The full reasoning can be found below the Sudoku.
May 09 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R2C3 can only be <6>
R5C5 can only be <6>
R5C9 can only be <4>
R7C3 can only be <9>
R5C1 can only be <9>
R1C5 can only be <7>
R9C5 can only be <5>
R3C3 can only be <3>
R6C3 can only be <8>
R8C3 can only be <5>
R1C1 can only be <1>
R4C3 can only be <4>
R1C9 can only be <8>
R2C2 can only be <7>
R3C2 can only be <9>
R3C6 can only be <1>
R3C8 can only be <2>
R2C6 can only be <8>
R2C4 can only be <2>
R3C4 can only be <5>
R3C7 can only be <7>
R2C8 can only be <1>
R2C7 can only be <4>
R4C9 is the only square in row 4 that can be <1>
R4C2 is the only square in row 4 that can be <5>
R6C1 is the only square in row 6 that can be <7>
R7C6 is the only square in row 7 that can be <7>
R8C6 is the only square in row 8 that can be <4>
Intersection of row 9 with block 8. The value <9> only appears in one or more of squares R9C4, R9C5 and R9C6 of row 9. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R8C4 - removing <9> from <169> leaving <16>
Squares R8C2 and R8C4 in row 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <16>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C7 - removing <6> from <689> leaving <89>
R8C8 - removing <6> from <689> leaving <89>
Intersection of block 9 with row 7. The value <6> only appears in one or more of squares R7C7, R7C8 and R7C9 of block 9. These squares are the ones that intersect with row 7. Thus, the other (non-intersecting) squares of row 7 cannot contain this value.
R7C2 - removing <6> from <126> leaving <12>
R7C4 - removing <6> from <136> leaving <13>
Squares R1C4, R1C6, R9C4 and R9C6 form a Type-1 Unique Rectangle on <69>.
R9C4 - removing <69> from <369> leaving <3>
R9C9 can only be <2>
R7C4 can only be <1>
R9C1 can only be <6>
R6C9 can only be <3>
R7C7 can only be <6>
R7C2 can only be <2>
R8C4 can only be <6>
R7C8 can only be <3>
R8C2 can only be <1>
R1C4 can only be <9>
R9C6 can only be <9>
R4C1 can only be <2>
R1C6 can only be <6>
R4C7 can only be <8>
R6C2 can only be <6>
R4C8 can only be <6>
R8C7 can only be <9>
R6C8 can only be <9>
R6C7 can only be <2>
R8C8 can only be <8>
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