The full reasoning can be found below the Sudoku.
Sep 02 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C5 is the only square in row 1 that can be <6>
R1C1 is the only square in row 1 that can be <9>
R2C7 is the only square in row 2 that can be <8>
R3C2 is the only square in row 3 that can be <2>
R5C4 is the only square in row 5 that can be <8>
R8C1 is the only square in row 8 that can be <8>
R8C5 is the only square in row 8 that can be <9>
R7C4 can only be <1>
R7C6 can only be <3>
R3C4 can only be <7>
R6C6 can only be <7>
R9C5 can only be <2>
R4C4 can only be <6>
R6C4 can only be <9>
R4C6 is the only square in row 4 that can be <2>
R7C8 is the only square in row 7 that can be <9>
R3C8 is the only square in column 8 that can be <4>
Squares R2C5 and R4C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C5 - removing <15> from <1345> leaving <34>
Squares R5C2 and R5C9 in row 5 and R9C2 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 2 and 9 can be removed.
R7C2 - removing <6> from <456> leaving <45>
Squares R4C3 and R8C3 in column 3 and R4C7 and R8C7 in column 7 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in rows 4 and 8 can be removed.
R8C9 - removing <7> from <157> leaving <15>
Squares R1C9 (XYZ), R1C8 (XZ) and R8C9 (YZ) form an XYZ-Wing pattern on <5>. All squares that are buddies of all three squares cannot be <5>.
R2C9 - removing <5> from <135> leaving <13>
Squares R8C7 (XYZ), R8C9 (XZ) and R4C7 (YZ) form an XYZ-Wing pattern on <5>. All squares that are buddies of all three squares cannot be <5>.
R7C7 - removing <5> from <56> leaving <6>
R6C7 can only be <3>
R6C5 can only be <4>
R6C3 can only be <6>
R5C5 can only be <3>
R3C3 is the only square in row 3 that can be <3>
R2C9 is the only square in row 2 that can be <3>
R5C9 is the only square in row 5 that can be <6>
R9C8 is the only square in row 9 that can be <3>
R9C2 is the only square in row 9 that can be <6>
Squares R7C2 and R7C3 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <45>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C3 - removing <5> from <157> leaving <17>
Squares R4C3 and R8C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C3 - removing <1> from <145> leaving <45>
Squares R3C7 and R4C5 form a remote naked pair. <15> can be removed from any square that is common to their groups.
R4C7 - removing <5> from <57> leaving <7>
R4C3 can only be <1>
R5C8 can only be <5>
R5C6 can only be <1>
R1C8 can only be <7>
R4C5 can only be <5>
R8C3 can only be <7>
R5C2 can only be <4>
R2C5 can only be <1>
R5C1 can only be <7>
R7C2 can only be <5>
R3C6 can only be <5>
R7C3 can only be <4>
R1C2 can only be <1>
R2C3 can only be <5>
R9C1 can only be <1>
R9C9 can only be <7>
R2C1 can only be <4>
R1C9 can only be <5>
R8C9 can only be <1>
R3C7 can only be <1>
R8C7 can only be <5>
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