The full reasoning can be found below the Sudoku.
Sep 06 - Super Hard
Puzzle Copyright © Kevin Stone
Reasoning
R3C4 can only be <3>
R4C4 can only be <1>
R1C5 is the only square in row 1 that can be <1>
R3C1 is the only square in row 3 that can be <7>
R2C8 is the only square in row 2 that can be <7>
R2C9 is the only square in row 2 that can be <2>
R4C5 is the only square in row 4 that can be <3>
R5C9 is the only square in row 5 that can be <3>
R5C1 is the only square in row 5 that can be <1>
R5C4 is the only square in row 5 that can be <9>
R6C7 is the only square in row 6 that can be <1>
R7C6 is the only square in row 7 that can be <7>
R6C5 is the only square in row 6 that can be <7>
R6C4 is the only square in row 6 that can be <2>
R7C4 can only be <8>
R7C3 is the only square in row 7 that can be <2>
R8C2 is the only square in row 8 that can be <3>
R1C3 is the only square in row 1 that can be <3>
R9C9 is the only square in row 9 that can be <1>
R9C2 is the only square in row 9 that can be <7>
R9C7 is the only square in row 9 that can be <9>
R3C9 is the only square in row 3 that can be <9>
R1C1 is the only square in row 1 that can be <9>
R1C7 is the only square in column 7 that can be <8>
R8C9 is the only square in column 9 that can be <8>
Squares R8C5 and R9C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <24>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R5C5 - removing <4> from <456> leaving <56>
Intersection of row 1 with block 3. The value <6> only appears in one or more of squares R1C7, R1C8 and R1C9 of row 1. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C7 - removing <6> from <456> leaving <45>
Intersection of column 1 with block 7. The value <4> only appears in one or more of squares R7C1, R8C1 and R9C1 of column 1. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C3 - removing <4> from <468> leaving <68>
Intersection of block 7 with column 1. The values <45> only appears in one or more of squares R7C1, R8C1 and R9C1 of block 7. These squares are the ones that intersect with column 1. Thus, the other (non-intersecting) squares of column 1 cannot contain these values.
R2C1 - removing <5> from <568> leaving <68>
Squares R3C3 and R3C7 in row 3, R4C3, R4C6 and R4C7 in row 4 and R6C3 and R6C6 in row 6 form a Swordfish pattern on possibility <4>. All other instances of this possibility in columns 3, 6 and 7 can be removed.
R5C6 - removing <4> from <4568> leaving <568>
R7C7 - removing <4> from <456> leaving <56>
Squares R8C5, R9C5, R8C8 and R9C8 form a Type-3 Unique Rectangle on <24>. Upon close inspection, it is clear that:
(R8C8 or R9C8)<56> and R7C7<56> form a naked pair on <56> in block 9. No other squares in the block can contain these possibilities
R7C9 - removing <6> from <46> leaving <4>
R1C9 can only be <6>
Squares R3C7 (XY), R4C7 (XZ) and R3C6 (YZ) form an XY-Wing pattern on <6>. All squares that are buddies of both the XZ and YZ squares cannot be <6>.
R4C6 - removing <6> from <456> leaving <45>
R4C7 is the only square in row 4 that can be <6>
R7C7 can only be <5>
R5C8 can only be <4>
R1C8 can only be <5>
R7C1 can only be <6>
R3C7 can only be <4>
R8C8 can only be <2>
R8C5 can only be <4>
R9C8 can only be <6>
R9C3 can only be <8>
R1C2 can only be <4>
R2C1 can only be <8>
R8C1 can only be <5>
R9C5 can only be <2>
R9C1 can only be <4>
R6C3 can only be <4>
R2C2 can only be <5>
R2C5 can only be <6>
R5C2 can only be <8>
R3C3 can only be <6>
R5C5 can only be <5>
R3C6 can only be <5>
R4C6 can only be <4>
R4C3 can only be <5>
R6C6 can only be <8>
R5C6 can only be <6>
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