The full reasoning can be found below the Sudoku.
May 02 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C5 can only be <6>
R6C2 can only be <9>
R4C8 can only be <5>
R6C5 can only be <2>
R4C2 can only be <8>
R5C9 can only be <2>
R5C5 can only be <3>
R5C7 can only be <9>
R6C8 can only be <6>
R5C1 can only be <5>
R5C3 can only be <7>
R1C5 is the only square in row 1 that can be <4>
R1C9 is the only square in row 1 that can be <7>
R1C1 is the only square in row 1 that can be <9>
R2C5 is the only square in row 2 that can be <9>
R7C8 is the only square in row 7 that can be <9>
R8C6 is the only square in row 8 that can be <4>
R7C6 is the only square in column 6 that can be <2>
Squares R2C6 and R3C6 in block 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <35>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R2C4 - removing <3> from <367> leaving <67>
R3C4 - removing <3> from <3678> leaving <678>
R3C5 - removing <5> from <578> leaving <78>
Squares R1C7 and R2C9 in block 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <56>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R3C7 - removing <56> from <2356> leaving <23>
Intersection of row 7 with block 8. The value <8> 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.
R8C4 - removing <8> from <378> leaving <37>
R8C5 - removing <8> from <578> leaving <57>
R9C5 - removing <8> from <158> leaving <15>
Squares R8C2<25>, R8C4<37>, R8C5<57> and R8C8<23> in row 8 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2357>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C1 - removing <2> from <268> leaving <68>
R8C9 - removing <5> from <568> leaving <68>
Squares R1C3 and R1C7 in row 1 and R7C3 and R7C7 in row 7 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 3 and 7 can be removed.
R3C3 - removing <6> from <156> leaving <15>
R3C4 is the only square in row 3 that can be <6>
R2C4 can only be <7>
R8C4 can only be <3>
R3C5 can only be <8>
R8C8 can only be <2>
R7C4 can only be <8>
R8C2 can only be <5>
R8C5 can only be <7>
R7C2 can only be <1>
R7C5 can only be <5>
R2C2 can only be <2>
R9C3 can only be <3>
R9C5 can only be <1>
R9C7 can only be <5>
R7C3 can only be <6>
R9C9 can only be <8>
R1C7 can only be <6>
R9C1 can only be <2>
R8C9 can only be <6>
R1C3 can only be <5>
R7C7 can only be <3>
R2C9 can only be <5>
R2C1 can only be <6>
R3C2 can only be <7>
R2C6 can only be <3>
R8C1 can only be <8>
R3C7 can only be <2>
R3C3 can only be <1>
R2C8 can only be <1>
R3C6 can only be <5>
R3C8 can only be <3>
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