The full reasoning can be found below the Sudoku.
May 05 - Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C5 is the only square in row 5 that can be <9>
R8C5 can only be <4>
R8C6 can only be <8>
R2C5 can only be <7>
R8C8 can only be <2>
R8C9 can only be <5>
R8C1 can only be <9>
R2C2 can only be <6>
R1C9 is the only square in row 1 that can be <9>
R3C3 is the only square in row 3 that can be <9>
R5C1 is the only square in row 5 that can be <6>
R7C4 is the only square in row 7 that can be <9>
R9C2 is the only square in column 2 that can be <4>
R9C1 is the only square in row 9 that can be <2>
R6C2 is the only square in column 2 that can be <2>
R3C2 is the only square in column 2 that can be <5>
R9C4 is the only square in column 4 that can be <5>
Squares R7C6 and R9C6 in column 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C6 - removing <1> from <126> leaving <26>
R3C6 - removing <1> from <1246> leaving <246>
R4C6 - removing <7> from <367> leaving <36>
R6C6 - removing <7> from <347> leaving <34>
Intersection of row 5 with block 6. The value <4> only appears in one or more of squares R5C7, R5C8 and R5C9 of row 5. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R6C7 - removing <4> from <345> leaving <35>
Intersection of row 6 with block 4. The value <8> only appears in one or more of squares R6C1, R6C2 and R6C3 of row 6. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R5C2 - removing <8> from <78> leaving <7>
R5C3 - removing <8> from <1378> leaving <137>
R1C2 can only be <8>
R3C1 can only be <1>
R2C1 can only be <3>
R1C1 can only be <7>
R3C4 is the only square in row 3 that can be <8>
R6C4 is the only square in row 6 that can be <7>
R4C4 can only be <6>
R4C6 can only be <3>
R1C4 can only be <1>
R4C8 can only be <1>
R6C6 can only be <4>
R4C3 can only be <5>
R2C4 can only be <4>
R2C9 can only be <1>
R4C7 can only be <2>
R6C1 can only be <8>
R4C9 can only be <7>
R6C3 can only be <3>
R7C1 can only be <5>
R6C7 can only be <5>
R5C3 can only be <1>
R1C6 is the only square in row 1 that can be <2>
R3C6 can only be <6>
R3C7 can only be <4>
R3C9 can only be <2>
R5C7 can only be <3>
R1C7 can only be <6>
R1C8 can only be <3>
R7C7 can only be <1>
R9C8 can only be <8>
R7C6 can only be <7>
R9C3 can only be <7>
R9C9 can only be <3>
R5C8 can only be <4>
R7C9 can only be <4>
R5C9 can only be <8>
R7C8 can only be <6>
R7C3 can only be <8>
R9C6 can only be <1>
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