The full reasoning can be found below the Sudoku.
Apr 25 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C5 is the only square in row 1 that can be <8>
R1C7 is the only square in row 1 that can be <6>
R4C5 is the only square in row 4 that can be <4>
R6C5 is the only square in row 6 that can be <6>
R5C9 is the only square in row 5 that can be <6>
R9C2 is the only square in row 9 that can be <6>
R8C9 is the only square in column 9 that can be <2>
R9C5 is the only square in row 9 that can be <2>
R5C5 can only be <9>
R3C5 can only be <7>
R7C5 can only be <1>
R9C3 is the only square in row 9 that can be <7>
R1C8 is the only square in row 1 that can be <7>
R2C1 is the only square in row 2 that can be <7>
R8C6 is the only square in row 8 that can be <7>
Squares R5C4 and R5C6 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <28>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C1 - removing <8> from <1358> leaving <135>
R5C3 - removing <8> from <1358> leaving <135>
R5C7 - removing <2> from <123> leaving <13>
Squares R1C2 and R1C3 in block 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <59>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R2C3 - removing <59> from <1459> leaving <14>
R3C1 - removing <5> from <135> leaving <13>
R3C2 - removing <59> from <3459> leaving <34>
Squares R9C7 and R9C8 in block 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R7C8 - removing <5> from <359> leaving <39>
Intersection of column 1 with block 7. The value <8> 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.
R8C3 - removing <8> from <348> leaving <34>
Intersection of column 9 with block 3. The value <1> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R2C7 - removing <1> from <1459> leaving <459>
R3C8 - removing <1> from <159> leaving <59>
Squares R3C4<259>, R3C6<29> and R3C8<59> in row 3 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <259>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R3C9 - removing <9> from <149> leaving <14>
Squares R3C2 and R3C9 in row 3 and R7C2 and R7C9 in row 7 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 2 and 9 can be removed.
R2C9 - removing <4> from <149> leaving <19>
Squares R2C3 (XY), R3C1 (XZ) and R8C3 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R8C1 - removing <3> from <38> leaving <8>
R7C1 - removing <3> from <358> leaving <58>
R8C4 can only be <9>
R7C1 can only be <5>
R5C3 is the only square in row 5 that can be <5>
R1C3 can only be <9>
R1C2 can only be <5>
Squares R3C2 and R7C2 in column 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C2 - removing <3> from <239> leaving <29>
Squares R4C2, R6C2, R4C7 and R6C7 form a Type-3 Unique Rectangle on <29>. Upon close inspection, it is clear that:
(R4C7 or R6C7)<13> and R5C7<13> form a naked pair on <13> in block 6. No other squares in the block can contain these possibilities
R4C8 - removing <3> from <389> leaving <89>
R6C8 - removing <1> from <189> leaving <89>
(R4C7 or R6C7)<13> and R5C7<13> form a naked pair on <13> in column 7. No other squares in the column can contain these possibilities
R8C7 - removing <3> from <34> leaving <4>
R9C7 - removing <1> from <15> leaving <5>
(R4C7 or R6C7)<13>, R8C7<34> and R5C7<13> form a naked triplet on <134> in column 7. No other squares in the column can contain these possibilities
R2C7 - removing <4> from <459> leaving <59>
(R4C7 or R6C7)<13>, R9C7<15> and R5C7<13> form a naked triplet on <135> in column 7. No other squares in the column can contain these possibilities
R2C7 - removing <5> from <59> leaving <9>
R2C6 can only be <6>
R2C9 can only be <1>
R3C8 can only be <5>
R2C3 can only be <4>
R3C9 can only be <4>
R3C4 can only be <2>
R9C8 can only be <1>
R3C2 can only be <3>
R7C9 can only be <9>
R7C8 can only be <3>
R8C3 can only be <3>
R2C4 can only be <5>
R7C6 can only be <8>
R3C1 can only be <1>
R7C2 can only be <4>
R3C6 can only be <9>
R5C4 can only be <8>
R5C6 can only be <2>
R7C4 can only be <6>
R4C3 can only be <8>
R5C1 can only be <3>
R4C8 can only be <9>
R6C3 can only be <1>
R4C2 can only be <2>
R6C8 can only be <8>
R5C7 can only be <1>
R6C7 can only be <2>
R6C2 can only be <9>
R4C7 can only be <3>
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