The full reasoning can be found below the Sudoku.
May 05 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C1 can only be <2>
R1C5 can only be <1>
R1C2 is the only square in row 1 that can be <6>
R8C7 is the only square in row 8 that can be <6>
R3C2 is the only square in column 2 that can be <4>
R3C3 is the only square in row 3 that can be <3>
R8C6 is the only square in row 8 that can be <3>
R4C5 is the only square in row 4 that can be <3>
R8C4 is the only square in row 8 that can be <4>
R9C2 is the only square in row 9 that can be <3>
R9C5 is the only square in row 9 that can be <2>
R7C6 can only be <8>
R7C4 can only be <5>
R9C8 is the only square in row 9 that can be <5>
R3C8 can only be <9>
R2C9 can only be <1>
R2C6 is the only square in row 2 that can be <9>
R2C4 is the only square in row 2 that can be <6>
R2C7 is the only square in row 2 that can be <2>
R3C7 can only be <5>
R6C6 is the only square in row 6 that can be <6>
R6C8 is the only square in row 6 that can be <4>
R1C8 can only be <8>
R1C9 can only be <4>
R4C6 is the only square in row 4 that can be <4>
R8C3 is the only square in row 8 that can be <1>
Intersection of row 5 with block 5. The value <2> only appears in one or more of squares R5C4, R5C5 and R5C6 of row 5. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R4C4 - removing <2> from <1278> leaving <178>
Intersection of column 3 with block 4. The value <8> only appears in one or more of squares R4C3, R5C3 and R6C3 of column 3. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R5C1 - removing <8> from <5789> leaving <579>
Intersection of column 7 with block 6. The value <8> only appears in one or more of squares R4C7, R5C7 and R6C7 of column 7. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R5C9 - removing <8> from <789> leaving <79>
R5C4 is the only square in row 5 that can be <8>
R5C6 is the only square in row 5 that can be <2>
R3C6 can only be <7>
R3C4 can only be <2>
Squares R4C4 and R4C8 in row 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C2 - removing <7> from <279> leaving <29>
R4C3 - removing <7> from <278> leaving <28>
R4C7 - removing <1> from <189> leaving <89>
Squares R5C1 and R5C9 in row 5 and R9C1 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in columns 1 and 9 can be removed.
R2C1 - removing <7> from <57> leaving <5>
R2C3 can only be <7>
R7C3 can only be <2>
R4C3 can only be <8>
R4C7 can only be <9>
R6C3 can only be <5>
R4C2 can only be <2>
R7C7 can only be <1>
R5C9 can only be <7>
R5C1 can only be <9>
R9C9 can only be <8>
R4C8 can only be <1>
R6C5 can only be <9>
R6C2 can only be <7>
R5C5 can only be <5>
R7C8 can only be <7>
R6C7 can only be <8>
R7C2 can only be <9>
R9C1 can only be <7>
R8C9 can only be <9>
R4C4 can only be <7>
R8C1 can only be <8>
R6C4 can only be <1>
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