The full reasoning can be found below the Sudoku.
Apr 24 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C3 is the only square in row 2 that can be <3>
R3C1 is the only square in row 3 that can be <9>
R1C7 is the only square in row 1 that can be <9>
R4C1 is the only square in row 4 that can be <4>
R5C7 is the only square in row 5 that can be <4>
R5C3 is the only square in row 5 that can be <9>
R7C6 is the only square in row 7 that can be <9>
R8C2 is the only square in row 8 that can be <7>
R8C3 is the only square in column 3 that can be <4>
Intersection of row 8 with block 9. The value <6> 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.
R7C8 - removing <6> from <1368> leaving <138>
R7C9 - removing <6> from <12368> leaving <1238>
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 <568> leaving <68>
Intersection of block 1 with row 1. The value <2> only appears in one or more of squares R1C1, R1C2 and R1C3 of block 1. These squares are the ones that intersect with row 1. Thus, the other (non-intersecting) squares of row 1 cannot contain this value.
R1C5 - removing <2> from <126> leaving <16>
R1C9 - removing <2> from <1268> leaving <168>
Squares R9C1<128>, R9C3<28> and R9C7<12> in row 9 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <128>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C5 - removing <12> from <1235> leaving <35>
R9C9 - removing <128> from <12358> leaving <35>
Intersection of row 9 with block 7. The value <8> 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.
R7C1 - removing <8> from <1268> leaving <126>
R7C2 - removing <8> from <168> leaving <16>
Squares R4C5 and R4C9 in row 4 and R9C5 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in columns 5 and 9 can be removed.
R5C5 - removing <3> from <23567> leaving <2567>
R5C9 - removing <3> from <1367> leaving <167>
R7C5 - removing <3> from <1234> leaving <124>
R7C9 - removing <3> from <1238> leaving <128>
Squares R1C5 and R1C9 in row 1 and R4C5 and R4C9 in row 4 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 5 and 9 can be removed.
R5C5 - removing <6> from <2567> leaving <257>
R5C9 - removing <6> from <167> leaving <17>
R6C5 - removing <6> from <567> leaving <57>
R6C9 - removing <6> from <67> leaving <7>
R6C5 can only be <5>
R5C9 can only be <1>
R6C1 can only be <6>
R9C5 can only be <3>
R9C9 can only be <5>
R4C5 can only be <6>
R4C9 can only be <3>
R1C5 can only be <1>
R5C8 can only be <6>
R5C2 can only be <8>
R8C8 can only be <1>
R8C4 can only be <2>
R9C7 can only be <2>
R9C3 can only be <8>
R8C7 can only be <6>
R7C9 can only be <8>
R5C1 can only be <5>
R7C8 can only be <3>
R1C9 can only be <6>
R3C9 can only be <2>
R8C6 can only be <5>
R2C4 can only be <6>
R3C4 can only be <4>
R5C4 can only be <3>
R7C5 can only be <4>
R2C7 can only be <1>
R9C1 can only be <1>
R1C3 can only be <2>
R1C1 can only be <8>
R2C2 can only be <5>
R7C4 can only be <1>
R3C5 can only be <7>
R7C1 can only be <2>
R7C2 can only be <6>
R2C8 can only be <8>
R3C2 can only be <1>
R2C6 can only be <2>
R3C8 can only be <5>
R3C6 can only be <8>
R5C5 can only be <2>
R5C6 can only be <7>
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