The full reasoning can be found below the Sudoku.
Sep 04 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C1 is the only square in row 2 that can be <1>
R2C2 is the only square in row 2 that can be <8>
R8C9 is the only square in row 8 that can be <1>
R8C2 is the only square in column 2 that can be <7>
R4C3 is the only square in column 3 that can be <1>
R6C3 is the only square in column 3 that can be <4>
R2C8 is the only square in column 8 that can be <5>
R4C5 is the only square in column 5 that can be <5>
Squares R7C4<479>, R7C6<79> and R8C5<49> in block 8 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <479>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R9C4 - removing <9> from <589> leaving <58>
R9C6 - removing <9> from <589> leaving <58>
Intersection of row 9 with block 9. The value <9> only appears in one or more of squares R9C7, R9C8 and R9C9 of row 9. 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 <9> from <249> leaving <24>
R8C8 - removing <9> from <469> leaving <46>
Squares R1C2, R5C2 and R9C2 in column 2, R1C3 and R9C3 in column 3 and R1C8, R5C8 and R9C8 in column 8 form a Swordfish pattern on possibility <2>. All other instances of this possibility in rows 1, 5 and 9 can be removed.
R5C5 - removing <2> from <2349> leaving <349>
R1C6 - removing <2> from <1269> leaving <169>
R5C6 - removing <2> from <129> leaving <19>
R1C7 - removing <2> from <2349> leaving <349>
R9C7 - removing <2> from <29> leaving <9>
Intersection of column 7 with block 6. The value <2> only appears in one or more of squares R4C7, R5C7 and R6C7 of column 7. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R4C9 - removing <2> from <2346> leaving <346>
R5C8 - removing <2> from <249> leaving <49>
R6C9 - removing <2> from <2369> leaving <369>
R5C2 is the only square in row 5 that can be <2>
R5C5 is the only square in row 5 that can be <3>
R8C5 is the only square in column 5 that can be <4>
R8C8 can only be <6>
R8C1 can only be <9>
R9C8 can only be <2>
R9C3 can only be <3>
R7C9 can only be <4>
R7C1 can only be <2>
R9C2 can only be <6>
R1C3 can only be <2>
R1C2 can only be <3>
R1C7 can only be <4>
R3C1 can only be <6>
R1C8 can only be <9>
R5C8 can only be <4>
R2C9 can only be <2>
R2C5 can only be <9>
R3C9 can only be <3>
R3C4 can only be <5>
R3C6 can only be <2>
R9C4 can only be <8>
R4C9 can only be <6>
R4C6 can only be <7>
R6C9 can only be <9>
R6C5 can only be <2>
R9C6 can only be <5>
R4C1 can only be <3>
R4C4 can only be <4>
R7C6 can only be <9>
R6C4 can only be <6>
R6C6 can only be <8>
R1C4 can only be <1>
R6C7 can only be <3>
R6C1 can only be <7>
R4C7 can only be <2>
R7C4 can only be <7>
R5C6 can only be <1>
R1C6 can only be <6>
R5C4 can only be <9>
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