Prelims
a) R2C23 = {16/25/34}, no 7,8,9
b) R45C7 = {29/38/47/56}, no 1
c) R5C23 = {39/48/57}, no 1,2,6
d) R89C3 = {19/28/37/46}, no 5
e) R89C7 = {39/48/57}, no 1,2,6
f) R9C45 = {18/27/36/45}, no 9
g) 11(3) cage at R1C1 = {128/137/146/236/245}, no 9
h) 21(3) cage at R2C9 = {489/579/678}, no 1,2,3
i) 39(6) cage at R7C2 = {456789}, no 1,2,3
1a. 45 rule on N3 2 innies R13C7 = 3 = {12}, locked for C7, N3 and 21(5) cage at R1C6, clean-up: no 9 in R45C7
1b. 45 rule in N6789 1 innie R7C5 = 1, R56C5 = 14 = {59/68}, clean-up: no 8 in R9C45
1c. 1 in C6 only in R456C6, locked for N5
1d. 13(3) cage at R4C6 = {139/148/157}, no 2,6
1e. Combined cage R456C6 + R56C5 = {139/68}/{148/59}/{157}{68}, 8 locked for N5
1f. 2 in C6 only in R789C6, locked for N8 and 27(6) cage at R7C6, clean-up: no 7 in R9C45
1g. 45 rule on N69 3 outies R789C6 = 14 contains 2 = {239/248/257}, no 6
1h. 2 in N9 only in R78C9 + R9C89, locked for 38(7) cage at R6C7
1i. 6 in C6 only in R123C6, locked for N2
1j. 45 rule on N23 2 innies R1C4 + R3C5 = 10 = [19]/{28/37}, no 4,5, no 9 in R1C4
1k. 11(3) cage at R1C1 = {128/137/146/245} (cannot be {236} which clashes with R2C23)
1l. 45 rule on N1 4 innies R13C23 = 27 = {3789/4689/5679}, no 1,2
1m. 17(4) cage at R1C5 = {1259/1349/1457/2348/2357} (cannot be {1358} which clashes with R1C4 + R3C5
1n. 14(3) cage at R3C5 = {239/257/347} (cannot be {356} which clashes with R56C5, cannot be {248} = 8{24} which clashes with combined cage R456C6 + R56C5), no 6,8, clean-up: no 2 in R1C4
1o. 4 of {347} must be in R4C4 (cannot be [374/734] which clashes with R1C4 + R3C5 = {37}, CCC), no 4 in R4C5
2a. 45 rule on N69 3 innies R7C78 + R8C8 = 13 = {139/148/157/346}
2b. Hidden killer pair 1,6 in R7C78 + R8C8 and R78C9 + R9C89 for N9, R7C78 + R8C8 contains one of 1,6 -> R78C9 + R9C89 must contain one of 1,6
2c. Since R78C9 + R9C89 contains one of 1,6, 38(7) cage at R6C7 must contain both of them, similarly 2 in R78C9 + R9C89 so 38(7) cage must contain both of 2,5 -> 38(7) cage = {1256789}, no 3,4
2d. 45 rule on N6 3 innies R6C789 = 18 = {189/567}
2e. 2 in N6 only in 16(4) cage at R4C8 = {1249/2347/2356} (cannot be {1258/1267} which clash with R6C789), no 8
2f. Consider combinations for R6C789
R6C789 = {189}, locked for 38(7) cage => R78C9 + R9C89 = {2567}, locked for N9 => R7C78 + R8C8 = {139/148}
or R6C789 = {567}, locked for 38(7) cage => R78C9 + R9C89 = {1289}, locked for N9 => R7C78 + R8C8 = {346}
-> R7C78 + R8C8 = {139/148/346}, no 5,7
2g. 1 of {139/148} must be in R8C8 -> no 8,9 in R8C8
[Eliminations can be made from R6C78, R7C7 and R8C8 using the forcing chain; they don’t seem to help at this stage.]
2h. 39(6) cage at R7C2 = {456789}
2i. Consider combinations for R9C45 = {36/45}
R9C45 = {36}, 6 locked for N8 => 6 in 39(6) cage in R7C23 + R8C2, locked for N7
or R9C45 = {45}, 4 locked for N8 => 4 in 39(6) cage in R7C23 + R8C2, locked for N7
-> R89C3 = {19/28/37}, no 4,6
2j. Hidden killer triple 1,2,3 in 17(4) cage at R7C1 and R89C3 for N7, R89C3 contains one of 1,2,3 -> 17(4) cage must contain two of 1,2,3 = {1259/1268/1349/1367/2348/2357} (cannot be {1457/2456} which only contain one of 1,2,3, cannot be {1358} which clashes with R89C3)
3a. 13(3) cage at R4C6 (step 1d) = {139/148/157}, R56C5 (step 1b) = {59/68}, 14(3) cage at R3C5 (step 1n) = {239/257/347}
3b. Consider combinations for 14(3) cage
14(3) cage = {239}, CPE no 9 in R56C5
or 14(3) cage = {257}, CPE no 5 in R56C5
or 14(3) cage = {347}, 4 locked for N5 => 13(3) cage = {139/157} => R56C5 = {68} (cannot be {59} which clashes with 13(3) cage
-> R56C5 = {68}, locked for C5 and N5, clean-up: no 4 in 13(3) cage, no 3 in R9C4
3c. 4 in N5 only in R456C4, locked for C4, clean-up: no 5 in R9C5
3d. Killer pair 6,8 in R6C5 and R6C789, locked for R6
3e. 6 in N4 only in R4C123 + R5C1, locked for 38(7) cage at R3C2, no 6 in R3C23
3f. 38(7) cage = {1256789/1346789}, 1 locked for N4
4a. 45 rule on R123 3 innies R3C235 = 19 = {289/379/478}, no 5
4b. Hidden killer pair 7,8,9 in R3C235 + R3C89 for R3, R3C235 contain two of 7,8,9 -> R3C89 must contain one of 7,8,9, 21(3) cage at R2C9 contains two of 7,8,9 -> R2C9 = {789}
4c. Killer triple 7,8,9 in R3C235 and R3C89, locked for R3
5a. 38(7) cage at R3C2 (step 3f) = {1256789/1346789}
5b. 45 rule on N1 2 innies R3C23 = 1 outie R1C4 + 9
5c. Consider combinations for R5C23 = {39/48/57}
R5C23 = {39/57} => 8 in N4 only in R4C123 + R5C1, locked for 38(7) cage, no 8 in R3C23 => max R3C23 = 16 => max R1C4 = 7
or R5C23 = {48}, locked for N4 => 8 in 38(7) cage only in R3C23 => caged X-Wing for 8 in R3C23 and R5C23, no other 8 in C23 => 8 in 39(6) cage at R7C2 only in R78C4, locked for C4
-> no 8 in R1C4, clean-up: no 2 in R3C5 (step 1j)
[A long way from solving this Assassin but things get a bit easier after this step.]
5d. 45 rule on N3 3 outies R123C6 = 18 contains 6 for C6 = {468/567} (cannot be {369} which clashes with R1C4 + R3C5), no 3,9
6a. 14(3) cage at R3C5 (step 1n) = {239/257/347} with {347} = [347/743], R9C45 (step 2i) = [54/63]
6b. 45 rule on N7 3 outies R7C4 + R8C45 = 21 = {489/579/678}
6c. Consider combinations for R7C4 + R8C45
R7C4 + R8C45 = {489}, locked for N8 => R9C45 = [63]
or R7C4 + R8C45 = {579}, locked for N8 => R9C45 = [63]
or R7C4 + R8C45 = {678} = {68}7
-> 14(3) cage at R3C5 = {239/257} (cannot be {347}), no 4, 2 locked for R4 and N5
6d. {257} = 7{25}, no 7 in R4C45
6e. R123C6 (step 5d) = {468/567}, 13(3) cage at R4C6 (step 1d) = {139/157}
6f. Consider combinations for 14(3) cage
14(3) cage = {239} = 3{29}/9{23} => 13(3) cage = {157} (cannot be {139} which clashes with R4C45), 5,7 locked for C6
or 14(3) cage = {257} = 7{25}
-> R123C6 = {468}, 4,8 locked for C6 and N2
also 5 in R4C45 or R456C6, locked for N5
6g. 8 in N8 only in R78C4, locked for 39(6) cage at R7C2, no 8 in R7C23 + R8C3
6h. R7C4 + R8C45 = {489/678}, no 5
6i. 4 of {489} must be in R8C5 -> no 9 in R8C5
6j. 7 of {678} must be in R8C5 -> no 7 in R78C4
6k. 39(6) cage = {456789} -> R7C23 + R8C3 = {459/567}, 5 locked for N7
6l. 4 in N5 only in R56C4, locked for 18(4) cage at R5C4
6m. 18(4) cage = {2349/2457}, 2 locked for N4
6n. 38(7) cage at R3C2 (step 3f) = {1346789}, no 5
6o. 45 rule on N56789 3 outies R3C5 + R6C23 = 14 contains 2 in R6C23 = 3{29}/7{25}/9{23}, no 7 in R6C23
7a. 5 in C1 only in 11(3) cage at R1C1 = {245}, locked for N1, 2,4 locked for C1
[Almost there]
7b. R13C23 (step 1l) = {3789}, 3 locked for N1
7c. Naked pair {16} in R2C23, locked for R2
[At long last, I can use a diagonal]
7d. R3C7 = 2 (hidden single on D/) -> R1C7 = 1, clean-up: no 9 in R3C5 (step 1j)
7e. Naked pair {37} in R1C4 + R3C5, locked for N2
7f. R3C235 (step 4a) = {379} (only remaining combination), 7,9 locked for R3, 9 locked for N1 and 38(7) cage at R3C2
7g. Naked triple {378} in 18(3) cage at R1C2, locked for R1
7h. 38(7) cage at R3C2 = {1346789}, 4,8 locked for N4, 4 locked for R4, clean-up: no 7 in R5C7
8a. R13C23 (step 7b) = {3789}
8b. Consider combinations for R5C23 = {39/57}
R5C23 = {39} => caged X-Wing with R13C23, no other 3,9 in C23 => R89C3 = {28}
or R5C23 = {57} => caged X-Wing with R13C23, no other 7 in C23 => R89C3 = {19/28}
-> R89C3 = {19/28}, no 3,7
[Cracked at last. Straightforward from here.]
8c. 3 in N7 only in 17(4) cage at R7C1 = {1349/1367} (cannot be {2348} because 2,4 only in R9C2), 1 locked for N7, clean-up: no 9 in R89C3
8d. Naked pair {28} in R89C3, locked for C3 and N7
8e. R1C2 = 8 (hidden single in R1)
8f. R2C6 = 8 (hidden single in C6)
8g. R6C24 = [24] (hidden pair in R6), 4 placed for D/
8h. R4C3 = 4 (hidden single in C3), clean-up: no 7 in R5C7
8i. R2C3 = 1 (hidden single in C3) -> R2C2 = 6, placed for D\
8j. R7C3 = 6 (hidden single in C3), placed for D/, R5C5 = 8, placed for D\, clean-up: no 3 in R4C7
8k. Naked pair {89} in R78C4, 9 locked for C4, N8 and 39(6) cage at R7C2 -> R8C8 = 4 (step 6b), R9C5 = 3 -> R9C4 = 6, clean-up: no 9 in R8C7, no 8 in R9C7
8l. Naked pair {57} in R78C2, locked for C2, 7 locked for N7, clean-up: no 5,7 in R5C3
8m. Naked pair {39} in R5C23, locked for R5 and N4, clean-up: no 8 in R4C7
9a. R6C5 = 6 -> R6C789 (step 2d) = {189}, locked for N6 and 38(7) cage at R6C7, 1,9 locked for R6)
9b. Naked quad {2567} in R78C9 + R9C89, 5,7 locked for N9
9c. 6 in C7 only in R45C7 = {56}, locked for N6
9d. R2C7 = 7 (hidden single in C7), R2C9 = 9, R3C89 = 12 = {48}, 4 locked for R3 and N3
9e. R1C9 = 5, R2C8 = 3, both placed for D/, R4C89 = [73]
10a. Naked pair {25} in R2C45, locked for R2 and N2 -> R1C5 = 9
10b. Naked pair {25} in R4C45, 5 locked for R4 and N5
10c. R5C4 = 7, R6C6 = 3, placed for D\, R3C5 = 7, R3C3 = 9, placed for D\, R7C7 = 4, placed for D\, R1C1 = 2, placed for D\
and the rest is naked singles, without using the diagonals.