The full reasoning can be found below the Sudoku.
Jun 10 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R8C6 can only be <3>
R3C4 is the only square in row 3 that can be <1>
R4C7 is the only square in row 4 that can be <1>
R4C5 is the only square in row 4 that can be <2>
R4C2 is the only square in row 4 that can be <8>
R7C1 is the only square in row 7 that can be <1>
R7C6 is the only square in row 7 that can be <8>
R5C5 is the only square in row 5 that can be <8>
R7C4 is the only square in row 7 that can be <7>
R8C5 is the only square in row 8 that can be <1>
R8C3 is the only square in row 8 that can be <4>
R6C2 is the only square in row 6 that can be <4>
R5C9 is the only square in row 5 that can be <4>
R9C7 is the only square in row 9 that can be <8>
Intersection of column 7 with block 3. The value <3> 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 <3> from <239> leaving <29>
R3C8 - removing <3> from <23569> leaving <2569>
R3C9 - removing <3> from <2359> leaving <259>
Intersection of row 3 with block 1. The value <3> only appears in one or more of squares R3C1, R3C2 and R3C3 of row 3. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R1C1 - removing <3> from <3679> leaving <679>
R1C3 - removing <3> from <23679> leaving <2679>
R2C3 - removing <3> from <23679> leaving <2679>
Intersection of column 9 with block 9. The value <3> only appears in one or more of squares R7C9, R8C9 and R9C9 of column 9. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C8 - removing <3> from <239> leaving <29>
Intersection of block 5 with row 5. The value <6> only appears in one or more of squares R5C4, R5C5 and R5C6 of block 5. These squares are the ones that intersect with row 5. Thus, the other (non-intersecting) squares of row 5 cannot contain this value.
R5C1 - removing <6> from <3679> leaving <379>
R5C2 - removing <6> from <369> leaving <39>
R5C8 - removing <6> from <3569> leaving <359>
R3C2 is the only square in column 2 that can be <6>
R3C6 can only be <2>
R1C7 is the only square in row 1 that can be <6>
R1C5 is the only square in row 1 that can be <3>
R2C3 is the only square in row 2 that can be <2>
R1C9 is the only square in row 1 that can be <2>
R2C7 is the only square in row 2 that can be <3>
R3C1 is the only square in row 3 that can be <3>
R7C8 is the only square in row 7 that can be <2>
R9C1 is the only square in column 1 that can be <6>
Squares R6C7 and R9C5 form a remote naked pair. <59> can be removed from any square that is common to their groups.
R6C5 - removing <5> from <57> leaving <7>
R2C5 can only be <9>
R5C6 can only be <6>
R2C4 can only be <6>
R9C5 can only be <5>
R5C4 can only be <5>
R2C6 can only be <7>
R8C4 can only be <9>
R8C7 can only be <5>
R6C7 can only be <9>
R6C3 can only be <6>
R5C8 can only be <3>
R5C2 can only be <9>
R4C8 can only be <6>
R6C8 can only be <5>
R4C3 can only be <3>
R3C8 can only be <9>
R3C9 can only be <5>
R9C3 can only be <9>
R5C1 can only be <7>
R7C2 can only be <3>
R7C9 can only be <9>
R9C9 can only be <3>
R1C3 can only be <7>
R1C1 can only be <9>
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