The full reasoning can be found below the Sudoku.
May 18 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R6C7 can only be <4>
R7C4 can only be <9>
R6C3 can only be <9>
R7C6 can only be <2>
R3C4 can only be <6>
R5C4 can only be <5>
R6C5 can only be <1>
R8C1 is the only square in row 8 that can be <3>
R9C8 is the only square in row 9 that can be <6>
R1C8 is the only square in column 8 that can be <1>
Squares R2C5 and R2C8 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <29>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C1 - removing <9> from <4579> leaving <457>
R2C2 - removing <29> from <2479> leaving <47>
R2C9 - removing <2> from <257> leaving <57>
Intersection of column 3 with block 7. The values <14> only appears in one or more of squares R7C3, R8C3 and R9C3 of column 3. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain these values.
R7C1 - removing <4> from <47> leaving <7>
R8C2 - removing <4> from <4789> leaving <789>
R9C1 - removing <4> from <4789> leaving <789>
R9C2 - removing <4> from <24789> leaving <2789>
R7C7 can only be <5>
R2C2 is the only square in column 2 that can be <7>
R2C9 can only be <5>
R2C1 can only be <4>
R1C1 is the only square in row 1 that can be <5>
R1C3 is the only square in row 1 that can be <6>
R4C3 can only be <8>
R4C7 can only be <3>
R3C3 can only be <2>
R5C1 can only be <6>
R5C2 can only be <4>
R4C5 can only be <6>
R9C2 is the only square in row 9 that can be <2>
Squares R1C2 and R1C7 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <89>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C5 - removing <9> from <239> leaving <23>
R1C9 - removing <8> from <238> leaving <23>
Squares R8C2 and R8C8 in row 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <89>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C9 - removing <8> from <478> leaving <47>
Squares R1C7 and R9C1 form a remote naked pair. <89> can be removed from any square that is common to their groups.
R9C7 - removing <89> from <789> leaving <7>
R9C5 can only be <4>
R8C9 can only be <4>
R8C5 can only be <7>
R7C9 can only be <1>
R9C3 can only be <1>
R7C3 can only be <4>
R9C9 can only be <8>
R9C1 can only be <9>
R5C9 can only be <2>
R8C8 can only be <9>
R5C8 can only be <8>
R1C9 can only be <3>
R8C2 can only be <8>
R2C8 can only be <2>
R3C1 can only be <8>
R1C5 can only be <2>
R3C9 can only be <7>
R2C5 can only be <9>
R3C7 can only be <9>
R1C2 can only be <9>
R3C6 can only be <3>
R1C7 can only be <8>
R5C5 can only be <3>
R5C6 can only be <9>
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