The full reasoning can be found below the Sudoku.
May 04 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R3C4 can only be <1>
R4C8 can only be <4>
R6C5 can only be <1>
R1C1 is the only square in row 1 that can be <1>
R1C9 is the only square in row 1 that can be <2>
R3C3 is the only square in row 3 that can be <3>
R5C1 is the only square in row 5 that can be <4>
R6C3 is the only square in row 6 that can be <5>
R7C5 is the only square in row 7 that can be <3>
R7C4 is the only square in row 7 that can be <2>
R8C5 is the only square in row 8 that can be <5>
R3C6 is the only square in row 3 that can be <5>
R1C8 is the only square in row 1 that can be <5>
R8C3 is the only square in row 8 that can be <9>
R9C6 is the only square in row 9 that can be <1>
R9C9 is the only square in row 9 that can be <4>
R2C9 is the only square in column 9 that can be <9>
Squares R4C5 and R5C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <26>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C5 - removing <6> from <6789> leaving <789>
R2C5 - removing <6> from <4678> leaving <478>
R9C5 - removing <6> from <689> leaving <89>
Squares R2C7 and R3C7 in column 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C7 - removing <8> from <1278> leaving <127>
R7C7 - removing <8> from <678> leaving <67>
R8C7 - removing <8> from <68> leaving <6>
R8C1 can only be <3>
R7C7 can only be <7>
R6C7 can only be <9>
R8C9 can only be <8>
R5C9 can only be <3>
R9C8 can only be <3>
R6C8 can only be <7>
R6C2 can only be <3>
R5C8 can only be <8>
R4C2 is the only square in row 4 that can be <9>
Intersection of row 2 with block 1. The value <6> only appears in one or more of squares R2C1, R2C2 and R2C3 of row 2. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R1C2 - removing <6> from <678> leaving <78>
Squares R3C5, R3C7, R2C5 and R2C7 form a Type-1 Unique Rectangle on <48>.
R2C5 - removing <48> from <478> leaving <7>
R2C1 can only be <6>
R2C3 can only be <8>
R9C1 can only be <7>
R2C7 can only be <4>
R7C3 can only be <6>
R1C2 can only be <7>
R3C7 can only be <8>
R3C5 can only be <4>
R7C6 can only be <8>
R4C3 can only be <1>
R1C6 can only be <6>
R9C5 can only be <9>
R9C2 can only be <8>
R9C4 can only be <6>
R1C5 can only be <8>
R5C2 can only be <6>
R1C4 can only be <9>
R4C7 can only be <2>
R5C3 can only be <7>
R4C5 can only be <6>
R5C7 can only be <1>
R5C5 can only be <2>
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