The full reasoning can be found below the Sudoku.
Aug 07 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C3 can only be <1>
R4C1 is the only square in row 4 that can be <1>
R7C3 is the only square in row 7 that can be <2>
R7C6 is the only square in row 7 that can be <6>
R9C3 is the only square in row 9 that can be <3>
R9C7 is the only square in row 9 that can be <4>
R1C3 is the only square in row 1 that can be <4>
R3C2 can only be <8>
R3C4 can only be <5>
R3C8 can only be <4>
R3C1 can only be <6>
R1C1 can only be <5>
R9C1 can only be <7>
R5C1 can only be <2>
R5C2 is the only square in row 5 that can be <4>
R6C4 is the only square in row 6 that can be <4>
R7C8 is the only square in row 7 that can be <5>
R6C2 is the only square in column 2 that can be <7>
R8C3 is the only square in column 3 that can be <5>
R5C7 is the only square in column 7 that can be <5>
R2C8 is the only square in column 8 that can be <8>
R6C9 is the only square in column 9 that can be <2>
Squares R4C8 and R5C8 in block 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <37>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C7 - removing <7> from <679> leaving <69>
R5C9 - removing <7> from <679> leaving <69>
Squares R5C3 and R5C9 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <69>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C5 - removing <6> from <367> leaving <37>
Squares R1C6<17>, R2C5<27> and R3C6<12> in block 2 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <127>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C4 - removing <7> from <378> leaving <38>
R1C5 - removing <17> from <1378> leaving <38>
Intersection of column 5 with block 8. The value <1> only appears in one or more of squares R7C5, R8C5 and R9C5 of column 5. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R9C6 - removing <1> from <159> leaving <59>
Squares R6C6 and R9C6 in column 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 column.
R4C6 - removing <9> from <279> leaving <27>
Squares R2C5 and R2C7 in row 2 and R8C5 and R8C7 in row 8 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in columns 5 and 7 can be removed.
R1C7 - removing <7> from <167> leaving <16>
R4C5 - removing <7> from <2367> leaving <236>
R5C5 - removing <7> from <37> leaving <3>
R5C8 can only be <7>
R1C5 can only be <8>
R4C8 can only be <3>
R1C4 can only be <3>
R9C4 is the only square in row 9 that can be <8>
Squares R5C3 and R5C9 in row 5, R6C3 and R6C6 in row 6 and R9C6 and R9C9 in row 9 form a Swordfish pattern on possibility <9>. All other instances of this possibility in columns 3, 6 and 9 can be removed.
R7C9 - removing <9> from <179> leaving <17>
Squares R7C4 (XY), R8C5 (XZ) and R7C2 (YZ) form an XY-Wing pattern on <1>. All squares that are buddies of both the XZ and YZ squares cannot be <1>.
R8C2 - removing <1> from <19> leaving <9>
R7C2 can only be <1>
R7C9 can only be <7>
R7C4 can only be <9>
R8C7 can only be <1>
R8C5 can only be <7>
R1C7 can only be <6>
R3C7 can only be <2>
R9C9 can only be <9>
R9C6 can only be <5>
R5C9 can only be <6>
R1C9 can only be <1>
R4C7 can only be <9>
R1C6 can only be <7>
R3C6 can only be <1>
R2C7 can only be <7>
R4C4 can only be <7>
R5C3 can only be <9>
R2C5 can only be <2>
R9C5 can only be <1>
R6C6 can only be <9>
R4C6 can only be <2>
R4C5 can only be <6>
R6C5 can only be <5>
R6C3 can only be <6>
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