The full reasoning can be found below the Sudoku.
Mar 29 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C5 can only be <4>
R6C1 can only be <4>
R1C6 is the only square in row 1 that can be <1>
R2C3 is the only square in row 2 that can be <4>
R3C8 is the only square in row 3 that can be <2>
R4C7 is the only square in row 4 that can be <1>
R5C9 is the only square in row 5 that can be <2>
R6C9 can only be <8>
R6C7 can only be <5>
R6C3 can only be <2>
R5C8 can only be <6>
R5C1 can only be <9>
R8C8 can only be <8>
R4C9 can only be <9>
R2C8 can only be <5>
R7C8 can only be <3>
R5C2 can only be <5>
R7C2 is the only square in row 7 that can be <1>
R9C6 is the only square in row 9 that can be <3>
R8C3 is the only square in column 3 that can be <5>
Squares R1C4 and R9C4 in column 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <57>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C4 - removing <7> from <479> leaving <49>
R7C4 - removing <7> from <479> leaving <49>
Squares R1C1 and R1C9 in row 1 and R9C1 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 1 and 9 can be removed.
R4C1 - removing <6> from <67> leaving <7>
R4C3 can only be <6>
Squares R1C4 and R9C4 in column 4 and R1C9 and R9C9 in column 9 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in rows 1 and 9 can be removed.
R1C5 - removing <7> from <578> leaving <58>
R9C5 - removing <7> from <578> leaving <58>
Squares R1C5 and R9C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C5 - removing <8> from <789> leaving <79>
R2C7 is the only square in row 2 that can be <8>
Squares R1C1 (XY), R1C9 (XZ) and R3C3 (YZ) form an XY-Wing pattern on <7>. All squares that are buddies of both the XZ and YZ squares cannot be <7>.
R3C7 - removing <7> from <67> leaving <6>
R1C9 can only be <7>
R1C4 can only be <5>
R9C9 can only be <6>
R9C1 can only be <8>
R1C5 can only be <8>
R9C4 can only be <7>
R1C1 can only be <6>
R9C5 can only be <5>
R3C6 can only be <4>
R3C4 can only be <9>
R7C6 can only be <8>
R7C3 can only be <7>
R8C5 can only be <9>
R3C2 can only be <7>
R7C4 can only be <4>
R2C5 can only be <7>
R7C7 can only be <9>
R3C3 can only be <8>
R8C2 can only be <6>
R8C7 can only be <7>
R2C2 can only be <9>
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