The full reasoning can be found below the Sudoku.
Apr 26 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C6 can only be <8>
R4C1 can only be <2>
R4C9 can only be <8>
R5C4 can only be <8>
R6C5 can only be <6>
R6C9 can only be <3>
R9C6 can only be <2>
R4C5 can only be <4>
R5C5 can only be <2>
R9C4 can only be <5>
R5C6 can only be <7>
R5C3 can only be <1>
R6C1 can only be <7>
R1C4 can only be <6>
R5C2 can only be <3>
R2C1 is the only square in row 2 that can be <3>
R1C7 is the only square in row 1 that can be <3>
R2C2 is the only square in row 2 that can be <7>
R3C1 is the only square in row 3 that can be <6>
R3C7 is the only square in row 3 that can be <8>
Intersection of row 2 with block 3. The value <4> only appears in one or more of squares R2C7, R2C8 and R2C9 of row 2. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C8 - removing <4> from <459> leaving <59>
R1C9 - removing <4> from <12459> leaving <1259>
Squares R1C8 and R5C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <59>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C8 - removing <5> from <45> leaving <4>
R8C8 - removing <59> from <45689> leaving <468>
R9C8 - removing <9> from <4689> leaving <468>
Intersection of column 7 with block 9. The value <1> only appears in one or more of squares R7C7, R8C7 and R9C7 of column 7. 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 <1579> leaving <579>
R9C9 - removing <1> from <1479> leaving <479>
Intersection of block 1 with row 1. The values <14> 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 these values.
R1C9 - removing <1> from <1259> leaving <259>
Squares R2C5, R2C9, R3C5 and R3C9 form a Type-1 Unique Rectangle on <15>.
R3C9 - removing <15> from <125> leaving <2>
R3C3 can only be <5>
R3C5 can only be <1>
R7C3 can only be <7>
R1C1 can only be <1>
R2C5 can only be <5>
R9C3 can only be <4>
R1C3 can only be <2>
R8C2 can only be <6>
R1C2 can only be <4>
R2C9 can only be <1>
R8C8 can only be <8>
R9C2 can only be <1>
R8C5 can only be <9>
R9C8 can only be <6>
R9C7 can only be <9>
R9C1 can only be <8>
R9C9 can only be <7>
R5C7 can only be <5>
R7C9 can only be <5>
R5C8 can only be <9>
R7C7 can only be <1>
R1C8 can only be <5>
R1C9 can only be <9>
R8C9 can only be <4>
R8C1 can only be <5>
R7C5 can only be <8>
R7C1 can only be <9>
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