The full reasoning can be found below the Sudoku.
May 13 - Hard
Puzzle Copyright © Kevin Stone
Reasoning
R2C3 is the only square in row 2 that can be <8>
R3C4 is the only square in row 3 that can be <8>
R6C5 is the only square in row 6 that can be <8>
R6C7 is the only square in row 6 that can be <9>
R2C8 is the only square in row 2 that can be <9>
R3C6 is the only square in row 3 that can be <9>
R5C5 is the only square in row 5 that can be <9>
R7C3 is the only square in row 7 that can be <9>
R9C7 is the only square in row 9 that can be <8>
Intersection of row 8 with block 7. The value <5> only appears in one or more of squares R8C1, R8C2 and R8C3 of row 8. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C2 - removing <5> from <2345> leaving <234>
R9C3 - removing <5> from <234567> leaving <23467>
Intersection of row 8 with block 9. The value <1> only appears in one or more of squares R8C7, R8C8 and R8C9 of row 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C7 - removing <1> from <12367> leaving <2367>
R7C9 - removing <1> from <1237> leaving <237>
Intersection of row 9 with block 7. The value <4> only appears in one or more of squares R9C1, R9C2 and R9C3 of row 9. 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 <4> from <23457> leaving <2357>
R8C2 - removing <4> from <2345> leaving <235>
R8C3 - removing <4> from <23457> leaving <2357>
R4C1 is the only square in column 1 that can be <4>
R5C7 is the only square in row 5 that can be <4>
R5C8 is the only square in row 5 that can be <7>
R8C9 is the only square in row 8 that can be <4>
Intersection of row 5 with block 4. The value <3> only appears in one or more of squares R5C1, R5C2 and R5C3 of row 5. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R4C3 - removing <3> from <1235> leaving <125>
Intersection of block 2 with row 1. The values <123> only appears in one or more of squares R1C4, R1C5 and R1C6 of block 2. These squares are the ones that intersect with row 1. Thus, the other (non-intersecting) squares of row 1 cannot contain these values.
R1C2 - removing <123> from <12345> leaving <45>
R1C3 - removing <123> from <123457> leaving <457>
R1C7 - removing <123> from <123567> leaving <567>
R1C8 - removing <13> from <136> leaving <6>
R9C8 can only be <3>
R9C5 can only be <2>
R8C8 can only be <1>
R9C2 can only be <4>
R9C4 can only be <5>
R4C5 can only be <1>
R1C5 can only be <3>
R1C2 can only be <5>
R9C6 can only be <7>
R5C4 can only be <2>
R9C3 can only be <6>
R7C6 can only be <1>
R1C7 can only be <7>
R1C3 can only be <4>
R8C7 can only be <2>
R5C6 can only be <5>
R1C4 can only be <1>
R7C4 can only be <3>
R1C6 can only be <2>
R8C2 can only be <3>
R7C7 can only be <6>
R7C9 can only be <7>
R7C1 can only be <2>
R5C2 can only be <1>
R5C3 can only be <3>
R2C2 can only be <2>
R6C3 can only be <5>
R6C1 can only be <6>
R6C9 can only be <1>
R4C3 can only be <2>
R8C3 can only be <7>
R3C1 can only be <3>
R8C1 can only be <5>
R2C9 can only be <3>
R3C3 can only be <1>
R2C1 can only be <7>
R2C7 can only be <1>
R4C9 can only be <5>
R3C7 can only be <5>
R3C9 can only be <2>
R4C7 can only be <3>
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