The full reasoning can be found below the Sudoku.
Jun 20 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C5 can only be <6>
R4C4 can only be <9>
R4C5 can only be <8>
R4C6 can only be <1>
R5C1 can only be <9>
R5C6 can only be <2>
R5C9 can only be <6>
R6C4 can only be <6>
R6C6 can only be <7>
R7C3 can only be <7>
R9C5 can only be <7>
R5C5 can only be <3>
R5C4 can only be <5>
R6C5 can only be <4>
R7C9 can only be <8>
R1C1 is the only square in row 1 that can be <8>
R7C7 is the only square in column 7 that can be <6>
R7C1 can only be <4>
Squares R2C8 and R8C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C8 - removing <1> from <149> leaving <49>
R9C8 - removing <1> from <149> leaving <49>
Squares R8C8<17>, R8C9<157> and R9C9<15> in block 9 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <157>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R9C7 - removing <5> from <459> leaving <49>
Intersection of column 7 with block 3. The value <5> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C9 - removing <5> from <125> leaving <12>
R3C9 - removing <5> from <235> leaving <23>
Squares R9C7, R9C8, R1C7 and R1C8 form a Type-1 Unique Rectangle on <49>.
R1C7 - removing <49> from <459> leaving <5>
R1C2 can only be <1>
R3C7 can only be <9>
R9C7 can only be <4>
R1C8 can only be <4>
R9C8 can only be <9>
R1C9 can only be <2>
R2C1 can only be <6>
R1C3 can only be <9>
R3C9 can only be <3>
R2C2 can only be <3>
R9C2 can only be <5>
R3C3 can only be <2>
R3C1 can only be <5>
R9C3 can only be <3>
R9C9 can only be <1>
R8C2 can only be <6>
R8C1 can only be <1>
R9C1 can only be <2>
R2C9 can only be <7>
R8C8 can only be <7>
R2C8 can only be <1>
R8C9 can only be <5>
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