The full reasoning can be found below the Sudoku.
Jun 15 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C5 is the only square in row 1 that can be <2>
R2C2 is the only square in row 2 that can be <1>
R6C3 is the only square in column 3 that can be <5>
R2C3 is the only square in column 3 that can be <6>
R2C1 is the only square in row 2 that can be <2>
R5C3 is the only square in column 3 that can be <2>
R5C1 is the only square in row 5 that can be <4>
R8C3 is the only square in column 3 that can be <9>
R1C8 is the only square in column 8 that can be <4>
R1C2 can only be <9>
R1C6 can only be <5>
R1C7 can only be <6>
Squares R2C7 and R2C9 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <57>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C4 - removing <7> from <4789> leaving <489>
R2C5 - removing <7> from <4789> leaving <489>
R2C6 - removing <7> from <79> leaving <9>
Squares R3C2 and R3C3 in row 3 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 row.
R3C5 - removing <48> from <3478> leaving <37>
Squares R5C9 and R6C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <39>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C9 - removing <39> from <2359> leaving <25>
R8C9 - removing <3> from <237> leaving <27>
R8C7 is the only square in block 9 that can be <3>
Intersection of row 8 with block 8. The values <46> only appears in one or more of squares R8C4, R8C5 and R8C6 of row 8. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain these values.
R9C4 - removing <4> from <3457> leaving <357>
R9C5 - removing <4> from <347> leaving <37>
Squares R3C5 and R9C5 in column 5 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 column.
R4C5 - removing <37> from <36789> leaving <689>
R6C5 - removing <37> from <136789> leaving <1689>
R7C5 - removing <7> from <179> leaving <19>
R8C5 - removing <7> from <1467> leaving <146>
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.
R4C8 - removing <8> from <2568> leaving <256>
R6C8 - removing <8> from <168> leaving <16>
Squares R3C2, R3C3, R9C2 and R9C3 form a Type-1 Unique Rectangle on <48>.
R9C2 - removing <48> from <478> leaving <7>
R9C5 can only be <3>
R8C1 can only be <8>
R9C4 can only be <5>
R3C5 can only be <7>
R3C6 can only be <3>
R5C6 can only be <1>
R5C7 can only be <8>
R4C7 can only be <5>
R9C3 can only be <4>
R3C3 can only be <8>
R9C8 can only be <8>
R3C2 can only be <4>
R4C9 can only be <2>
R2C7 can only be <7>
R4C8 can only be <6>
R8C9 can only be <7>
R8C4 can only be <4>
R8C6 can only be <6>
R2C9 can only be <5>
R7C7 can only be <1>
R6C8 can only be <1>
R7C8 can only be <5>
R8C8 can only be <2>
R7C5 can only be <9>
R2C4 can only be <8>
R8C5 can only be <1>
R6C6 can only be <7>
R2C5 can only be <4>
R6C1 can only be <9>
R7C4 can only be <7>
R4C5 can only be <8>
R4C2 can only be <3>
R6C5 can only be <6>
R6C9 can only be <3>
R4C1 can only be <7>
R6C2 can only be <8>
R5C9 can only be <9>
R4C4 can only be <9>
R5C4 can only be <3>
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