The full reasoning can be found below the Sudoku.
Jun 08 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R5C5 can only be <6>
R2C5 is the only square in row 2 that can be <8>
R6C1 is the only square in row 6 that can be <6>
R7C6 is the only square in row 7 that can be <8>
R7C5 is the only square in row 7 that can be <1>
R7C4 is the only square in row 7 that can be <6>
R9C6 can only be <3>
R3C6 can only be <6>
R1C6 can only be <1>
R1C9 is the only square in row 1 that can be <6>
R1C8 is the only square in row 1 that can be <4>
R9C9 is the only square in row 9 that can be <4>
R9C2 is the only square in row 9 that can be <6>
R5C8 is the only square in column 8 that can be <3>
R8C8 is the only square in column 8 that can be <8>
R3C7 is the only square in block 3 that can be <9>
R4C7 can only be <2>
Squares R2C8 and R2C9 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <27>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C1 - removing <7> from <157> leaving <15>
Squares R2C2 and R5C2 in column 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C2 - removing <5> from <359> leaving <39>
R8C2 - removing <5> from <359> leaving <39>
Squares R2C1 and R2C2 in block 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C1 - removing <5> from <3579> leaving <379>
Intersection of column 3 with block 4. The values <49> only appears in one or more of squares R4C3, R5C3 and R6C3 of column 3. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain these values.
R4C1 - removing <9> from <389> leaving <38>
Squares R2C1, R2C2, R5C1 and R5C2 form a Type-1 Unique Rectangle on <15>.
R5C1 - removing <15> from <158> leaving <8>
R4C1 can only be <3>
R4C3 can only be <9>
R4C9 can only be <8>
R5C2 is the only square in row 5 that can be <1>
R2C2 can only be <5>
R2C1 can only be <1>
R6C9 is the only square in row 6 that can be <9>
R8C2 is the only square in row 8 that can be <3>
R1C2 can only be <9>
R1C1 can only be <7>
R1C5 can only be <5>
R3C3 can only be <3>
R1C4 can only be <3>
R3C4 can only be <2>
R3C5 can only be <7>
R9C4 can only be <5>
R9C1 can only be <9>
R9C5 can only be <2>
R8C1 can only be <5>
R9C8 can only be <7>
R8C5 can only be <9>
R2C8 can only be <2>
R7C7 can only be <5>
R2C9 can only be <7>
R5C9 can only be <5>
R5C3 can only be <4>
R8C9 can only be <2>
R6C7 can only be <4>
R6C3 can only be <5>
R5C7 can only be <7>
R7C3 can only be <7>
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