The full reasoning can be found below the Sudoku.
Sep 18 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R3C3 can only be <2>
R4C6 can only be <2>
R5C1 can only be <9>
R7C7 can only be <3>
R4C4 can only be <5>
R4C8 can only be <8>
R6C6 can only be <3>
R5C5 can only be <4>
R4C2 can only be <1>
R6C8 can only be <2>
R5C2 can only be <2>
R6C2 can only be <8>
R6C4 can only be <9>
R8C6 can only be <1>
R3C7 can only be <7>
R9C8 can only be <1>
R2C6 can only be <7>
R1C2 is the only square in row 1 that can be <7>
R2C5 is the only square in row 2 that can be <1>
R2C8 is the only square in row 2 that can be <5>
R7C3 is the only square in row 7 that can be <1>
Squares R1C1 and R9C1 in column 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <38>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C1 - removing <38> from <3468> leaving <46>
R8C1 - removing <38> from <3468> leaving <46>
Squares R8C7 and R9C9 in block 9 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 block.
R8C9 - removing <28> from <2789> leaving <79>
Intersection of row 1 with block 3. The value <9> only appears in one or more of squares R1C7, R1C8 and R1C9 of row 1. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R2C9 - removing <9> from <2689> leaving <268>
Squares R1C5 and R1C9 in row 1 and R9C5 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 5 and 9 can be removed.
R2C9 - removing <2> from <268> leaving <68>
R8C5 - removing <2> from <235> leaving <35>
Squares R1C1 and R1C9 in row 1 and R9C1 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 1 and 9 can be removed.
R2C9 - removing <8> from <68> leaving <6>
R2C1 can only be <4>
R5C9 can only be <7>
R3C8 can only be <3>
R3C2 can only be <6>
R1C8 can only be <9>
R5C8 can only be <6>
R8C9 can only be <9>
R8C3 can only be <8>
R7C8 can only be <4>
R8C1 can only be <6>
R7C2 can only be <9>
R8C8 can only be <7>
R8C7 can only be <2>
R2C3 can only be <9>
R9C1 can only be <3>
R8C4 can only be <3>
R2C7 can only be <8>
R9C9 can only be <8>
R9C2 can only be <5>
R1C1 can only be <8>
R9C5 can only be <2>
R8C2 can only be <4>
R1C5 can only be <3>
R1C9 can only be <2>
R8C5 can only be <5>
R2C4 can only be <2>
R2C2 can only be <3>
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