Prelims
a) R1C12 = {19/28/37/46}, no 5
b) R1C34 = {49/58/67}, no 1,2,3
c) R1C67 = {19/28/37/46}, no 5
d) R2C67 = {19/28/37/46}, no 5
e) R6C56 = {16/25/34}, no 7,8,9
f) R6C89 = {19/28/37/46}, no 5
g) R89C1 = {18/27/36/45}, no 1
h) R9C23 = {18/27/36/45}, no 1
i) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9
j) 19(3) cage at R7C9 = {289/379/469/478/568}, no 1
k) 21(3) cage at R8C5 = {489/579/678}, no 1,2,3
l) 41(8) cage at R6C2 = {12356789}, no 4
1a. 45 rule on N89 1 outie R6C7 = 8, clean-up: no 2 in R1C6, no 2 in R2C6, no 2 in R6C89
1b. 4 in N7 only in R89C1 = {45} or R9C23 = {45} (locking cages), 5 locked for N9
1c. 41(8) cage at R6C2 = {12356789}, 5 in R6C234, locked for R6, clean-up: no 2 in R6C56
1d. 41(8) cage at R6C2 = {12356789}, 8 in R7C123 + R8C23, locked for N7, clean-up: no 1 in R89C1, no 1 in R9C23
1e. 45 rule on N7 3 outies R6C234 = 14 contains 5 = {257/356}, no 1,9
1f. R6C89 = {19} (cannot be {37} which clashes with R6C234, cannot be {46} which clashes with R6C56), locked for R6 and N6, clean-up: no 6 in R6C56
1g. Naked pair {34} in R6C56, locked for R6 and N5, clean-up: no 6 in R6C234
1h. Naked triple {257} in R6C234, 2,7 locked for R6 and locked for 41(8) cage -> R6C1 = 6, clean-up: no 4 in R1C2
1i. 41(8) cage at R6C2 = {12356789}, 3,6 locked for N7
1j. 35(7) cage at R2C9 = {2345678} {cannot be {1235789/1245689/1345679} which clash with R6C9, no 1,9, 8 locked for C9 and N3
1k. 45 rule on C1234 1 innie R5C4 = 1 outie R9C5, no 3,4 in R9C5
1l. 45 rule on N36 2 innies R12C7 = 12 = {39}, locked for C7 and N3 -> R12C6 = {17}, locked for C6 and N2, clean-up: no 6 in R1C3
1m. 13(3) cage at R1C5 = {238/256} (cannot be {346} which clashes with R6C5), no 4,9, 2 locked for C5 and N2, clean-up: no 2 in R5C4
1n. 14(3) cage at R3C7 = {167/257}, no 4
1o. 45 rule on N6 2 outies R23C9 must contain 8 = 1 innie R4C7 + 8, no 4 in R4C7 -> no 4 in R23C9
1p. 4 in N3 only in 11(3) cage at R1C8 = {146/245}, no 7
1q. 45 rule on N9 3 innies R7C78 + R8C7 = 13 = {157/247/256/346} (cannot be {139} because R8C7 only contains 4,5,6,7), no 9
1r. 21(3) cage at R8C5 = {579/678}/{89}4, no 4 in R8C5 + R9C6
1s. 33(6) cage at R3C6 = {126789/135789/145689/235689} (cannot be {234789/345678} because 3,4 only in R3C6, cannot be {245679} because 1,8 in N5 cannot both be in R4C4)
1t. 3,4 of {135789/145689/235689} must be in R3C6 -> no 5 in R3C6
1u. 3 on D/ only in R7C3 + R8C2, locked for N7
1v. 9 in N2 only in R123C4 + R3C6, CPE no 9 in R5C4, clean-up: no 9 in R9C5
1w. 45 rule on R1 1 innie R1C5 = 1 outie R2C8 +1, no 8 in R1C5, no 6 in R2C8
1x. 45 rule on R1 3 innies R1C589 = 12 = {156/246/345}
1y. 2 in R1 only in R1C12 = {28} or R1C589 = {246} -> R1C12 = {19/28/37} (cannot be [46], locking-out cages), no 4,6
2a. R5C4 = R9C5 (step 1k)
2b. 45 rule on N5 2 innies R46C4 = 1 outie R3C6 + 5
2c. Min R3C6 = 3 -> min R46C4 = 8
2d. 16(4) cage at R7C4 = R5789C4 = {1249/1258/1267/1348/1357/1456/2356} (cannot be {2347} because R4C4 = 1 (hidden single in C4), R6C4 = 5 total less than 8)
2e. Consider combinations for 13(3) cage at R1C5 (step 1m) = {238/256}
13(3) cage = {238}, locked for N2 => min R3C6 = 4, min R46C4 = 9 => R5789C4 cannot be {2356} because R4C4 = 1 (hidden single in C4), R6C4 = 7 total less than 9)
or 13(3) cage = {256}, locked for C5 => 16(4) cage cannot be {2356} because R9C5 only contains 1,7,8)
-> 16(4) cage at R7C4 = R5789C4 = {1249/1258/1267/1348/1357/1456}, 1 locked for C4 and N8
2f. Hidden killer pair 4,9 in R123C4 and R3C6 for N2, R3C6 cannot contain both of 4,9 -> R123C4 must contain at least one of 4,9
2g. 16(4) cage at R7C4 = R5789C4 = {1258/1267/1348/1357/1456} (cannot be {1249} which clashes with R123C4), no 9
2h. 1 in N5 only in 33(6) cage at R3C6 (step 1s) = {126789/135789/145689}
2i. R12C6 = {17}, 13(3) cage at R1C5 = {238/256} -> R123C4 + R3C6 = {3489/4569}
2j. 6 of {4569} must be in R123C4 (cannot be {459}6 which clashes with R46C4 = 11 = [65/92]), no 6 in R3C6
2k. R3C6 = {3489} -> R46C4 = 8,9,13,14 = [62]/{27}/[85/95] (cannot be {67} which clashes with the 33(6) cage)
2l. 16(4) cage at R7C4 = R5789C4 = {1267/1348/1357/1456} (cannot be {1258} which clashes with R46C4
3a. R7C78 + R8C7 (step 1q) = {157/247/256/346}
3b. R6C7 = 8 -> 29(6) cage at R6C7 = {124589/125678/234578} (cannot be {123689} because R7C78 with 1 only contain 1,5,7, cannot be {134678} because 8{346}{17} clashes with 16(4) cage at R7C4)
3c. Consider placement for 9 in N8
9 in 29(6) cage = {124589} => R7C78 + R8C7 = {15}7
or 9 in 21(3) cage at R8C5 = {579}/{89}4
-> no 6 in R8C7
3d. Consider combinations for 29(6) cage
29(6) cage = {124589/125678} => R7C78 + R8C7 = {157}
or 29(6) cage = {234578}, no 6 => R7C78 + R8C7 = {247}
-> R7C78 + R8C7 = {157/247}, no 3,6, 7 locked for N9
3e. 19(3) cage at R7C9 = {289/469/568}, no 3
3f. 8 of {289/568} only in R8C8 -> no 2,5 in R8C8
3g. 3 in N9 only in 13(3) cage at R9C7, locked for R9
3h. 13(3) cage = {139/238/346}, no 5
3i. 1,2 of {139/238} must be in R9C7 -> no 1,2, in R9C89
3j. R46C4 (step 2k) = [62]/{27}/[85/95]
3k. Consider combinations for 13(3) cage
13(3) cage = {139}, 9 locked for N9 => 19(3) cage = {568}, 8 placed for D\ => R46C4 = {27}/[62/95]
or 13(3) cage = {238/346} => 1 in N9 in R7C78 + R8C7 = {157} => 29(6) cage = {124589/125678}, 2 locked for C6 and N8 => 2 in C4 only in R46C4
-> R46C4 = {27}/[62/95], no 8
3l. R46C4 = {27}/[62/95]= 8,9,14 -> R3C3 = {349} (step 2b)
3m. 45 rule on N1 3 innies R1C3 + R3C23 = 14
[Ignore the difficult step 3n. Much easier is R46C4 = 1 outie R3C6 + 5 (step 2b), 45 rule on N47 3(1+2) outies R3C3 + R46C4 = 14 which I eventually used at step 4f; I’d seen that 45 much earlier but not added a note under my worksheet as it hadn’t been useful then, I ought to have remembered it at this stage -> max R46C4 = 13 -> no 9 in R3C6
Even simpler is wellbeback’s 45 rule on N12 2 remaining innies R3C26 = 9, no 9 in R3C26]
3n. Consider combinations for R123C4 + R3C6 (step 2i) = {3489/4569}
R123C4 + R3C6 = {3489} cannot be 4{38}9 because then R1C3 = 9, R3C2 = 5 (cage sum) so R1C3 + R3C23 greater than 14
or R123C4 + R3C6 = {3489} cannot be 8{34}9 because then R1C1267 = {1379}, locked for R1, R6C9 = 1 (hidden single in C9), R6C8 = 9 and either 1 in R1C78 => 29(6) cage = {124589/125678}, 2 locked for C6 => R46C4 = {27}/[62], no 9 in R4C4 => no 9 in R3C6 or 1 in 13(3) cage = [139], 9 placed for D\, no 9 in R4C4 => no 9 in R3C6
or R123C4 + R3C6 = {4569} cannot be {456}9 which clashes with R46C4 = 14 = [95] -> 9 in R123C4, locked for C4 and N2
3o. R3C6 = {34} -> 33(6) cage (step 2h) = {135789/145689}, no 2, 5 locked for N5
3p. Naked pair {34} in R36C6, locked for C6
3q. 2 in C6 only in R78C6, locked for N8 and 29(6) cage, no 2 in R7C78
3r. R7C78 + R8C7 = {157} (only remaining combination), 1,5 locked for N9, 1 locked for R7 -> 19(3) cage = {289/469}, 9 locked for N9, 29(6) cage = {124589/125678}, no 3
[Cracked, it gets easier from here.]
3s. 3 in N8 only in R78C4, locked for C4
3s. 16(4) cage at R7C4 = R5789C4 (step 2l) = {1348/1357}, no 6
3t. R9C6 = 9 (hidden single in R9) -> 21(3) cage at R8C5 = {579}, R8C57 = {57} locked for R8, clean-up: no 2,4 in R9C1
3u. 29(6) cage = {125678}, no 4
3v. 8 in C6 only in R45C6, locked for N5, clean-up: no 8 in R9C5 (step 1k)
3w. R789C4 + R9C5 = {348}1 (hidden quad in N8), 4,8 locked for C4, R5C4 = 1 (hidden single in N5), clean-up: no 5,9 in R1C3
3x. Naked triple {569} in R123C4, 5,6 locked for C4 , 6,9 locked for N2
3y. Naked triple {238} in 13(3) cage at R1C5, 3 locked for C5 and N2 -> R3C6 = 4, R6C56 = [43], 3 placed for D\, clean-up: no 7 in R1C2
3z. R46C4 = {27} (hidden pair in C4), 7 locked for N5
4a. R9C8 = 3 (hidden single in R9) -> R9C79 = 10 = {46}, locked for N9, 4 locked for R9 -> R9C4 = 8, clean-up: no 5 in R9C23
4b. Naked pair {29} in R78C9, locked for C9, 9 locked for N9 -> R8C8 = 8, placed for D\, R6C89 = [91], clean-up: no 2 in R1C2
4c. R89C1 = [45] (hidden pair in N7), 5 placed for D/, R78C4 = [43]
4d. 5 in R6 only in R6C23, locked for N4
4e. R23C4 = {569} -> 16(3) disjoint cage at R2C4 = {169/259}, 9 locked for C4, R3C2 = {12}, clean-up: no 4 in R1C3
4f. 45 rule on N47 3(1+2) outies R3C3 + R46C4 = 14, R46C4 = {27} -> R3C3 = 5, placed for D\
4g. 9 in R1 only in R1C12 = {19} or R1C67 = [19] (locking cages), 1 locked for R1
4h. 4 in R1 only in R1C89, locked for N3
4i. R1C9 = 4 (hidden single on D/) -> R12C8 = 7 = [52/61], R9C9 = 6, placed for D\ -> R5C5 = 9, placed for both diagonals, R2C2 = 4 (hidden single on D\), clean-up: no 1 in R1C2
4j. R7C3 = 3 (hidden single on D/) -> R4C6 = 8 (hidden single on D/)
4k. 18(4) cage at R4C1 = {1368} (cannot be {1289} because 1,9 only in R4C1, cannot be {2378} which clashes with R6C3) -> R4C1 = 1, R5C12 = {38}, locked for N4, 3 locked for R5, clean-up: no 9 in R1C2
4l. R1C67 = [19] (hidden pair in R1) -> R2C67 = [73]
4m. R2C2 = 4, R2C3 = 6 (hidden single in N1) = 10, 9 in N1 only in R23C1 = 11 = {29}, 2 locked for N1, 9 locked for C1, R1C1 = 7, placed for D\, R1C3 = 8 -> R1C4 = 5, R1C89 = [64] -> R2C8 = 1 (cage sum), placed for D/
4n. R4C4 = 2 -> R6C4 = 7, placed for D/, R3C78 = [27] -> R4C7 = 5 (cage sum)
and the rest is naked singles, without using the diagonals.