The full reasoning can be found below the Sudoku.
Jun 07 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R6C6 can only be <3>
R3C6 can only be <5>
R3C5 can only be <8>
R4C6 can only be <9>
R7C6 can only be <7>
R3C4 can only be <3>
R5C5 can only be <2>
R5C7 can only be <3>
R7C5 can only be <5>
R4C4 can only be <5>
R6C4 can only be <8>
R5C3 can only be <8>
R4C7 can only be <2>
R6C3 can only be <2>
R7C4 can only be <2>
R4C3 can only be <3>
R6C7 can only be <7>
R2C2 is the only square in row 2 that can be <4>
R9C8 is the only square in row 9 that can be <4>
Squares R9C3 and R9C7 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <56>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C1 - removing <56> from <2356> leaving <23>
R9C9 - removing <5> from <2357> leaving <237>
Squares R3C1 and R3C8 in row 3 and R7C1 and R7C8 in row 7 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 1 and 8 can be removed.
R2C1 - removing <6> from <3568> leaving <358>
R2C8 - removing <6> from <136> leaving <13>
Squares R3C1 and R3C9 in row 3 and R7C1 and R7C9 in row 7 form a Simple X-Wing pattern on possibility <9>. All other instances of this possibility in columns 1 and 9 can be removed.
R1C1 - removing <9> from <23589> leaving <2358>
R1C9 - removing <9> from <235789> leaving <23578>
R8C1 - removing <9> from <2589> leaving <258>
R8C9 - removing <9> from <25789> leaving <2578>
Squares R1C1<2358>, R2C1<358>, R8C1<258> and R9C1<23> in column 1 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2358>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C1 - removing <2> from <269> leaving <69>
R7C1 - removing <38> from <3689> leaving <69>
Squares R1C3 (XY), R9C3 (XZ) and R3C1 (YZ) form an XY-Wing pattern on <6>. All squares that are buddies of both the XZ and YZ squares cannot be <6>.
R7C1 - removing <6> from <69> leaving <9>
R2C3 - removing <6> from <156> leaving <15>
R3C1 can only be <6>
R2C7 is the only square in row 2 that can be <6>
R9C7 can only be <5>
R9C3 can only be <6>
R3C9 is the only square in row 3 that can be <9>
R1C7 can only be <8>
R8C7 can only be <9>
R1C3 is the only square in row 1 that can be <9>
R2C1 is the only square in row 2 that can be <8>
R7C8 is the only square in row 7 that can be <6>
Intersection of row 2 with block 3. The value <3> 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 <3> from <237> leaving <27>
R1C9 - removing <3> from <2357> leaving <257>
R2C8 is the only square in column 8 that can be <3>
R2C9 can only be <5>
R2C3 can only be <1>
R8C3 can only be <5>
R3C2 can only be <2>
R3C8 can only be <1>
R1C2 can only be <3>
R8C1 can only be <2>
R1C1 can only be <5>
R7C2 can only be <8>
R9C2 can only be <7>
R7C9 can only be <3>
R8C8 can only be <7>
R9C1 can only be <3>
R8C2 can only be <1>
R8C9 can only be <8>
R1C8 can only be <2>
R9C9 can only be <2>
R1C9 can only be <7>
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