The full reasoning can be found below the Sudoku.
May 07 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R2C4 can only be <5>
R6C8 can only be <2>
R2C6 can only be <4>
R8C4 can only be <8>
R4C4 can only be <2>
R6C4 can only be <1>
R5C4 can only be <7>
R5C7 is the only square in row 5 that can be <1>
R6C6 is the only square in column 6 that can be <3>
R5C6 is the only square in column 6 that can be <6>
R6C5 can only be <8>
R6C2 can only be <6>
R4C2 is the only square in row 4 that can be <8>
R2C2 can only be <9>
R2C8 can only be <3>
R7C2 can only be <7>
R3C3 is the only square in row 3 that can be <3>
R3C1 is the only square in row 3 that can be <7>
R4C5 is the only square in row 4 that can be <4>
R2C3 is the only square in block 1 that can be <8>
R2C7 can only be <2>
R7C9 is the only square in row 7 that can be <2>
Squares R1C1 and R5C1 in column 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <24>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R9C1 - removing <4> from <489> leaving <89>
Intersection of column 9 with block 3. The value <6> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C8 - removing <6> from <456> leaving <45>
Squares R3C2 and R8C2 in column 2 and R3C8 and R8C8 in column 8 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in rows 3 and 8 can be removed.
R8C3 - removing <4> from <1456> leaving <156>
R3C7 - removing <4> from <458> leaving <58>
R8C7 - removing <4> from <459> leaving <59>
Squares R8C6 and R8C7 in row 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <59>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C3 - removing <5> from <156> leaving <16>
R8C8 - removing <59> from <4569> leaving <46>
Squares R4C8 and R8C7 form a remote naked pair. <59> can be removed from any square that is common to their groups.
R7C8 - removing <59> from <569> leaving <6>
R7C3 can only be <5>
R8C8 can only be <4>
R8C2 can only be <1>
R3C8 can only be <5>
R3C7 can only be <8>
R4C8 can only be <9>
R4C6 can only be <5>
R5C9 can only be <5>
R5C5 can only be <9>
R9C3 can only be <4>
R8C3 can only be <6>
R3C2 can only be <4>
R5C3 can only be <2>
R1C1 can only be <2>
R3C9 can only be <6>
R3C5 can only be <1>
R1C9 can only be <9>
R8C6 can only be <9>
R5C1 can only be <4>
R1C3 can only be <1>
R7C5 can only be <3>
R7C7 can only be <9>
R9C5 can only be <5>
R7C1 can only be <8>
R1C7 can only be <4>
R8C7 can only be <5>
R9C9 can only be <8>
R9C7 can only be <3>
R9C1 can only be <9>
R1C5 can only be <6>
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