The full reasoning can be found below the Sudoku.
Dec 26 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R6C5 can only be <2>
R4C5 can only be <3>
R5C6 can only be <9>
R4C9 can only be <5>
R4C8 can only be <2>
R5C4 can only be <8>
R4C2 can only be <6>
R5C7 can only be <1>
R6C8 can only be <4>
R6C9 can only be <7>
R5C9 can only be <3>
R4C1 can only be <8>
R1C4 is the only square in row 1 that can be <3>
R1C9 is the only square in row 1 that can be <9>
R1C3 is the only square in row 1 that can be <8>
R2C2 is the only square in row 2 that can be <3>
R2C8 is the only square in row 2 that can be <8>
R7C1 is the only square in row 7 that can be <4>
R7C9 is the only square in row 7 that can be <8>
R3C2 is the only square in column 2 that can be <2>
R8C2 is the only square in column 2 that can be <7>
Intersection of block 3 with row 3. The value <1> only appears in one or more of squares R3C7, R3C8 and R3C9 of block 3. These squares are the ones that intersect with row 3. Thus, the other (non-intersecting) squares of row 3 cannot contain this value.
R3C1 - removing <1> from <167> leaving <67>
R3C5 - removing <1> from <1567> leaving <567>
Squares R7C5<15>, R8C6<145> and R9C6<145> in block 8 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <145>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R9C5 - removing <15> from <1567> leaving <67>
Squares R3C5 and R3C8 in row 3 and R7C5 and R7C8 in row 7 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in columns 5 and 8 can be removed.
R1C5 - removing <5> from <156> leaving <16>
R8C8 - removing <5> from <159> leaving <19>
Squares R1C1 and R1C5 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <16>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C6 - removing <1> from <125> leaving <25>
R1C7 - removing <6> from <256> leaving <25>
Squares R2C4 and R2C7 in row 2 and R8C4 and R8C7 in row 8 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 4 and 7 can be removed.
R9C4 - removing <6> from <679> leaving <79>
R9C7 - removing <6> from <456> leaving <45>
Squares R2C3 (XY), R1C1 (XZ) and R2C4 (YZ) form an XY-Wing pattern on <6>. All squares that are buddies of both the XZ and YZ squares cannot be <6>.
R1C5 - removing <6> from <16> leaving <1>
R1C1 can only be <6>
R7C5 can only be <5>
R2C6 can only be <2>
R2C7 can only be <6>
R1C6 can only be <5>
R2C4 can only be <7>
R3C9 can only be <1>
R3C8 can only be <5>
R9C9 can only be <6>
R9C5 can only be <7>
R3C1 can only be <7>
R1C7 can only be <2>
R2C3 can only be <1>
R9C4 can only be <9>
R3C5 can only be <6>
R5C1 can only be <2>
R5C3 can only be <7>
R9C1 can only be <1>
R8C4 can only be <6>
R8C3 can only be <5>
R8C7 can only be <4>
R9C3 can only be <2>
R8C6 can only be <1>
R9C7 can only be <5>
R9C6 can only be <4>
R6C1 can only be <9>
R7C2 can only be <9>
R6C2 can only be <1>
R7C8 can only be <1>
R8C8 can only be <9>
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