The full reasoning can be found below the Sudoku.
May 06 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C4 can only be <1>
R6C7 can only be <1>
R4C6 can only be <5>
R4C7 can only be <4>
R5C5 can only be <9>
R6C6 can only be <7>
R4C3 can only be <3>
R6C4 can only be <2>
R7C6 can only be <1>
R6C3 can only be <5>
R5C8 can only be <2>
R5C9 can only be <6>
R3C6 can only be <9>
R2C1 is the only square in row 2 that can be <2>
R2C3 is the only square in row 2 that can be <8>
R3C2 is the only square in row 3 that can be <5>
R1C9 is the only square in row 1 that can be <5>
R9C5 is the only square in row 9 that can be <5>
R8C9 is the only square in column 9 that can be <4>
Squares R3C4 and R3C7 in row 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <37>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R3C8 - removing <37> from <1347> leaving <14>
Squares R7C4 and R7C8 in row 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <37>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C2 - removing <7> from <479> leaving <49>
R7C7 - removing <37> from <3679> leaving <69>
Intersection of row 8 with block 7. The value <1> only appears in one or more of squares R8C1, R8C2 and R8C3 of row 8. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C1 - removing <1> from <137> leaving <37>
R9C2 - removing <1> from <179> leaving <79>
Intersection of block 1 with row 1. The value <7> 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 <7> from <137> leaving <13>
R1C8 - removing <7> from <1347> leaving <134>
Intersection of column 8 with block 9. The value <7> only appears in one or more of squares R7C8, R8C8 and R9C8 of column 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R8C7 - removing <7> from <367> leaving <36>
Squares R3C7 and R8C5 form a remote naked pair. <37> can be removed from any square that is common to their groups.
R8C7 - removing <3> from <36> leaving <6>
R8C3 can only be <1>
R7C7 can only be <9>
R7C2 can only be <4>
R2C7 can only be <7>
R9C9 can only be <1>
R3C3 can only be <4>
R2C9 can only be <9>
R2C5 can only be <1>
R3C7 can only be <3>
R3C8 can only be <1>
R7C3 can only be <6>
R3C4 can only be <7>
R1C8 can only be <4>
R5C2 can only be <1>
R1C5 can only be <3>
R7C4 can only be <3>
R5C1 can only be <4>
R1C2 can only be <7>
R7C8 can only be <7>
R8C5 can only be <7>
R9C8 can only be <3>
R8C1 can only be <3>
R9C1 can only be <7>
R1C1 can only be <1>
R9C2 can only be <9>
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