The full reasoning can be found below the Sudoku.
Apr 15 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C4 is the only square in column 4 that can be <5>
R3C5 is the only square in column 5 that can be <6>
R3C6 is the only square in row 3 that can be <2>
R7C6 can only be <7>
R7C5 is the only square in row 7 that can be <2>
R5C6 is the only square in block 5 that can be <1>
Intersection of column 4 with block 8. The value <4> only appears in one or more of squares R7C4, R8C4 and R9C4 of column 4. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R9C5 - removing <4> from <3489> leaving <389>
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 <1235> leaving <235>
R2C7 - removing <1> from <1236> leaving <236>
Intersection of block 5 with column 5. The values <479> only appears in one or more of squares R4C5, R5C5 and R6C5 of block 5. These squares are the ones that intersect with column 5. Thus, the other (non-intersecting) squares of column 5 cannot contain these values.
R9C5 - removing <9> from <389> leaving <38>
R9C7 is the only square in row 9 that can be <9>
Intersection of row 9 with block 7. The value <4> only appears in one or more of squares R9C1, R9C2 and R9C3 of row 9. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R7C1 - removing <4> from <146> leaving <16>
R7C2 - removing <4> from <145> leaving <15>
R8C2 - removing <4> from <4578> leaving <578>
R8C3 - removing <4> from <4678> leaving <678>
Squares R1C1<18>, R2C2<189>, R3C1<1489> and R3C2<1489> in block 1 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1489>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C3 - removing <18> from <1238> leaving <23>
R2C3 - removing <18> from <1238> leaving <23>
Intersection of column 3 with block 4. The value <1> only appears in one or more of squares R4C3, R5C3 and R6C3 of column 3. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R4C2 - removing <1> from <147> leaving <47>
R6C2 - removing <1> from <1479> leaving <479>
Squares R4C2 and R4C5 in row 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <47>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C3 - removing <47> from <1467> leaving <16>
R4C7 - removing <4> from <1456> leaving <156>
R4C8 - removing <47> from <4567> leaving <56>
Squares R5C1 and R7C1 in column 1 and R5C9 and R7C9 in column 9 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in rows 5 and 7 can be removed.
R5C3 - removing <6> from <4678> leaving <478>
R5C7 - removing <6> from <46> leaving <4>
R7C8 - removing <6> from <3456> leaving <345>
Squares R1C1 and R1C5 in row 1, R5C1 and R5C3 in row 5 and R9C1, R9C3 and R9C5 in row 9 form a Swordfish pattern on possibility <8>. All other instances of this possibility in columns 1, 3 and 5 can be removed.
R3C1 - removing <8> from <1489> leaving <149>
R8C3 - removing <8> from <678> leaving <67>
Squares R1C3, R2C3, R1C7 and R2C7 form a Type-3 Unique Rectangle on <23>. Upon close inspection, it is clear that:
(R1C7 or R2C7)<56> and R8C7<56> form a naked pair on <56> in column 7. No other squares in the column can contain these possibilities
R4C7 - removing <56> from <156> leaving <1>
(R1C7 or R2C7)<56>, R8C7<56> and R4C7<156> form a naked triplet on <156> in column 7. No other squares in the column can contain these possibilities
R6C7 - removing <1> from <13> leaving <3>
R4C3 can only be <6>
R6C8 can only be <7>
R5C9 can only be <6>
R4C8 can only be <5>
R8C3 can only be <7>
R5C3 can only be <8>
R5C1 can only be <9>
R9C3 can only be <4>
R9C1 can only be <8>
R6C3 can only be <1>
R5C5 can only be <7>
R6C2 can only be <4>
R4C5 can only be <4>
R6C5 can only be <9>
R4C2 can only be <7>
R9C5 can only be <3>
R1C1 can only be <1>
R8C2 can only be <5>
R9C9 can only be <7>
R1C5 can only be <8>
R7C4 can only be <4>
R3C1 can only be <4>
R7C1 can only be <6>
R2C6 can only be <9>
R2C2 can only be <8>
R8C6 can only be <8>
R7C8 can only be <3>
R8C4 can only be <9>
R7C9 can only be <5>
R3C8 can only be <8>
R7C2 can only be <1>
R1C9 can only be <3>
R8C7 can only be <6>
R8C8 can only be <4>
R2C7 can only be <2>
R1C3 can only be <2>
R3C9 can only be <1>
R2C8 can only be <6>
R3C2 can only be <9>
R2C3 can only be <3>
R1C7 can only be <5>
R3C4 can only be <3>
R2C4 can only be <1>
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