The full reasoning can be found below the Sudoku.
May 11 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R6C3 can only be <4>
R1C3 is the only square in row 1 that can be <9>
R2C2 is the only square in row 2 that can be <4>
R3C8 is the only square in row 3 that can be <8>
R7C8 is the only square in row 7 that can be <4>
R9C3 is the only square in row 9 that can be <8>
R4C3 can only be <2>
R4C5 can only be <1>
R2C3 can only be <6>
R4C7 can only be <8>
R5C2 is the only square in row 5 that can be <8>
Squares R5C8 and R6C7 in block 6 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 block.
R5C7 - removing <59> from <1569> leaving <16>
Intersection of row 3 with block 2. The values <67> only appears in one or more of squares R3C4, R3C5 and R3C6 of row 3. These squares are the ones that intersect with block 2. Thus, the other (non-intersecting) squares of block 2 cannot contain these values.
R1C4 - removing <6> from <2456> leaving <245>
R2C5 - removing <7> from <257> leaving <25>
Intersection of row 7 with block 8. The values <79> 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 these values.
R8C5 - removing <79> from <23679> leaving <236>
R9C6 - removing <7> from <2357> leaving <235>
Squares R2C5<25>, R5C5<259> and R6C5<59> in column 5 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <259>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C5 - removing <2> from <2367> leaving <367>
R7C5 - removing <29> from <2679> leaving <67>
R8C5 - removing <2> from <236> leaving <36>
R7C4 is the only square in row 7 that can be <9>
R3C4 is the only square in column 4 that can be <6>
Squares R3C2 and R3C6 in row 3 and R7C2 and R7C6 in row 7 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 2 and 6 can be removed.
R1C6 - removing <2> from <2345> leaving <345>
R5C6 - removing <2> from <245> leaving <45>
R8C2 - removing <2> from <236> leaving <36>
R9C6 - removing <2> from <235> leaving <35>
R8C7 is the only square in row 8 that can be <2>
R9C9 can only be <1>
R9C7 can only be <7>
R5C9 can only be <6>
R5C7 can only be <1>
R1C9 can only be <2>
R2C7 can only be <5>
R8C8 can only be <9>
R1C1 can only be <3>
R2C5 can only be <2>
R2C8 can only be <7>
R1C7 can only be <6>
R6C7 can only be <9>
R6C5 can only be <5>
R5C8 can only be <5>
R5C1 can only be <7>
R9C1 can only be <2>
R3C2 can only be <2>
R7C2 can only be <6>
R5C3 can only be <3>
R8C3 can only be <7>
R5C5 can only be <9>
R5C6 can only be <4>
R7C5 can only be <7>
R8C2 can only be <3>
R7C6 can only be <2>
R3C5 can only be <3>
R9C4 can only be <5>
R8C5 can only be <6>
R9C6 can only be <3>
R1C4 can only be <4>
R3C6 can only be <7>
R1C6 can only be <5>
R5C4 can only be <2>
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