The full reasoning can be found below the Sudoku.
Jan 12 - Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R3C8 can only be <5>
R7C2 can only be <2>
R3C3 is the only square in row 3 that can be <2>
R1C5 is the only square in row 1 that can be <2>
R4C7 is the only square in row 4 that can be <2>
R4C5 is the only square in row 4 that can be <1>
R5C4 is the only square in row 5 that can be <2>
R8C8 is the only square in row 8 that can be <2>
R8C7 is the only square in row 8 that can be <1>
R5C8 is the only square in row 5 that can be <1>
R5C6 is the only square in column 6 that can be <8>
R8C6 is the only square in column 6 that can be <9>
Squares R5C5 and R6C5 in column 5 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.
R2C5 - removing <4> from <4567> leaving <567>
R8C5 - removing <3> from <356> leaving <56>
R9C5 - removing <34> from <34567> leaving <567>
Intersection of row 1 with block 1. The values <57> only appears in one or more of squares R1C1, R1C2 and R1C3 of row 1. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain these values.
R2C1 - removing <5> from <14589> leaving <1489>
R2C2 - removing <57> from <4578> leaving <48>
R2C3 - removing <7> from <1789> leaving <189>
Squares R2C2 and R4C2 in column 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C2 - removing <8> from <3578> leaving <357>
R3C2 - removing <4> from <34> leaving <3>
R5C2 - removing <4> from <34567> leaving <3567>
R8C2 - removing <8> from <68> leaving <6>
R3C6 can only be <4>
R2C6 can only be <6>
R8C5 can only be <5>
R8C4 can only be <3>
R2C5 can only be <7>
R1C6 can only be <3>
R8C9 can only be <8>
R1C9 can only be <9>
R9C8 can only be <7>
R9C4 can only be <4>
R9C5 can only be <6>
R7C8 can only be <9>
R2C8 can only be <8>
R2C4 can only be <5>
R2C2 can only be <4>
R6C8 can only be <6>
R1C7 can only be <6>
R7C7 can only be <4>
R9C7 can only be <5>
R7C4 can only be <7>
R9C9 can only be <3>
R4C2 can only be <8>
R4C3 can only be <9>
R6C3 can only be <3>
R4C1 can only be <4>
R2C3 can only be <1>
R6C5 can only be <4>
R6C7 can only be <8>
R6C9 can only be <5>
R5C5 can only be <3>
R5C9 can only be <4>
R5C7 can only be <9>
R2C1 can only be <9>
R9C3 can only be <8>
R5C1 can only be <5>
R5C2 can only be <7>
R1C1 can only be <8>
R5C3 can only be <6>
R1C2 can only be <5>
R9C1 can only be <1>
R1C3 can only be <7>
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