The full reasoning can be found below the Sudoku.
May 08 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R4C4 is the only square in row 4 that can be <2>
R5C5 is the only square in row 5 that can be <7>
R5C6 is the only square in row 5 that can be <3>
R7C3 is the only square in row 7 that can be <7>
R8C1 is the only square in row 8 that can be <9>
R6C6 is the only square in column 6 that can be <5>
R5C7 is the only square in column 7 that can be <5>
R5C9 is the only square in row 5 that can be <2>
R7C7 is the only square in column 7 that can be <2>
Squares R5C4 and R8C4 in column 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <68>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C4 - removing <68> from <4689> leaving <49>
R6C4 - removing <6> from <469> leaving <49>
Squares R4C8 and R6C8 in column 8 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 column.
R1C8 - removing <1> from <1568> leaving <568>
R7C8 - removing <1> from <136> leaving <36>
R9C8 - removing <14> from <1345> leaving <35>
Squares R1C5 and R2C6 in block 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <68>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R3C5 - removing <68> from <4689> leaving <49>
Intersection of row 5 with block 4. The value <1> only appears in one or more of squares R5C1, R5C2 and R5C3 of row 5. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R4C2 - removing <1> from <189> leaving <89>
R6C2 - removing <1> from <169> leaving <69>
Squares R8C4<68>, R8C6<168> and R8C7<16> in row 8 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <168>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C3 - removing <1> from <134> leaving <34>
R8C9 - removing <1> from <134> leaving <34>
Squares R2C3 and R2C9 in row 2 and R8C3 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in columns 3 and 9 can be removed.
R3C3 - removing <3> from <348> leaving <48>
R9C9 - removing <3> from <1345> leaving <145>
Squares R2C3 and R2C6 in row 2, R5C3 and R5C4 in row 5 and R8C4 and R8C6 in row 8 form a Swordfish pattern on possibility <8>. All other instances of this possibility in columns 3, 4 and 6 can be removed.
R3C3 - removing <8> from <48> leaving <4>
R4C6 - removing <8> from <18> leaving <1>
R3C5 can only be <9>
R8C3 can only be <3>
R3C7 can only be <6>
R2C4 can only be <4>
R8C7 can only be <1>
R4C8 can only be <4>
R4C5 can only be <8>
R6C8 can only be <1>
R8C9 can only be <4>
R7C2 can only be <1>
R2C7 can only be <9>
R9C9 can only be <5>
R9C8 can only be <3>
R1C9 can only be <1>
R2C9 can only be <3>
R6C4 can only be <9>
R3C8 can only be <8>
R3C2 can only be <3>
R1C8 can only be <5>
R4C2 can only be <9>
R1C5 can only be <6>
R5C4 can only be <6>
R5C1 can only be <1>
R8C4 can only be <8>
R6C5 can only be <4>
R6C2 can only be <6>
R9C2 can only be <2>
R8C6 can only be <6>
R2C6 can only be <8>
R7C5 can only be <3>
R9C1 can only be <4>
R9C5 can only be <1>
R7C8 can only be <6>
R1C1 can only be <2>
R1C2 can only be <8>
R2C3 can only be <1>
R5C3 can only be <8>
R2C1 can only be <6>
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