The full reasoning can be found below the Sudoku.
Dec 23 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C1 is the only square in row 2 that can be <6>
R6C9 is the only square in row 6 that can be <4>
R6C7 is the only square in row 6 that can be <2>
R9C4 is the only square in row 9 that can be <2>
R7C9 is the only square in row 7 that can be <2>
R9C3 is the only square in row 9 that can be <6>
R9C2 is the only square in row 9 that can be <8>
Squares R4C5 and R6C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C5 - removing <58> from <13458> leaving <134>
R2C5 - removing <58> from <158> leaving <1>
R9C6 is the only square in column 6 that can be <1>
R9C7 can only be <7>
R1C6 is the only square in block 2 that can be <8>
Squares R4C5 and R6C5 in block 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R5C4 - removing <5> from <567> leaving <67>
Squares R5C4 and R5C6 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <67>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C1 - removing <7> from <1378> leaving <138>
R5C2 - removing <7> from <357> leaving <35>
R5C8 - removing <7> from <178> leaving <18>
R5C9 - removing <7> from <1578> leaving <158>
R4C9 is the only square in block 6 that can be <7>
Intersection of row 8 with block 7. The value <7> 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.
R7C1 - removing <7> from <347> leaving <34>
Intersection of block 6 with row 5. The value <8> 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.
R5C1 - removing <8> from <138> leaving <13>
Squares R3C1 and R3C6 in row 3 and R7C1 and R7C6 in row 7 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 1 and 6 can be removed.
R8C1 - removing <4> from <347> leaving <37>
Squares R2C2 and R2C9 in row 2 and R5C2 and R5C9 in row 5 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in columns 2 and 9 can be removed.
R1C2 - removing <5> from <457> leaving <47>
R3C9 - removing <5> from <59> leaving <9>
R3C1 can only be <4>
R8C9 can only be <1>
R3C6 can only be <6>
R7C1 can only be <3>
R1C2 can only be <7>
R3C4 can only be <5>
R5C6 can only be <7>
R5C4 can only be <6>
R7C6 can only be <4>
R7C4 can only be <7>
R5C1 can only be <1>
R8C1 can only be <7>
R9C5 can only be <9>
R8C2 can only be <4>
R6C1 can only be <8>
R8C8 can only be <9>
R8C5 can only be <3>
R9C8 can only be <4>
R1C8 can only be <1>
R2C2 can only be <5>
R1C7 can only be <5>
R5C8 can only be <8>
R2C9 can only be <8>
R5C2 can only be <3>
R1C3 can only be <9>
R2C8 can only be <7>
R5C9 can only be <5>
R1C4 can only be <3>
R4C7 can only be <1>
R6C5 can only be <5>
R4C1 can only be <9>
R6C3 can only be <7>
R4C5 can only be <8>
R1C5 can only be <4>
R4C3 can only be <5>
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