The full reasoning can be found below the Sudoku.
Apr 14 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C7 is the only square in row 2 that can be <4>
R2C5 is the only square in row 2 that can be <6>
R1C9 is the only square in row 1 that can be <6>
R8C5 is the only square in row 8 that can be <4>
R4C4 is the only square in row 4 that can be <4>
R9C9 is the only square in row 9 that can be <4>
R9C7 is the only square in row 9 that can be <7>
R1C6 is the only square in row 1 that can be <7>
R4C5 is the only square in row 4 that can be <7>
Squares R1C4 and R9C4 in column 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <28>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C4 - removing <28> from <2368> leaving <36>
R6C4 - removing <2> from <236> leaving <36>
Intersection of row 8 with block 7. The value <2> only appears in one or more of squares R8C1, R8C2 and R8C3 of row 8. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C1 - removing <2> from <238> leaving <38>
R9C3 - removing <2> from <123> leaving <13>
Squares R6C5<25>, R6C6<29> and R6C9<259> in row 6 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <259>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R6C1 - removing <25> from <2356> leaving <36>
Squares R1C3 and R5C3 in column 3 and R1C7 and R5C7 in column 7 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in rows 1 and 5 can be removed.
R1C1 - removing <5> from <2589> leaving <289>
R5C5 - removing <5> from <1258> leaving <128>
R5C8 - removing <5> from <259> leaving <29>
R6C5 is the only square in column 5 that can be <5>
Squares R5C8 and R6C9 in block 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <29>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C9 - removing <2> from <125> leaving <15>
R5C7 - removing <9> from <159> leaving <15>
Squares R1C3 and R8C3 in column 3 and R1C7 and R8C7 in column 7 form a Simple X-Wing pattern on possibility <9>. All other instances of this possibility in rows 1 and 8 can be removed.
R1C1 - removing <9> from <289> leaving <28>
R8C8 - removing <9> from <89> leaving <8>
R8C2 can only be <2>
Squares R1C1 and R1C4 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <28>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C3 - removing <2> from <259> leaving <59>
Squares R5C2 and R5C4 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <36>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C3 - removing <3> from <235> leaving <25>
Squares R1C1<28>, R2C2<37>, R2C3<23> and R3C2<78> in block 1 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2378>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R3C1 - removing <28> from <2589> leaving <59>
Squares R1C1 and R1C4 in row 1, R4C1 and R4C6 in row 4 and R9C4 and R9C6 in row 9 form a Swordfish pattern on possibility <2>. All other instances of this possibility in columns 1, 4 and 6 can be removed.
R5C6 - removing <2> from <1289> leaving <189>
R6C6 - removing <2> from <29> leaving <9>
R6C9 can only be <2>
R5C8 can only be <9>
R7C8 can only be <5>
Squares R1C1 (XY), R2C3 (XZ) and R9C1 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R9C3 - removing <3> from <13> leaving <1>
R8C3 can only be <9>
R8C7 can only be <1>
R1C3 can only be <5>
R5C7 can only be <5>
R7C9 can only be <9>
R1C7 can only be <9>
R5C3 can only be <2>
R3C1 can only be <9>
R3C9 can only be <5>
R4C9 can only be <1>
R4C6 can only be <2>
R2C3 can only be <3>
R4C1 can only be <5>
R2C2 can only be <7>
R9C6 can only be <8>
R9C1 can only be <3>
R9C4 can only be <2>
R5C6 can only be <1>
R7C5 can only be <1>
R2C8 can only be <2>
R3C2 can only be <8>
R3C8 can only be <7>
R3C5 can only be <2>
R7C2 can only be <6>
R1C1 can only be <2>
R1C4 can only be <8>
R5C5 can only be <8>
R7C1 can only be <8>
R5C2 can only be <3>
R6C1 can only be <6>
R5C4 can only be <6>
R6C4 can only be <3>
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