The full reasoning can be found below the Sudoku.
Apr 20 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C4 can only be <1>
R8C2 can only be <1>
R5C6 can only be <9>
R5C3 can only be <2>
R4C5 can only be <6>
R6C5 can only be <3>
R5C5 can only be <8>
R1C9 is the only square in row 1 that can be <2>
R1C1 is the only square in row 1 that can be <6>
R1C3 is the only square in row 1 that can be <3>
R9C3 can only be <9>
R5C1 is the only square in row 5 that can be <4>
Squares R1C6 and R1C7 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C2 - removing <5> from <589> leaving <89>
R1C5 - removing <15> from <14579> leaving <479>
R1C8 - removing <5> from <4578> leaving <478>
Squares R7C1 and R9C2 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <78>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R9C1 - removing <78> from <2378> leaving <23>
Intersection of column 2 with block 1. The value <9> only appears in one or more of squares R1C2, R2C2 and R3C2 of column 2. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R2C1 - removing <9> from <1589> leaving <158>
Squares R1C6 and R9C6 in column 6 and R1C7 and R9C7 in column 7 form a Simple X-Wing pattern on possibility <1>. All other instances of this possibility in rows 1 and 9 can be removed.
R9C5 - removing <1> from <12457> leaving <2457>
R9C9 - removing <1> from <134568> leaving <34568>
Squares R3C5 and R3C9 in row 3, R4C1 and R4C9 in row 4 and R7C1 and R7C5 in row 7 form a Swordfish pattern on possibility <7>. All other instances of this possibility in columns 1, 5 and 9 can be removed.
R1C5 - removing <7> from <479> leaving <49>
R5C9 - removing <7> from <3567> leaving <356>
R9C5 - removing <7> from <2457> leaving <245>
Squares R3C5 (XYZ), R7C5 (XZ) and R1C6 (YZ) form an XYZ-Wing pattern on <1>. All squares that are buddies of all three squares cannot be <1>.
R2C5 - removing <1> from <1459> leaving <459>
Squares R1C5<49>, R2C5<459>, R8C5<245> and R9C5<245> in column 5 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2459>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C5 - removing <5> from <157> leaving <17>
Squares R3C1 and R3C9 in row 3 and R6C1 and R6C9 in row 6 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in columns 1 and 9 can be removed.
R2C1 - removing <5> from <158> leaving <18>
R2C9 - removing <5> from <1458> leaving <148>
R5C9 - removing <5> from <356> leaving <36>
R8C9 - removing <5> from <345> leaving <34>
R9C9 - removing <5> from <34568> leaving <3468>
Squares R2C9<148>, R5C9<36>, R7C9<18>, R8C9<34> and R9C9<3468> in column 9 form a comprehensive naked set. These 5 squares can only contain the 5 possibilities <13468>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C9 - removing <1> from <157> leaving <57>
Squares R3C5 (XY), R1C6 (XZ) and R3C9 (YZ) form an XY-Wing pattern on <5>. All squares that are buddies of both the XZ and YZ squares cannot be <5>.
R1C7 - removing <5> from <15> leaving <1>
R1C6 can only be <5>
R9C6 can only be <1>
R7C5 can only be <7>
R7C1 can only be <8>
R3C5 can only be <1>
R9C4 can only be <4>
R1C4 can only be <7>
R3C1 can only be <5>
R7C9 can only be <1>
R2C1 can only be <1>
R9C2 can only be <7>
R5C2 can only be <5>
R3C9 can only be <7>
R6C1 can only be <9>
R4C9 can only be <9>
R4C1 can only be <7>
R6C9 can only be <5>
R5C7 can only be <3>
R5C9 can only be <6>
R9C7 can only be <5>
R5C8 can only be <7>
R9C5 can only be <2>
R8C8 can only be <4>
R8C9 can only be <3>
R1C8 can only be <8>
R8C1 can only be <2>
R9C9 can only be <8>
R9C1 can only be <3>
R8C5 can only be <5>
R9C8 can only be <6>
R2C9 can only be <4>
R1C2 can only be <9>
R2C8 can only be <5>
R2C5 can only be <9>
R1C5 can only be <4>
R2C2 can only be <8>
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