The full reasoning can be found below the Sudoku.
Sep 04 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C9 can only be <7>
R2C6 can only be <2>
R5C5 can only be <4>
R8C4 can only be <6>
R8C6 can only be <1>
R1C5 can only be <5>
R2C5 can only be <6>
R2C4 can only be <4>
R3C5 can only be <7>
R1C1 is the only square in row 1 that can be <2>
R7C2 is the only square in row 7 that can be <4>
R8C7 is the only square in row 8 that can be <4>
Intersection of column 1 with block 7. The value <8> only appears in one or more of squares R7C1, R8C1 and R9C1 of column 1. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C3 - removing <8> from <368> leaving <36>
Intersection of column 3 with block 1. The value <8> only appears in one or more of squares R1C3, R2C3 and R3C3 of column 3. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R2C2 - removing <8> from <1389> leaving <139>
Intersection of column 7 with block 3. The value <8> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R2C8 - removing <8> from <1389> leaving <139>
Squares R1C3, R1C7, R2C3 and R2C7 form a Type-4 Unique Rectangle on <38>.
R2C3 - removing <3> from <389> leaving <89>
R2C7 - removing <3> from <138> leaving <18>
Squares R9C3 (XY), R9C5 (XZ) and R7C1 (YZ) form an XY-Wing pattern on <8>. All squares that are buddies of both the XZ and YZ squares cannot be <8>.
R7C5 - removing <8> from <238> leaving <23>
R9C1 - removing <8> from <568> leaving <56>
R7C1 is the only square in row 7 that can be <8>
R9C5 is the only square in row 9 that can be <8>
Intersection of row 7 with block 9. The value <6> only appears in one or more of squares R7C7, R7C8 and R7C9 of row 7. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R9C9 - removing <6> from <16> leaving <1>
Squares R1C3 and R1C7 in row 1 and R9C3 and R9C7 in row 9 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in columns 3 and 7 can be removed.
R8C3 - removing <3> from <37> leaving <7>
R5C2 is the only square in row 5 that can be <7>
Squares R5C1 (XY), R5C7 (XZ) and R9C1 (YZ) form an XY-Wing pattern on <5>. All squares that are buddies of both the XZ and YZ squares cannot be <5>.
R9C7 - removing <5> from <35> leaving <3>
R9C3 can only be <6>
R1C7 can only be <8>
R1C3 can only be <3>
R2C7 can only be <1>
R5C7 can only be <5>
R9C1 can only be <5>
R5C3 can only be <9>
R2C2 can only be <9>
R2C3 can only be <8>
R2C8 can only be <3>
R4C2 can only be <8>
R4C8 can only be <9>
R6C2 can only be <1>
R3C8 can only be <6>
R5C9 can only be <2>
R5C8 can only be <1>
R7C9 can only be <6>
R6C8 can only be <8>
R3C2 can only be <5>
R5C1 can only be <6>
R7C8 can only be <2>
R3C9 can only be <9>
R3C1 can only be <1>
R8C2 can only be <3>
R7C5 can only be <3>
R8C8 can only be <5>
R8C5 can only be <2>
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