The full reasoning can be found below the Sudoku.
Apr 22 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R7C6 can only be <5>
R3C6 can only be <4>
R3C8 is the only square in row 3 that can be <9>
R3C2 is the only square in row 3 that can be <3>
R5C5 is the only square in row 5 that can be <9>
R8C7 is the only square in row 8 that can be <1>
R8C3 is the only square in row 8 that can be <4>
R8C5 is the only square in row 8 that can be <7>
R9C6 can only be <8>
R9C1 is the only square in column 1 that can be <3>
R4C5 is the only square in column 5 that can be <3>
R4C9 can only be <7>
R2C9 can only be <2>
R9C9 can only be <4>
R1C9 can only be <3>
R1C4 is the only square in block 2 that can be <7>
Squares R4C8 and R6C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <18>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C8 - removing <8> from <458> leaving <45>
R2C8 - removing <8> from <578> leaving <57>
R5C8 - removing <18> from <1348> leaving <34>
Squares R5C6 and R6C6 in block 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R6C5 - removing <1> from <156> leaving <56>
Squares R4C8 and R6C8 in block 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <18>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R5C7 - removing <8> from <348> leaving <34>
Squares R8C1 and R8C2 in block 7 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.
R7C2 - removing <6> from <269> leaving <29>
R9C2 - removing <6> from <2569> leaving <259>
R9C3 - removing <6> from <256> leaving <25>
R1C3 is the only square in column 3 that can be <6>
Squares R3C3 and R9C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <25>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C3 - removing <5> from <158> leaving <18>
R5C3 - removing <5> from <158> leaving <18>
Squares R1C1<58>, R1C7<48> and R1C8<45> in row 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <458>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C2 - removing <58> from <1258> leaving <12>
R1C5 - removing <5> from <125> leaving <12>
Squares R3C3 and R3C4 in row 3, R5C2 and R5C4 in row 5 and R9C2 and R9C3 in row 9 form a Swordfish pattern on possibility <5>. All other instances of this possibility in columns 2, 3 and 4 can be removed.
R6C2 - removing <5> from <15678> leaving <1678>
Squares R5C6, R6C6, R5C2 and R6C2 form a Type-4 Unique Rectangle on <17>.
R5C2 - removing <1> from <1578> leaving <578>
R6C2 - removing <1> from <1678> leaving <678>
Squares R4C2 (XYZ), R8C2 (XZ) and R5C3 (YZ) form an XYZ-Wing pattern on <8>. All squares that are buddies of all three squares cannot be <8>.
R5C2 - removing <8> from <578> leaving <57>
R6C2 - removing <8> from <678> leaving <67>
Squares R6C2 (XY), R6C5 (XZ) and R5C2 (YZ) form an XY-Wing pattern on <5>. All squares that are buddies of both the XZ and YZ squares cannot be <5>.
R5C4 - removing <5> from <58> leaving <8>
R6C1 - removing <5> from <568> leaving <68>
R5C3 can only be <1>
R4C4 can only be <6>
R7C4 can only be <2>
R6C5 can only be <5>
R5C6 can only be <7>
R2C3 can only be <8>
R4C2 can only be <8>
R5C2 can only be <5>
R6C6 can only be <1>
R2C5 can only be <1>
R6C8 can only be <8>
R6C1 can only be <6>
R4C8 can only be <1>
R7C2 can only be <9>
R3C4 can only be <5>
R9C5 can only be <6>
R9C8 can only be <7>
R9C7 can only be <9>
R2C8 can only be <5>
R2C7 can only be <7>
R1C1 can only be <5>
R1C5 can only be <2>
R1C8 can only be <4>
R3C3 can only be <2>
R8C2 can only be <6>
R6C2 can only be <7>
R8C1 can only be <8>
R7C7 can only be <3>
R9C2 can only be <2>
R7C8 can only be <6>
R5C7 can only be <4>
R9C3 can only be <5>
R1C2 can only be <1>
R1C7 can only be <8>
R5C8 can only be <3>
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