The full reasoning can be found below the Sudoku.
May 09 - Super Hard
Puzzle Copyright © Kevin Stone
Reasoning
R1C9 is the only square in row 1 that can be <3>
R1C6 is the only square in row 1 that can be <2>
R2C3 is the only square in row 2 that can be <1>
R1C4 is the only square in row 1 that can be <1>
R3C2 is the only square in row 3 that can be <3>
R5C6 is the only square in row 5 that can be <6>
R4C7 is the only square in row 4 that can be <6>
R6C6 is the only square in row 6 that can be <1>
R8C8 is the only square in row 8 that can be <1>
R7C1 is the only square in row 7 that can be <1>
R7C2 is the only square in row 7 that can be <5>
R9C4 is the only square in row 9 that can be <3>
R4C5 is the only square in row 4 that can be <3>
R4C3 is the only square in row 4 that can be <2>
R6C5 is the only square in row 6 that can be <2>
R6C4 is the only square in row 6 that can be <8>
R9C1 is the only square in row 9 that can be <6>
Intersection of row 5 with block 5. The value <5> only appears in one or more of squares R5C4, R5C5 and R5C6 of row 5. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R4C4 - removing <5> from <4579> leaving <479>
Intersection of column 1 with block 4. The value <9> only appears in one or more of squares R4C1, R5C1 and R6C1 of column 1. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R6C3 - removing <9> from <579> leaving <57>
Squares R3C6 and R3C9 in row 3 and R7C6 and R7C9 in row 7 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 6 and 9 can be removed.
R9C6 - removing <8> from <4789> leaving <479>
R9C9 - removing <8> from <478> leaving <47>
Squares R4C9<457>, R6C9<457> and R9C9<47> in column 9 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <457>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C9 - removing <457> from <24578> leaving <28>
R7C9 - removing <7> from <278> leaving <28>
Intersection of column 9 with block 6. The value <5> only appears in one or more of squares R4C9, R5C9 and R6C9 of column 9. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R6C7 - removing <5> from <4579> leaving <479>
Squares R1C3 and R1C7 in row 1 and R9C3 and R9C7 in row 9 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 3 and 7 can be removed.
R2C7 - removing <8> from <4578> leaving <457>
R8C3 - removing <8> from <789> leaving <79>
R8C7 - removing <8> from <4789> leaving <479>
Squares R8C2 (XY), R5C2 (XZ) and R8C5 (YZ) form an XY-Wing pattern on <4>. All squares that are buddies of both the XZ and YZ squares cannot be <4>.
R5C5 - removing <4> from <45> leaving <5>
R2C7 is the only square in row 2 that can be <5>
R3C4 is the only square in column 4 that can be <5>
Intersection of column 4 with block 5. The value <4> only appears in one or more of squares R4C4, R5C4 and R6C4 of column 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R4C6 - removing <4> from <479> leaving <79>
Intersection of block 3 with column 8. The value <7> only appears in one or more of squares R1C8, R2C8 and R3C8 of block 3. These squares are the ones that intersect with column 8. Thus, the other (non-intersecting) squares of column 8 cannot contain this value.
R5C8 - removing <7> from <479> leaving <49>
R7C8 - removing <7> from <279> leaving <29>
Intersection of row 7 with block 8. The value <7> only appears in one or more of squares R7C4, R7C5 and R7C6 of row 7. 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 <7> from <479> leaving <49>
Squares R9C6 (XY), R9C9 (XZ) and R4C6 (YZ) form an XY-Wing pattern on <7>. All squares that are buddies of both the XZ and YZ squares cannot be <7>.
R4C9 - removing <7> from <457> leaving <45>
Intersection of block 6 with row 6. The value <7> only appears in one or more of squares R6C7, R6C8 and R6C9 of block 6. These squares are the ones that intersect with row 6. Thus, the other (non-intersecting) squares of row 6 cannot contain this value.
R6C1 - removing <7> from <4579> leaving <459>
R6C3 - removing <7> from <57> leaving <5>
R1C3 can only be <8>
R1C7 can only be <4>
R1C1 can only be <5>
R2C8 can only be <7>
R2C2 can only be <4>
R3C8 can only be <2>
R3C9 can only be <8>
R7C8 can only be <9>
R3C6 can only be <4>
R7C9 can only be <2>
R7C4 can only be <7>
R5C8 can only be <4>
R8C7 can only be <7>
R8C2 can only be <8>
R8C3 can only be <9>
R6C7 can only be <9>
R9C7 can only be <8>
R9C9 can only be <4>
R9C6 can only be <9>
R4C9 can only be <5>
R6C9 can only be <7>
R2C5 can only be <8>
R5C2 can only be <7>
R3C1 can only be <7>
R8C5 can only be <4>
R5C4 can only be <9>
R6C1 can only be <4>
R7C6 can only be <8>
R9C3 can only be <7>
R4C6 can only be <7>
R4C4 can only be <4>
R4C1 can only be <9>
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