The full reasoning can be found below the Sudoku.
May 13 - Super Hard
Puzzle Copyright © Kevin Stone
Reasoning
R5C5 can only be <2>
R5C9 can only be <8>
R5C7 can only be <3>
R3C2 is the only square in row 3 that can be <3>
R2C4 is the only square in row 2 that can be <3>
R4C2 is the only square in row 4 that can be <7>
R6C8 is the only square in row 6 that can be <1>
R2C7 is the only square in row 2 that can be <1>
R7C8 is the only square in row 7 that can be <3>
R8C6 is the only square in row 8 that can be <1>
R8C3 is the only square in row 8 that can be <7>
R6C1 is the only square in column 1 that can be <8>
R1C2 is the only square in column 2 that can be <8>
R4C8 is the only square in column 8 that can be <4>
R8C4 is the only square in block 8 that can be <4>
R1C4 can only be <6>
R9C4 can only be <2>
R1C6 is the only square in row 1 that can be <4>
R8C8 is the only square in column 8 that can be <2>
R8C7 is the only square in row 8 that can be <8>
R3C8 is the only square in row 3 that can be <8>
Squares R7C5 and R7C7 in row 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <69>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C2 - removing <69> from <2469> leaving <24>
R7C3 - removing <6> from <246> leaving <24>
Squares R4C9 and R6C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <26>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C9 - removing <6> from <569> leaving <59>
R8C9 - removing <6> from <569> leaving <59>
Intersection of row 8 with block 7. The value <6> only appears in one or more of squares R8C1, R8C2 and R8C3 of row 8. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C2 - removing <6> from <569> leaving <59>
Intersection of column 3 with block 1. The value <6> only appears in one or more of squares R1C3, R2C3 and R3C3 of column 3. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R2C1 - removing <6> from <2456> leaving <245>
R2C2 - removing <6> from <2456> leaving <245>
Squares R3C5 and R3C7 in row 3 and R7C5 and R7C7 in row 7 form a Simple X-Wing pattern on possibility <9>. All other instances of this possibility in columns 5 and 7 can be removed.
R9C5 - removing <9> from <679> leaving <67>
Squares R7C2, R7C3, R2C2 and R2C3 form a Type-3 Unique Rectangle on <24>. Upon close inspection, it is clear that:
(R2C2 or R2C3)<56> and R3C3<56> form a naked pair on <56> in block 1. No other squares in the block can contain these possibilities
R2C1 - removing <5> from <245> leaving <24>
Squares R2C6 (XY), R1C5 (XZ) and R2C9 (YZ) form an XY-Wing pattern on <5>. All squares that are buddies of both the XZ and YZ squares cannot be <5>.
R1C8 - removing <5> from <57> leaving <7>
R1C5 can only be <5>
R3C5 can only be <9>
R3C7 can only be <6>
R7C5 can only be <6>
R2C6 can only be <7>
R3C3 can only be <5>
R7C7 can only be <9>
R2C8 can only be <5>
R9C5 can only be <7>
R8C9 can only be <5>
R8C1 can only be <6>
R2C9 can only be <9>
R9C8 can only be <6>
R9C6 can only be <9>
R9C2 can only be <5>
R5C3 can only be <4>
R5C1 can only be <5>
R7C3 can only be <2>
R7C2 can only be <4>
R2C3 can only be <6>
R8C2 can only be <9>
R4C1 can only be <2>
R4C9 can only be <6>
R2C1 can only be <4>
R6C2 can only be <6>
R6C9 can only be <2>
R2C2 can only be <2>
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