The full reasoning can be found below the Sudoku.
Jun 08 - Super Hard
Puzzle Copyright © Kevin Stone
Reasoning
R3C6 can only be <3>
R4C3 can only be <6>
R5C5 can only be <5>
R7C6 can only be <5>
R1C5 can only be <6>
R4C7 can only be <9>
R6C3 can only be <3>
R6C7 can only be <6>
R5C2 can only be <7>
R1C9 can only be <3>
R5C9 can only be <1>
R5C1 can only be <2>
R5C8 can only be <3>
R1C7 is the only square in row 1 that can be <4>
R3C2 is the only square in row 3 that can be <4>
R7C8 is the only square in row 7 that can be <4>
R8C3 is the only square in row 8 that can be <5>
R1C3 can only be <7>
R1C1 can only be <5>
R3C7 is the only square in row 3 that can be <5>
Intersection of column 8 with block 3. The value <8> only appears in one or more of squares R1C8, R2C8 and R3C8 of column 8. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C9 - removing <8> from <268> leaving <26>
Squares R2C5 and R2C7 in row 2 and R8C5 and R8C7 in row 8 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 5 and 7 can be removed.
R7C7 - removing <2> from <237> leaving <37>
R9C5 - removing <2> from <239> leaving <39>
R9C7 - removing <2> from <237> leaving <37>
Squares R7C7 and R9C7 in column 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <37>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R8C7 - removing <3> from <123> leaving <12>
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.
R3C8 - removing <6> from <168> leaving <18>
R7C2 - removing <6> from <369> leaving <39>
Squares R3C9 (XY), R9C9 (XZ) and R3C1 (YZ) form an XY-Wing pattern on <8>. All squares that are buddies of both the XZ and YZ squares cannot be <8>.
R9C1 - removing <8> from <78> leaving <7>
R9C7 can only be <3>
R9C5 can only be <9>
R7C7 can only be <7>
R2C5 can only be <2>
R7C4 can only be <2>
R2C7 can only be <1>
R8C5 can only be <3>
R3C4 can only be <9>
R8C7 can only be <2>
R3C8 can only be <8>
R3C1 can only be <6>
R3C3 can only be <1>
R2C8 can only be <6>
R8C2 can only be <6>
R9C9 can only be <8>
R9C3 can only be <2>
R7C9 can only be <6>
R2C2 can only be <9>
R8C8 can only be <1>
R3C9 can only be <2>
R7C1 can only be <8>
R7C3 can only be <9>
R7C2 can only be <3>
R2C3 can only be <8>
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