The full reasoning can be found below the Sudoku.
Sep 18 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C3 can only be <2>
R5C7 can only be <9>
R4C1 can only be <7>
R5C4 can only be <3>
R4C9 can only be <8>
R6C1 can only be <3>
R5C2 can only be <5>
R6C6 can only be <1>
R6C4 can only be <9>
R5C6 can only be <8>
R2C8 is the only square in row 2 that can be <9>
R2C9 is the only square in row 2 that can be <4>
R6C9 can only be <7>
R6C5 can only be <4>
R5C8 can only be <4>
R5C5 can only be <7>
R8C1 is the only square in row 8 that can be <4>
R9C2 is the only square in row 9 that can be <9>
R2C1 is the only square in column 1 that can be <8>
Squares R7C5 and R9C4 in block 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <12>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C5 - removing <12> from <1235> leaving <35>
R9C6 - removing <2> from <235> leaving <35>
Squares R2C2 and R2C5 in row 2 and R7C2 and R7C5 in row 7 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 2 and 5 can be removed.
R1C2 - removing <2> from <1237> leaving <137>
R4C5 - removing <2> from <256> leaving <56>
R8C2 - removing <2> from <128> leaving <18>
R8C9 is the only square in row 8 that can be <2>
Squares R9C6 and R9C9 in row 9 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 row.
R9C8 - removing <5> from <156> leaving <16>
Squares R3C5<36>, R4C5<56> and R8C5<35> in column 5 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <356>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C5 - removing <36> from <1236> leaving <12>
Squares R1C6 and R9C6 in column 6 and R1C9 and R9C9 in column 9 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in rows 1 and 9 can be removed.
R1C2 - removing <3> from <137> leaving <17>
Squares R2C3 and R8C3 in column 3 and R2C7 and R8C7 in column 7 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in rows 2 and 8 can be removed.
R8C8 - removing <6> from <1568> leaving <158>
Squares R1C6 (XY), R1C1 (XZ) and R3C5 (YZ) form an XY-Wing pattern on <6>. All squares that are buddies of both the XZ and YZ squares cannot be <6>.
R1C4 - removing <6> from <126> leaving <12>
R4C4 is the only square in column 4 that can be <6>
R4C5 can only be <5>
R4C6 can only be <2>
R8C5 can only be <3>
R1C6 can only be <3>
R8C7 can only be <6>
R3C5 can only be <6>
R9C6 can only be <5>
R8C3 can only be <1>
R2C7 can only be <3>
R9C8 can only be <1>
R9C9 can only be <3>
R9C4 can only be <2>
R7C8 can only be <8>
R1C9 can only be <5>
R3C8 can only be <7>
R3C2 can only be <3>
R1C8 can only be <6>
R8C8 can only be <5>
R8C2 can only be <8>
R2C3 can only be <6>
R7C2 can only be <2>
R9C1 can only be <6>
R1C4 can only be <1>
R7C5 can only be <1>
R1C2 can only be <7>
R2C5 can only be <2>
R1C1 can only be <2>
R2C2 can only be <1>
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