The full reasoning can be found below the Sudoku.
Apr 21 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R3C1 is the only square in row 3 that can be <3>
R6C5 is the only square in row 6 that can be <1>
R6C4 is the only square in row 6 that can be <7>
R7C1 is the only square in row 7 that can be <1>
Squares R7C3 and R8C1 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C2 - removing <4> from <24> leaving <2>
R9C2 - removing <4> from <2459> leaving <259>
R9C3 - removing <4> from <459> leaving <59>
Intersection of block 2 with row 1. The value <8> 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 <8> from <4589> leaving <459>
R1C7 - removing <8> from <1489> leaving <149>
Intersection of block 4 with column 1. The value <9> only appears in one or more of squares R4C1, R5C1 and R6C1 of block 4. These squares are the ones that intersect with column 1. Thus, the other (non-intersecting) squares of column 1 cannot contain this value.
R2C1 - removing <9> from <4589> leaving <458>
Squares R2C1 and R2C9 in row 2 and R8C1 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 1 and 9 can be removed.
R3C9 - removing <8> from <678> leaving <67>
R7C9 - removing <8> from <248> leaving <24>
Squares R4C9<245>, R6C9<45> and R7C9<24> in column 9 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <245>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C9 - removing <4> from <478> leaving <78>
R8C9 - removing <4> from <4678> leaving <678>
Squares R9C2, R9C3, R1C2 and R1C3 form a Type-2 Unique Rectangle on <59>.
R2C1 - removing <4> from <458> leaving <58>
R2C2 - removing <4> from <4579> leaving <579>
R1C7 - removing <4> from <149> leaving <19>
R1C8 - removing <4> from <146> leaving <16>
R2C8 is the only square in row 2 that can be <4>
R8C8 is the only square in column 8 that can be <7>
Squares R9C6 and R9C8 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <26>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C4 - removing <6> from <346> leaving <34>
Squares R3C5 and R3C9 in row 3 and R8C5 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 5 and 9 can be removed.
R4C5 - removing <6> from <24569> leaving <2459>
Squares R7C7 (XYZ), R7C3 (XZ) and R9C7 (YZ) form an XYZ-Wing pattern on <4>. All squares that are buddies of all three squares cannot be <4>.
R7C9 - removing <4> from <24> leaving <2>
R9C8 can only be <6>
R9C6 can only be <2>
R1C8 can only be <1>
R8C9 can only be <8>
R1C7 can only be <9>
R5C8 can only be <2>
R8C1 can only be <4>
R2C9 can only be <7>
R3C7 can only be <8>
R3C9 can only be <6>
R3C5 can only be <9>
R8C5 can only be <6>
R7C3 can only be <8>
R3C3 can only be <7>
R2C5 can only be <5>
R2C1 can only be <8>
R2C2 can only be <9>
R9C2 can only be <5>
R9C3 can only be <9>
R1C2 can only be <4>
R1C3 can only be <5>
R5C2 can only be <7>
R5C3 can only be <4>
R5C5 can only be <3>
R5C7 can only be <1>
R5C4 can only be <8>
R7C5 can only be <4>
R7C7 can only be <3>
R4C5 can only be <2>
R9C4 can only be <3>
R9C7 can only be <4>
R5C6 can only be <5>
R1C4 can only be <6>
R6C6 can only be <9>
R6C1 can only be <5>
R4C6 can only be <6>
R1C6 can only be <8>
R4C4 can only be <4>
R4C9 can only be <5>
R4C1 can only be <9>
R6C9 can only be <4>
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