The full reasoning can be found below the Sudoku.
May 01 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R6C7 can only be <5>
R7C6 can only be <9>
R7C4 can only be <6>
R7C7 can only be <3>
R6C4 can only be <4>
R1C5 is the only square in row 1 that can be <1>
R4C5 is the only square in row 4 that can be <5>
R5C4 is the only square in row 5 that can be <3>
R5C1 is the only square in row 5 that can be <4>
R8C5 is the only square in row 8 that can be <4>
R9C5 can only be <8>
R8C2 is the only square in row 8 that can be <8>
R9C7 is the only square in row 9 that can be <4>
Intersection of column 1 with block 7. The value <1> only appears in one or more of squares R7C1, R8C1 and R9C1 of column 1. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R7C3 - removing <1> from <12> leaving <2>
Intersection of column 7 with block 3. The value <7> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C8 - removing <7> from <279> leaving <29>
R2C8 - removing <7> from <2579> leaving <259>
Intersection of row 2 with block 1. The value <7> only appears in one or more of squares R2C1, R2C2 and R2C3 of row 2. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R1C1 - removing <7> from <3679> leaving <369>
R1C2 - removing <7> from <267> leaving <26>
R1C3 - removing <7> from <6789> leaving <689>
R3C1 - removing <7> from <5679> leaving <569>
R3C3 - removing <7> from <6789> leaving <689>
R1C7 is the only square in row 1 that can be <7>
Intersection of block 6 with row 5. The value <1> only appears in one or more of squares R5C7, R5C8 and R5C9 of block 6. These squares are the ones that intersect with row 5. Thus, the other (non-intersecting) squares of row 5 cannot contain this value.
R5C6 - removing <1> from <178> leaving <78>
Squares R3C1 and R3C9 in row 3 and R7C1 and R7C9 in row 7 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in columns 1 and 9 can be removed.
R2C1 - removing <5> from <3579> leaving <379>
R2C9 - removing <5> from <2359> leaving <239>
R9C1 - removing <5> from <5679> leaving <679>
R9C9 - removing <5> from <259> leaving <29>
Squares R3C4 and R4C4 in column 4 and R3C7 and R4C7 in column 7 form a Simple X-Wing pattern on possibility <9>. All other instances of this possibility in rows 3 and 4 can be removed.
R3C1 - removing <9> from <569> leaving <56>
R3C3 - removing <9> from <689> leaving <68>
R3C9 - removing <9> from <25689> leaving <2568>
Squares R4C4 (XY), R5C6 (XZ) and R4C7 (YZ) form an XY-Wing pattern on <8>. All squares that are buddies of both the XZ and YZ squares cannot be <8>.
R4C6 - removing <8> from <178> leaving <17>
R5C9 - removing <8> from <189> leaving <19>
R4C7 is the only square in row 4 that can be <8>
R3C7 can only be <9>
R3C4 can only be <7>
R1C8 can only be <2>
R1C2 can only be <6>
R2C8 can only be <5>
R2C9 can only be <3>
R3C6 can only be <2>
R4C4 can only be <9>
R6C6 can only be <1>
R2C5 can only be <9>
R5C5 can only be <6>
R5C2 can only be <7>
R6C5 can only be <2>
R6C3 can only be <6>
R4C6 can only be <7>
R1C9 can only be <8>
R3C1 can only be <5>
R3C3 can only be <8>
R1C3 can only be <9>
R3C9 can only be <6>
R2C1 can only be <7>
R7C1 can only be <1>
R4C3 can only be <1>
R5C6 can only be <8>
R2C2 can only be <2>
R9C2 can only be <5>
R7C9 can only be <5>
R1C1 can only be <3>
R9C3 can only be <7>
R8C1 can only be <9>
R8C9 can only be <1>
R9C1 can only be <6>
R8C8 can only be <7>
R5C9 can only be <9>
R9C8 can only be <9>
R9C9 can only be <2>
R5C8 can only be <1>
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