The full reasoning can be found below the Sudoku.
Apr 04 - Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C1 is the only square in row 1 that can be <1>
R9C1 can only be <7>
R8C3 can only be <1>
R1C5 is the only square in row 1 that can be <4>
R1C2 is the only square in row 1 that can be <7>
R2C7 is the only square in row 2 that can be <3>
R2C1 is the only square in row 2 that can be <4>
R4C3 is the only square in row 4 that can be <7>
R8C5 is the only square in row 8 that can be <7>
R8C2 is the only square in row 8 that can be <6>
R7C2 can only be <3>
R8C9 is the only square in row 8 that can be <3>
Intersection of column 4 with block 8. The value <1> only appears in one or more of squares R7C4, R8C4 and R9C4 of column 4. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R7C5 - removing <1> from <1569> leaving <569>
R9C5 - removing <1> from <189> leaving <89>
Intersection of column 7 with block 6. The value <2> 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.
R4C8 - removing <2> from <2459> leaving <459>
R4C9 - removing <2> from <2589> leaving <589>
R5C9 - removing <2> from <12689> leaving <1689>
R6C9 - removing <2> from <125689> leaving <15689>
Intersection of column 8 with block 9. The value <1> 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.
R7C9 - removing <1> from <159> leaving <59>
R9C9 - removing <1> from <1289> leaving <289>
Squares R1C9<25>, R4C9<589>, R7C9<59> and R9C9<289> in column 9 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2589>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C9 - removing <89> from <1689> leaving <16>
R6C9 - removing <589> from <15689> leaving <16>
Intersection of block 6 with row 4. The values <49> only appears in one or more of squares R4C7, R4C8 and R4C9 of block 6. These squares are the ones that intersect with row 4. Thus, the other (non-intersecting) squares of row 4 cannot contain these values.
R4C4 - removing <9> from <89> leaving <8>
R4C5 - removing <9> from <2389> leaving <238>
R9C9 is the only square in column 9 that can be <8>
R9C5 can only be <9>
R9C4 can only be <1>
R9C8 can only be <2>
R7C4 can only be <6>
R7C5 can only be <5>
R2C4 can only be <9>
R7C9 can only be <9>
R8C6 can only be <8>
R7C8 can only be <1>
R4C9 can only be <5>
R4C1 can only be <3>
R1C9 can only be <2>
R1C6 can only be <5>
R4C5 can only be <2>
R4C7 can only be <4>
R5C5 can only be <1>
R6C6 can only be <9>
R4C8 can only be <9>
R8C7 can only be <5>
R5C9 can only be <6>
R6C5 can only be <3>
R5C1 can only be <8>
R6C9 can only be <1>
R8C8 can only be <4>
R3C6 can only be <2>
R5C7 can only be <2>
R3C1 can only be <5>
R6C2 can only be <2>
R5C3 can only be <9>
R6C7 can only be <8>
R6C3 can only be <5>
R2C2 can only be <8>
R6C1 can only be <6>
R2C3 can only be <2>
R2C5 can only be <6>
R3C2 can only be <9>
R2C8 can only be <5>
R3C5 can only be <8>
R3C8 can only be <6>
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