The full reasoning can be found below the Sudoku.
Apr 06 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R7C7 can only be <3>
R7C5 can only be <4>
R3C5 can only be <5>
R9C6 can only be <1>
R2C5 can only be <6>
R1C4 can only be <4>
R1C6 can only be <2>
R2C1 is the only square in row 2 that can be <9>
R3C9 is the only square in row 3 that can be <2>
R5C7 is the only square in row 5 that can be <1>
R3C7 can only be <8>
R3C3 can only be <4>
R3C1 can only be <1>
R1C9 is the only square in row 1 that can be <1>
R7C1 is the only square in row 7 that can be <2>
R8C2 is the only square in row 8 that can be <1>
R9C2 is the only square in column 2 that can be <4>
R8C8 is the only square in row 8 that can be <4>
R6C4 is the only square in column 4 that can be <5>
Squares R5C5 and R5C8 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <38>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C3 - removing <8> from <578> leaving <57>
R1C3 is the only square in column 3 that can be <8>
Intersection of column 2 with block 1. The value <3> only appears in one or more of squares R1C2, R2C2 and R3C2 of column 2. 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 <357> leaving <57>
Intersection of block 6 with column 9. The value <6> only appears in one or more of squares R4C9, R5C9 and R6C9 of block 6. These squares are the ones that intersect with column 9. Thus, the other (non-intersecting) squares of column 9 cannot contain this value.
R9C9 - removing <6> from <56789> leaving <5789>
Squares R5C8 and R9C4 form a remote naked pair. <38> can be removed from any square that is common to their groups.
R9C8 - removing <8> from <568> leaving <56>
R5C8 is the only square in column 8 that can be <8>
R5C5 can only be <3>
R6C9 can only be <6>
R6C6 can only be <4>
R4C9 can only be <3>
R4C4 can only be <8>
R8C5 can only be <8>
R6C1 can only be <8>
R4C6 can only be <6>
R8C9 can only be <7>
R9C4 can only be <3>
R8C1 can only be <3>
R2C9 can only be <5>
R9C7 can only be <6>
R9C8 can only be <5>
R1C7 can only be <7>
R9C1 can only be <7>
R2C8 can only be <3>
R7C9 can only be <9>
R1C1 can only be <5>
R1C8 can only be <6>
R2C2 can only be <7>
R4C1 can only be <4>
R7C3 can only be <5>
R9C9 can only be <8>
R9C3 can only be <9>
R1C2 can only be <3>
R5C2 can only be <5>
R5C3 can only be <7>
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