The full reasoning can be found below the Sudoku.
Apr 28 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C2 is the only square in row 2 that can be <4>
R2C8 is the only square in row 2 that can be <2>
R1C2 is the only square in row 1 that can be <2>
R2C1 is the only square in row 2 that can be <7>
R5C6 is the only square in row 5 that can be <2>
R9C1 is the only square in row 9 that can be <3>
R4C6 is the only square in column 6 that can be <3>
Squares R9C7 and R9C8 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C2 - removing <8> from <678> leaving <67>
R9C3 - removing <5> from <5679> leaving <679>
R9C5 - removing <8> from <4789> leaving <479>
R9C9 - removing <5> from <4567> leaving <467>
R7C1 is the only square in block 7 that can be <5>
Squares R9C7 and R9C8 in block 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C8 - removing <8> from <18> leaving <1>
Intersection of row 1 with block 1. The value <6> only appears in one or more of squares R1C1, R1C2 and R1C3 of row 1. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R3C1 - removing <6> from <169> leaving <19>
Squares R3C1 and R3C9 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.
R3C4 - removing <1> from <168> leaving <68>
Squares R3C4 and R3C6 in block 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <68>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C5 - removing <8> from <138> leaving <13>
Squares R1C5 and R2C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <13>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C5 - removing <1> from <1578> leaving <578>
R6C5 - removing <1> from <1459> leaving <459>
Intersection of row 4 with block 5. The value <8> only appears in one or more of squares R4C4, R4C5 and R4C6 of row 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R5C4 - removing <8> from <6789> leaving <679>
R5C5 - removing <8> from <5789> leaving <579>
Intersection of row 7 with block 8. The value <8> only appears in one or more of squares R7C4, R7C5 and R7C6 of row 7. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R8C5 - removing <8> from <789> leaving <79>
R4C5 is the only square in column 5 that can be <8>
Intersection of column 4 with block 5. The values <19> only appears in one or more of squares R4C4, R5C4 and R6C4 of column 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain these values.
R5C5 - removing <9> from <579> leaving <57>
R6C5 - removing <9> from <459> leaving <45>
Intersection of column 9 with block 3. The value <1> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C7 - removing <1> from <1589> leaving <589>
Squares R6C3<56>, R6C5<45> and R6C6<46> in row 6 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <456>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R6C4 - removing <6> from <169> leaving <19>
R6C7 - removing <5> from <159> leaving <19>
Squares R2C5, R2C9, R1C5 and R1C9 form a Type-1 Unique Rectangle on <13>.
R1C9 - removing <13> from <1359> leaving <59>
Squares R9C7, R9C8, R1C7 and R1C8 form a Type-3 Unique Rectangle on <58>. Upon close inspection, it is clear that:
(R1C7 or R1C8)<39>, R3C9<19> and R2C9<13> form a naked triplet on <139> in block 3. No other squares in the block can contain these possibilities
R1C9 - removing <9> from <59> leaving <5>
Squares R4C4 (XY), R4C7 (XZ) and R5C5 (YZ) form an XY-Wing pattern on <5>. All squares that are buddies of both the XZ and YZ squares cannot be <5>.
R5C8 - removing <5> from <35> leaving <3>
R5C9 can only be <9>
R1C8 can only be <8>
R3C9 can only be <1>
R6C7 can only be <1>
R6C4 can only be <9>
R4C7 can only be <5>
R1C7 can only be <9>
R9C8 can only be <5>
R3C1 can only be <9>
R2C9 can only be <3>
R4C3 can only be <7>
R9C7 can only be <8>
R1C3 can only be <6>
R2C5 can only be <1>
R4C4 can only be <1>
R1C1 can only be <1>
R6C3 can only be <5>
R9C3 can only be <9>
R1C5 can only be <3>
R6C5 can only be <4>
R6C6 can only be <6>
R3C6 can only be <8>
R5C4 can only be <7>
R9C5 can only be <7>
R9C2 can only be <6>
R5C5 can only be <5>
R8C5 can only be <9>
R7C4 can only be <8>
R3C4 can only be <6>
R7C6 can only be <4>
R7C9 can only be <7>
R8C9 can only be <6>
R8C1 can only be <8>
R9C9 can only be <4>
R5C2 can only be <8>
R5C1 can only be <6>
R8C2 can only be <7>
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