The full reasoning can be found below the Sudoku.
Apr 27 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R6C1 can only be <9>
R9C6 can only be <9>
R4C6 is the only square in row 4 that can be <6>
R7C5 is the only square in row 7 that can be <5>
R7C2 is the only square in column 2 that can be <8>
R6C4 is the only square in column 4 that can be <7>
R6C5 can only be <1>
R8C5 can only be <2>
R9C4 can only be <1>
Intersection of block 2 with row 1. The value <5> only appears in one or more of squares R1C4, R1C5 and R1C6 of block 2. These squares are the ones that intersect with row 1. Thus, the other (non-intersecting) squares of row 1 cannot contain this value.
R1C3 - removing <5> from <2569> leaving <269>
R1C7 - removing <5> from <1589> leaving <189>
R1C9 - removing <5> from <2589> leaving <289>
Intersection of block 4 with row 5. The value <5> only appears in one or more of squares R5C1, R5C2 and R5C3 of block 4. These squares are the ones that intersect with row 5. Thus, the other (non-intersecting) squares of row 5 cannot contain this value.
R5C4 - removing <5> from <259> leaving <29>
R5C6 - removing <5> from <358> leaving <38>
R5C7 - removing <5> from <3569> leaving <369>
Intersection of column 7 with block 3. The value <5> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C9 - removing <5> from <34589> leaving <3489>
Squares R4C5 (XY), R4C1 (XZ) and R5C4 (YZ) form an XY-Wing pattern on <2>. All squares that are buddies of both the XZ and YZ squares cannot be <2>.
R5C2 - removing <2> from <25> leaving <5>
R5C3 - removing <2> from <245> leaving <45>
R4C4 - removing <2> from <259> leaving <59>
R5C3 can only be <4>
R4C1 can only be <2>
R4C5 is the only square in row 4 that can be <4>
R5C4 is the only square in row 5 that can be <2>
R2C2 is the only square in column 2 that can be <2>
R1C9 is the only square in row 1 that can be <2>
R9C3 is the only square in row 9 that can be <2>
R7C8 is the only square in row 7 that can be <2>
Squares R1C3 and R7C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <69>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C3 - removing <9> from <3579> leaving <357>
R3C3 - removing <9> from <3579> leaving <357>
R8C3 - removing <69> from <369> leaving <3>
R9C1 can only be <4>
R3C1 is the only square in column 1 that can be <3>
Squares R3C2 and R3C8 in row 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <19>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R3C5 - removing <9> from <789> leaving <78>
R3C7 - removing <19> from <14589> leaving <458>
R3C9 - removing <9> from <489> leaving <48>
Intersection of row 8 with block 9. The value <6> only appears in one or more of squares R8C7, R8C8 and R8C9 of row 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C7 - removing <6> from <1469> leaving <149>
Squares R1C1 and R1C7 in row 1 and R7C1 and R7C7 in row 7 form a Simple X-Wing pattern on possibility <1>. All other instances of this possibility in columns 1 and 7 can be removed.
R8C7 - removing <1> from <169> leaving <69>
Squares R1C3 and R7C3 in column 3, R1C4 and R4C4 in column 4 and R4C9 and R7C9 in column 9 form a Swordfish pattern on possibility <9>. All other instances of this possibility in rows 1, 4 and 7 can be removed.
R1C7 - removing <9> from <189> leaving <18>
R7C7 - removing <9> from <149> leaving <14>
Squares R1C7 (XY), R7C7 (XZ) and R3C9 (YZ) form an XY-Wing pattern on <4>. All squares that are buddies of both the XZ and YZ squares cannot be <4>.
R3C7 - removing <4> from <458> leaving <58>
R7C9 - removing <4> from <49> leaving <9>
R7C3 can only be <6>
R4C9 can only be <5>
R8C7 can only be <6>
R8C8 can only be <1>
R8C2 can only be <9>
R3C8 can only be <9>
R7C7 can only be <4>
R3C2 can only be <1>
R2C8 can only be <3>
R4C4 can only be <9>
R6C9 can only be <3>
R6C6 can only be <5>
R9C9 can only be <8>
R5C7 can only be <9>
R5C8 can only be <6>
R7C1 can only be <1>
R1C3 can only be <9>
R9C7 can only be <3>
R3C9 can only be <4>
R1C4 can only be <5>
R1C6 can only be <8>
R1C7 can only be <1>
R5C6 can only be <3>
R3C5 can only be <7>
R1C1 can only be <6>
R2C7 can only be <5>
R3C3 can only be <5>
R2C5 can only be <9>
R5C5 can only be <8>
R2C3 can only be <7>
R3C7 can only be <8>
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