The full reasoning can be found below the Sudoku.
May 03 - Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R8C4 is the only square in row 8 that can be <4>
R6C5 is the only square in row 6 that can be <4>
R5C1 is the only square in row 5 that can be <4>
R8C8 is the only square in row 8 that can be <8>
R3C9 is the only square in row 3 that can be <8>
R7C3 is the only square in row 7 that can be <8>
Intersection of row 3 with block 1. The values <35> only appears in one or more of squares R3C1, R3C2 and R3C3 of row 3. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain these values.
R1C3 - removing <3> from <123> leaving <12>
Intersection of row 4 with block 5. The values <59> only appears in one or more of squares R4C4, R4C5 and R4C6 of row 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain these values.
R5C4 - removing <5> from <1357> leaving <137>
R5C6 - removing <5> from <12356> leaving <1236>
Squares R2C4<179>, R2C6<19> and R3C5<179> in block 2 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <179>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C4 - removing <1> from <138> leaving <38>
R1C6 - removing <1> from <138> leaving <38>
Squares R1C3<12>, R4C3<27> and R6C3<127> in column 3 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <127>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C3 - removing <17> from <1357> leaving <35>
R9C3 - removing <7> from <357> leaving <35>
R9C1 is the only square in row 9 that can be <7>
R7C1 is the only square in column 1 that can be <9>
Squares R3C1 and R3C7 in row 3 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 row.
R3C2 - removing <1> from <1357> leaving <357>
R3C5 - removing <1> from <179> leaving <79>
R3C8 - removing <6> from <69> leaving <9>
R3C5 can only be <7>
R2C8 can only be <2>
R7C8 can only be <5>
R5C8 can only be <6>
R1C3 is the only square in row 1 that can be <2>
R4C3 can only be <7>
R4C7 can only be <2>
R6C3 can only be <1>
R4C5 can only be <9>
R6C7 can only be <7>
R5C9 can only be <5>
R5C2 can only be <2>
R6C4 can only be <8>
R4C4 can only be <5>
R5C5 can only be <1>
R5C6 can only be <3>
R7C5 can only be <2>
R5C4 can only be <7>
R1C6 can only be <8>
R6C6 can only be <2>
R1C4 can only be <3>
R7C9 can only be <4>
R7C7 can only be <3>
R1C9 can only be <6>
R1C1 can only be <1>
R9C9 can only be <2>
R3C7 can only be <1>
R3C1 can only be <6>
R1C7 can only be <4>
R4C6 can only be <6>
R9C4 can only be <9>
R7C2 can only be <1>
R9C7 can only be <6>
R9C6 can only be <5>
R2C4 can only be <1>
R9C3 can only be <3>
R8C6 can only be <1>
R2C2 can only be <7>
R2C6 can only be <9>
R8C2 can only be <5>
R3C2 can only be <3>
R3C3 can only be <5>
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