The full reasoning can be found below the Sudoku.
Apr 14 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C7 is the only square in row 1 that can be <1>
R5C8 is the only square in row 5 that can be <9>
R8C1 is the only square in row 8 that can be <2>
R9C4 is the only square in row 9 that can be <9>
R5C2 is the only square in column 2 that can be <4>
R6C3 is the only square in column 3 that can be <9>
R3C4 is the only square in column 4 that can be <4>
R7C4 is the only square in column 4 that can be <5>
R9C6 is the only square in column 6 that can be <1>
R8C2 is the only square in row 8 that can be <1>
R4C7 is the only square in column 7 that can be <5>
Squares R3C6 and R7C6 in column 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <68>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C6 - removing <68> from <2678> leaving <27>
R5C6 - removing <8> from <278> leaving <27>
Intersection of row 5 with block 5. The values <278> only appears in one or more of squares R5C4, R5C5 and R5C6 of row 5. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain these values.
R4C5 - removing <8> from <148> leaving <14>
R6C5 - removing <78> from <1478> leaving <14>
Intersection of column 8 with block 9. The value <8> only appears in one or more of squares R7C8, R8C8 and R9C8 of column 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C9 - removing <8> from <678> leaving <67>
R8C9 - removing <8> from <568> leaving <56>
Squares R3C1 and R3C6 in row 3 and R7C1 and R7C6 in row 7 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 1 and 6 can be removed.
R4C1 - removing <8> from <168> leaving <16>
R6C1 - removing <8> from <18> leaving <1>
R6C5 can only be <4>
R4C1 can only be <6>
R6C7 can only be <7>
R4C5 can only be <1>
R6C9 can only be <8>
R9C7 can only be <4>
R4C9 can only be <4>
R4C3 can only be <8>
R2C8 is the only square in row 2 that can be <4>
R9C3 is the only square in row 9 that can be <7>
R1C3 can only be <6>
R7C1 can only be <8>
R1C8 can only be <5>
R2C2 can only be <5>
R1C2 can only be <8>
R7C6 can only be <6>
R3C1 can only be <9>
R9C2 can only be <6>
R7C9 can only be <7>
R3C6 can only be <8>
R8C5 can only be <8>
R8C8 can only be <6>
R5C5 can only be <7>
R8C9 can only be <5>
R9C8 can only be <8>
R1C4 can only be <2>
R1C6 can only be <7>
R5C4 can only be <8>
R5C6 can only be <2>
R2C5 can only be <6>
R2C9 can only be <9>
R2C1 can only be <7>
R3C9 can only be <6>
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