35 -> no 0 in R8C5
1h. 45 rule on N9 cages total 32 -> min R8C8 = 4
2a. Combined cages R89C9 + R9C78 = {05}{12}/{14}{03}, 1 locked for N9, no 2,3 in R89C9, clean-up: no 9 in R7C89
2b. Combined cages R7C45 + R78C6 = {04}{15}/{13}{06} (cannot be {13}{24} which clashes with R89C4), 1 locked for N8, no 2,4 in R78C6, clean-up: no 9 in R89C4
[Alternatively combined cages R7C45 + R78C6 = {04}{15}/{13}{06} must contain 0 for N8. I originally had these two cages as steps 1a and 1b, then realised how important the 45s on nonets are so moved those to become step 1.]
2c. R89C4 = {28/37} (cannot be {46} which clashes with combined cages), no 4,6
2d. R9C56 = {69} (cannot be {78} which clashes with R89C4), locked for N8, 6 locked for R9, clean-up: no 0 in R78C6, no 2 in R9C23
2e. Naked pair {15} in R78C6, locked for C6 and N8, clean-up: no 2,6 in R1C5, no 8 in R23C6, no 2 in R45C6, no 4,8 in R6C5, no 3 in R7C45
2f. Naked pair {03} in R45C6, locked for C6 and N5, clean-up: no 4,7 in R1C5, no 9 in R4C45, no 6,9 in R6C56
2g. R56C4 = {69} (cannot be {78} which clashes with R4C45), locked for C4, clean-up: no 0 in R12C4, no 0,3 in R3C5
2h. Naked pair {04} in R7C45, 4 locked for R7 and N8, clean-up: no 6 in R7C12, no 6 in R7C89, no 6 in R8C3
2i. Killer pair 6,9 in R23C6 and R9C6, locked for C6, clean-up: no 1 in R1C5
2j. R1C56 = [07/34] (cannot be [52] which clashes with R12C4), no 2,5
2k. R1C12 = {19/28/46} (cannot be {37} which clashes with R1C56), no 3,7
2l. Killer pair 4,7 in R1C6 and R23C6, locked for C6 and N2, clean-up: no 2 in R12C4, no 2,5 in R3C45, no 2,5 in R6C5
2m. Naked pair {15} in R12C4, locked for N2, 5 locked for C4, clean-up: no 8 in R3C45, no 7 in R4C5
2n. R1C89 = {06/24} (cannot be {15} which clashes with R1C4), no 1,5
2o. R1C12 = {19/28} (cannot be {46} which clashes with R1C89), no 4,6
2p. R23C9 = {19/28/37} (cannot be {46} which clashes with R1C89), no 4,6
2q. R3C78 = {19/28/37} (cannot be {46} which clashes with R1C89), no 4,6
2r. Naked pair {03} in R1C5 + R3C4, 3 locked for N2
2s. Killer pair 6,9 in R23C6 and R3C45, locked for N2
3a. 0 in N1 only in R12C3 = {07} or R3C23 = {08} -> R3C23 = {08/26/35} (cannot be {17}, locking-out cages), no 1,7
3b. 0 in N3 only in R12C7 = {09) or R1C89 = {06) -> R12C7 = {09/18/27/45} (cannot be {36}, locking-out cages), no 3,6
3c. 0 in N7 only in R89C1 = {05} or R9C23 = {08} -> R9C23 = {08/17} (cannot be {35}, locking-out cages), no 3,5
3d. Killer pair 0,1 in R9C23 and R9C78, locked for R9, clean-up: no 4,5 in R8C1, no 4,5 in R8C9
3e. Killer pair 0,4 in R1C56 and R1C89, locked for R1, clean-up: no 3,7 in R2C3, no 5,9 in R2C7
[Time to start looking at forcing chains.]
4a. R1C89 = {06/24}
4b. Consider combinations for R1C12 = {19/28}
R1C12 = {19}, 9 locked for R1, no 9 in R1C7 => no 0 in R2C7, 0 in N3 only in R1C89 = {06}
or R1C12 = {28}, 2 locked for R1 => R1C89 = {06}
-> R1C89 = {06}, locked for R1 and N3, R1C5 = 3 -> R1C6 = 4, R3C4 = 0 -> R3C5 = 9, R7C45 = [40], R9C56 = [69], clean-up: no 9 in R1C7, no 1 in R2C1, no 4 in R2C3, no 1 in R2C9, no 8 in R3C23, no 1 in R3C78, no 8 in R4C5
4c. R2C3 = 0 (hidden single in N1) -> R1C3 = 7, clean-up: no 3 in R23C1, no 2 in R2C7, no 4 in R45C3, no 2,9 in R6C2, no 3 in R78C3, no 1,8 in R9C2
4d. Killer pair 2,3 in R3C23 and R3C78, locked for R3, clean-up: no 8 in R2C1, no 7,8 in R2C9
5a. R7C12 = {19/28/37}, R7C78 = {28/37}, R9C23 = [08/71], R9C78 = {03/12}
5b. Consider combinations for R9C23 = [08/71]
R9C23 = [08] => R9C78 = {12}, 2 locked for N9
or R9C23 = [71], R7C12 = {28}, 2 locked for R7
-> R7C78 = {37}, locked for R7 and N9, clean-up: no 0 in R9C78
5c. Naked pair {12} locked for R9, 1 locked for N9, R8C9 = 0 -> R9C9 = 5, R1C89 = [06], R9C3 = 8 -> R9C2 = 0, clean-up: no 8 in R4C1, no 3 in R45C3, no 1 in R6C2, no 9 in R6C3, no no 3,4,9 in R6C8, no 9 in R6C9, no 2 in R7C12, no 2 in R78C3, no 9 in R78C7, no 3 in R8C1, no 2,8 in R8C4
5d. Naked pair {19} in R7C12, locked for N7, 1 locked for R7 -> R78C3 = [64], R78C6 = [51], R7C87 = [86], R89C9 = [23], R89C4 = [37], R4C4 = 8 -> R4C5 = 4, R6C6 = 2 -> R6C5 = 7, R28C5 = [28], clean-up: no 8 in R1C2, no 1 in R12C7, no 8 in R3C1, no 2 in R3C2, no 2 in R3C8, no 8 in R3C9, no 0 in R4C1, no 5,6 in R4C2, no 5 in R45C3, no 0 in R4C7, no 2 in R4C8, no 0,4,5 in R5C1, no 1,2,3,7 in R5C7, no 2,4 in R5C9, no 4,5 in R6C1, no 3,5 in R6C2, no 1,3 in R6C7
5e. R23C9 = [91] (cannot be [37] which clashes with R7C9), clean-up: no 0,8 in R6C8
5f. R45C9 = [28] (cannot be {37} which clashes with R7C9), R45C3 = [92], clean-up: no 6 in R3C2, no 6 in R4C1, no 6 in R4C8, no 1 in R6C8
5g. Naked pair {46}, locked for N1, 6 locked for C1, clean-up: no 1 in R56C1
5h. R56C1 = [70], clean-up: no 1 in R4C12, no 9 in R5C7
5i. R4C12 = [53], R45C6 = [03], R3C23 = [53], R6C3 = 1 -> R6C2 = 8, R2C2 = 1, R7C12 = [19], R1C2 = 2 -> R1C1 = 8, R1C7 = 5 -> R2C7 = 4, R56C7 = [09], R56C4 = [96], R6C8 = 5 -> R6C9 = 4, clean-up: no 7 in R3C78
5j. R3C78 = [28], R23C1 = [64], R23C6 = [76], R2C8 = 3, R9C78 = [12], R4C78 = [71], R5C28 = [46]
5k. 45 rule on the whole grid 1 remaining innie R5C5 = 5
and the rest is naked singles.