The full reasoning can be found below the Sudoku.
Jun 15 - Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C6 can only be <6>
R5C4 can only be <1>
R4C6 can only be <2>
R6C4 can only be <4>
R5C5 can only be <9>
R5C1 can only be <3>
R5C9 can only be <5>
R3C4 is the only square in row 3 that can be <3>
R6C7 is the only square in row 6 that can be <3>
R6C3 is the only square in row 6 that can be <5>
R9C5 is the only square in row 9 that can be <2>
R7C6 is the only square in column 6 that can be <3>
R7C5 is the only square in row 7 that can be <5>
R3C9 is the only square in column 9 that can be <7>
R8C5 is the only square in block 8 that can be <4>
Squares R6C8 and R9C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <69>. 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 <1589> leaving <158>
R3C8 - removing <9> from <159> leaving <15>
R8C8 - removing <6> from <1568> leaving <158>
Intersection of row 3 with block 3. The value <5> only appears in one or more of squares R3C7, R3C8 and R3C9 of row 3. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C7 - removing <5> from <1458> leaving <148>
R2C7 - removing <5> from <14589> leaving <1489>
R2C8 - removing <5> from <158> leaving <18>
Intersection of row 7 with block 7. The values <16> only appears in one or more of squares R7C1, R7C2 and R7C3 of row 7. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain these values.
R8C2 - removing <16> from <136> leaving <3>
R8C3 - removing <16> from <1367> leaving <37>
R9C3 - removing <6> from <34679> leaving <3479>
R8C3 can only be <7>
R8C9 can only be <8>
R8C4 can only be <6>
R9C4 can only be <7>
R4C9 can only be <9>
R7C9 can only be <4>
R6C8 can only be <6>
R6C1 can only be <9>
R9C8 can only be <9>
R4C7 can only be <8>
R9C9 can only be <3>
R9C3 can only be <4>
R9C7 can only be <6>
R1C1 is the only square in row 1 that can be <8>
R1C5 is the only square in row 1 that can be <7>
R2C5 can only be <1>
R2C1 can only be <7>
R2C8 can only be <8>
R3C5 can only be <6>
R3C1 can only be <1>
R3C8 can only be <5>
R7C1 can only be <6>
R1C3 can only be <6>
R3C7 can only be <9>
R8C8 can only be <1>
R8C7 can only be <5>
R1C2 can only be <5>
R4C3 can only be <1>
R2C7 can only be <4>
R4C2 can only be <6>
R7C3 can only be <9>
R7C2 can only be <1>
R2C3 can only be <3>
R1C6 can only be <4>
R2C2 can only be <9>
R1C7 can only be <1>
R2C6 can only be <5>
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