The full reasoning can be found below the Sudoku.
Oct 31 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C8 can only be <2>
R6C2 can only be <8>
R4C9 can only be <1>
R4C7 can only be <8>
R4C2 can only be <4>
R5C1 can only be <2>
R6C1 can only be <9>
R6C3 can only be <6>
R3C3 is the only square in row 3 that can be <1>
R2C5 is the only square in row 2 that can be <1>
R5C4 is the only square in row 5 that can be <4>
R6C6 is the only square in row 6 that can be <2>
R3C6 can only be <9>
R1C9 is the only square in row 1 that can be <9>
R3C9 is the only square in row 3 that can be <2>
R2C4 is the only square in row 2 that can be <2>
R8C1 is the only square in row 8 that can be <1>
R9C7 is the only square in row 9 that can be <1>
R9C3 is the only square in row 9 that can be <2>
R8C8 is the only square in column 8 that can be <9>
R1C5 is the only square in block 2 that can be <7>
Squares R1C6 and R2C6 in column 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <68>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C6 - removing <8> from <358> leaving <35>
R8C6 - removing <68> from <3568> leaving <35>
R5C5 is the only square in row 5 that can be <8>
Squares R5C9 and R9C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <35>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C9 - removing <35> from <3456> leaving <46>
R8C9 - removing <35> from <3456> leaving <46>
Intersection of row 1 with block 1. The value <4> only appears in one or more of squares R1C1, R1C2 and R1C3 of row 1. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R2C1 - removing <4> from <4578> leaving <578>
Intersection of column 3 with block 7. The value <3> only appears in one or more of squares R7C3, R8C3 and R9C3 of column 3. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R8C2 - removing <3> from <357> leaving <57>
Squares R6C5 and R9C9 form a remote naked pair. <35> can be removed from any square that is common to their groups.
R9C5 - removing <35> from <359> leaving <9>
R9C4 can only be <8>
R4C5 can only be <6>
R4C4 can only be <9>
R9C1 can only be <5>
R9C9 can only be <3>
R8C2 can only be <7>
R5C9 can only be <5>
R5C6 can only be <3>
R6C8 can only be <3>
R6C5 can only be <5>
R8C4 can only be <6>
R3C2 can only be <3>
R7C1 can only be <4>
R8C9 can only be <4>
R7C4 can only be <7>
R2C9 can only be <6>
R7C8 can only be <5>
R2C6 can only be <8>
R1C7 can only be <5>
R3C7 can only be <7>
R2C2 can only be <5>
R8C6 can only be <5>
R8C5 can only be <3>
R7C3 can only be <3>
R1C1 can only be <8>
R8C3 can only be <8>
R7C7 can only be <6>
R2C8 can only be <4>
R1C3 can only be <4>
R1C6 can only be <6>
R2C1 can only be <7>
R2C7 can only be <3>
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