The full reasoning can be found below the Sudoku.
Jun 08 - Hard
Puzzle Copyright © Kevin Stone
Reasoning
R2C5 is the only square in row 2 that can be <1>
R2C3 is the only square in row 2 that can be <7>
R8C5 is the only square in row 8 that can be <8>
R6C1 is the only square in row 6 that can be <8>
R6C7 is the only square in row 6 that can be <4>
R3C8 is the only square in row 3 that can be <4>
R9C2 is the only square in row 9 that can be <3>
R2C1 is the only square in row 2 that can be <3>
R7C7 is the only square in row 7 that can be <3>
R7C8 is the only square in row 7 that can be <2>
R1C5 is the only square in row 1 that can be <2>
R1C6 is the only square in row 1 that can be <7>
R9C6 can only be <5>
R7C6 can only be <6>
R3C3 is the only square in column 3 that can be <2>
R7C5 is the only square in column 5 that can be <4>
R9C5 is the only square in column 5 that can be <7>
Squares R4C1 and R4C7 in row 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <59>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C3 - removing <59> from <13569> leaving <136>
R4C5 - removing <59> from <3569> leaving <36>
R4C9 - removing <59> from <1569> leaving <16>
Squares R7C4 and R9C4 in column 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <19>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C4 - removing <9> from <5689> leaving <568>
Intersection of row 7 with block 7. The value <5> only appears in one or more of squares R7C1, R7C2 and R7C3 of row 7. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R8C1 - removing <5> from <459> leaving <49>
R8C2 - removing <5> from <14569> leaving <1469>
R8C3 - removing <5> from <1569> leaving <169>
Intersection of column 1 with block 4. The value <5> only appears in one or more of squares R4C1, R5C1 and R6C1 of column 1. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R5C2 - removing <5> from <14569> leaving <1469>
R5C3 - removing <5> from <13569> leaving <1369>
R6C3 - removing <5> from <569> leaving <69>
R7C3 is the only square in column 3 that can be <5>
Intersection of block 1 with column 2. The values <589> only appears in one or more of squares R1C2, R2C2 and R3C2 of block 1. These squares are the ones that intersect with column 2. Thus, the other (non-intersecting) squares of column 2 cannot contain these values.
R5C2 - removing <9> from <1469> leaving <146>
R7C2 - removing <9> from <19> leaving <1>
R8C2 - removing <9> from <1469> leaving <146>
R7C4 can only be <9>
R9C4 can only be <1>
R9C8 can only be <9>
R1C8 can only be <5>
R1C4 can only be <8>
R2C8 can only be <6>
R2C9 can only be <2>
R2C7 can only be <8>
R1C2 can only be <9>
R3C6 can only be <3>
R2C2 can only be <5>
R3C7 can only be <9>
R5C6 can only be <8>
R4C7 can only be <5>
R4C1 can only be <9>
R8C7 can only be <7>
R8C8 can only be <1>
R5C7 can only be <2>
R8C9 can only be <5>
R5C8 can only be <7>
R3C2 can only be <8>
R8C1 can only be <4>
R6C3 can only be <6>
R6C9 can only be <9>
R8C3 can only be <9>
R5C2 can only be <4>
R6C5 can only be <5>
R8C2 can only be <6>
R5C1 can only be <5>
R5C4 can only be <6>
R5C9 can only be <1>
R3C4 can only be <5>
R4C5 can only be <3>
R5C3 can only be <3>
R4C9 can only be <6>
R3C5 can only be <6>
R4C3 can only be <1>
R5C5 can only be <9>
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