The full reasoning can be found below the Sudoku.
May 01 - Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C7 is the only square in row 5 that can be <8>
R1C4 is the only square in row 1 that can be <8>
R3C4 can only be <2>
R7C8 is the only square in row 7 that can be <4>
R5C4 is the only square in column 4 that can be <9>
R7C9 is the only square in column 9 that can be <3>
R7C4 can only be <6>
R9C4 can only be <3>
Squares R3C5 and R3C6 in row 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <14>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R3C2 - removing <1> from <1679> leaving <679>
R3C9 - removing <1> from <178> leaving <78>
Squares R3C5 and R3C6 in block 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <14>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C6 - removing <1> from <135> leaving <35>
R2C5 - removing <1> from <135> leaving <35>
Intersection of row 7 with block 8. The value <2> only appears in one or more of squares R7C4, R7C5 and R7C6 of row 7. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R8C5 - removing <2> from <1257> leaving <157>
R9C6 - removing <2> from <12> leaving <1>
R3C6 can only be <4>
R3C5 can only be <1>
R7C2 is the only square in row 7 that can be <1>
R5C2 is the only square in column 2 that can be <5>
R4C7 is the only square in row 4 that can be <5>
R2C5 is the only square in row 2 that can be <5>
R8C5 can only be <7>
R1C6 can only be <3>
R7C5 can only be <2>
R5C6 can only be <2>
R5C8 can only be <7>
R7C6 can only be <5>
R3C8 can only be <6>
R4C9 can only be <9>
R7C1 can only be <7>
R1C8 can only be <5>
R9C8 can only be <2>
R6C9 can only be <2>
R6C7 can only be <4>
R8C9 can only be <1>
R4C1 can only be <6>
R2C9 can only be <8>
R8C7 can only be <6>
R2C1 can only be <3>
R3C9 can only be <7>
R3C2 can only be <9>
R1C7 can only be <1>
R4C5 can only be <4>
R8C1 can only be <5>
R4C3 can only be <7>
R5C5 can only be <3>
R5C3 can only be <4>
R6C5 can only be <6>
R8C3 can only be <2>
R9C7 can only be <7>
R2C7 can only be <2>
R2C3 can only be <1>
R6C1 can only be <9>
R3C1 can only be <8>
R9C2 can only be <6>
R1C3 can only be <6>
R6C3 can only be <3>
R9C3 can only be <9>
R1C2 can only be <7>
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