The full reasoning can be found below the Sudoku.
Sep 07 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C5 can only be <4>
R8C5 can only be <7>
R4C5 can only be <8>
R4C1 can only be <4>
R8C3 can only be <4>
R5C5 can only be <3>
R6C5 can only be <1>
R9C6 can only be <9>
R9C4 can only be <1>
R6C4 can only be <4>
R1C7 is the only square in row 1 that can be <1>
R7C2 is the only square in row 7 that can be <1>
R9C7 is the only square in row 9 that can be <4>
R3C8 is the only square in row 3 that can be <4>
Squares R1C4 and R1C6 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <26>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C1 - removing <26> from <2368> leaving <38>
R1C9 - removing <6> from <3689> leaving <389>
Squares R5C8 and R7C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <89>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C8 - removing <9> from <569> leaving <56>
Intersection of block 6 with column 9. The value <5> only appears in one or more of squares R4C9, R5C9 and R6C9 of block 6. These squares are the ones that intersect with column 9. Thus, the other (non-intersecting) squares of column 9 cannot contain this value.
R3C9 - removing <5> from <3568> leaving <368>
Squares R2C2 and R2C8 in row 2 and R8C2 and R8C8 in row 8 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 2 and 8 can be removed.
R3C2 - removing <6> from <268> leaving <28>
Squares R1C1, R1C3 and R1C9 in row 1, R6C1 and R6C9 in row 6 and R9C1, R9C3 and R9C9 in row 9 form a Swordfish pattern on possibility <8>. All other instances of this possibility in columns 1, 3 and 9 can be removed.
R3C1 - removing <8> from <2368> leaving <236>
R3C3 - removing <8> from <358> leaving <35>
R3C9 - removing <8> from <368> leaving <36>
R7C1 - removing <8> from <2378> leaving <237>
R7C3 - removing <8> from <378> leaving <37>
R7C9 - removing <8> from <789> leaving <79>
Intersection of row 7 with block 9. The values <89> 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 these values.
R9C9 - removing <8> from <678> leaving <67>
Squares R3C3 (XY), R1C1 (XZ) and R3C7 (YZ) form an XY-Wing pattern on <8>. All squares that are buddies of both the XZ and YZ squares cannot be <8>.
R3C2 - removing <8> from <28> leaving <2>
R1C9 - removing <8> from <389> leaving <39>
R8C2 can only be <6>
R8C8 can only be <5>
R2C2 can only be <7>
R8C7 can only be <2>
R2C8 can only be <6>
R5C2 can only be <8>
R3C9 can only be <3>
R3C1 can only be <6>
R3C3 can only be <5>
R1C9 can only be <9>
R5C8 can only be <9>
R6C1 can only be <7>
R5C4 can only be <2>
R7C8 can only be <8>
R4C9 can only be <5>
R6C6 can only be <5>
R9C1 can only be <8>
R6C9 can only be <8>
R4C6 can only be <6>
R7C7 can only be <9>
R9C3 can only be <7>
R1C1 can only be <3>
R9C9 can only be <6>
R7C3 can only be <3>
R7C1 can only be <2>
R1C3 can only be <8>
R7C9 can only be <7>
R2C7 can only be <5>
R2C3 can only be <9>
R3C7 can only be <8>
R4C4 can only be <9>
R1C6 can only be <2>
R5C6 can only be <7>
R1C4 can only be <6>
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