The full reasoning can be found below the Sudoku.
May 13 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R4C8 is the only square in row 4 that can be <5>
R4C9 is the only square in row 4 that can be <3>
R4C2 is the only square in row 4 that can be <7>
R8C8 is the only square in row 8 that can be <2>
R2C6 is the only square in column 6 that can be <3>
R2C4 is the only square in block 2 that can be <5>
R9C4 can only be <8>
R4C4 can only be <1>
Squares R7C5 and R8C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <16>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C5 - removing <6> from <268> leaving <28>
Squares R7C5 and R8C5 in block 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <16>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C6 - removing <6> from <4569> leaving <459>
R9C6 - removing <6> from <56> leaving <5>
Squares R8C4 and R8C6 in row 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <49>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C2 - removing <4> from <1456> leaving <156>
R8C3 - removing <4> from <14567> leaving <1567>
R8C7 - removing <9> from <5679> leaving <567>
Intersection of row 5 with block 6. The value <9> only appears in one or more of squares R5C7, R5C8 and R5C9 of row 5. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R6C8 - removing <9> from <179> leaving <17>
R6C9 - removing <9> from <679> leaving <67>
Intersection of block 2 with row 1. The value <4> 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 this value.
R1C3 - removing <4> from <1248> leaving <128>
R1C7 - removing <4> from <14> leaving <1>
Squares R2C5 and R2C8 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <78>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C2 - removing <8> from <1468> leaving <146>
R2C3 - removing <8> from <1468> leaving <146>
R2C7 - removing <7> from <47> leaving <4>
Intersection of block 9 with column 7. The values <57> only appears in one or more of squares R7C7, R8C7 and R9C7 of block 9. These squares are the ones that intersect with column 7. Thus, the other (non-intersecting) squares of column 7 cannot contain these values.
R3C7 - removing <7> from <379> leaving <39>
Squares R2C2 and R2C3 in row 2, R5C2, R5C3 and R5C7 in row 5 and R9C3 and R9C7 in row 9 form a Swordfish pattern on possibility <6>. All other instances of this possibility in columns 2, 3 and 7 can be removed.
R6C2 - removing <6> from <1246> leaving <124>
R7C2 - removing <6> from <14568> leaving <1458>
R7C3 - removing <6> from <14568> leaving <1458>
R7C7 - removing <6> from <3569> leaving <359>
R8C2 - removing <6> from <156> leaving <15>
R8C3 - removing <6> from <1567> leaving <157>
R8C7 - removing <6> from <567> leaving <57>
R8C5 is the only square in row 8 that can be <6>
R7C5 can only be <1>
R6C1 is the only square in column 1 that can be <1>
R6C8 can only be <7>
R6C9 can only be <6>
R2C8 can only be <8>
R6C6 can only be <9>
R7C9 can only be <9>
R5C7 can only be <9>
R7C8 can only be <3>
R3C9 can only be <7>
R2C5 can only be <7>
R5C8 can only be <1>
R3C7 can only be <3>
R6C4 can only be <2>
R8C6 can only be <4>
R7C7 can only be <5>
R3C8 can only be <9>
R8C4 can only be <9>
R1C6 can only be <8>
R1C3 can only be <2>
R4C6 can only be <6>
R3C5 can only be <2>
R5C5 can only be <8>
R1C4 can only be <4>
R4C1 can only be <8>
R6C2 can only be <4>
R8C7 can only be <7>
R9C7 can only be <6>
R9C3 can only be <7>
R5C3 can only be <6>
R3C1 can only be <4>
R5C2 can only be <2>
R2C3 can only be <1>
R7C2 can only be <8>
R7C3 can only be <4>
R3C2 can only be <5>
R7C1 can only be <6>
R2C2 can only be <6>
R8C3 can only be <5>
R3C3 can only be <8>
R8C2 can only be <1>
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