The full reasoning can be found below the Sudoku.
Apr 04 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C5 can only be <9>
R3C4 can only be <1>
R4C7 can only be <4>
R6C7 can only be <6>
R7C4 can only be <7>
R3C7 can only be <7>
R4C4 can only be <2>
R4C3 can only be <8>
R6C4 can only be <5>
R5C4 can only be <9>
R4C5 can only be <7>
R4C6 can only be <1>
R6C3 can only be <2>
R3C3 can only be <3>
R3C6 can only be <4>
R2C5 can only be <3>
R6C5 can only be <8>
R6C6 can only be <3>
R5C6 can only be <6>
R5C5 can only be <4>
R7C6 can only be <8>
R2C8 is the only square in row 2 that can be <4>
R9C2 is the only square in row 9 that can be <4>
Squares R9C5 and R9C9 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.
R9C8 - removing <26> from <2567> leaving <57>
Squares R1C8<56>, R1C9<16> and R2C7<15> in block 3 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <156>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R2C9 - removing <1> from <128> leaving <28>
Squares R1C1<157>, R5C1<15> and R9C1<57> in column 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <157>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C1 - removing <15> from <1589> leaving <89>
R8C1 - removing <7> from <789> leaving <89>
Squares R2C3 and R7C3 in column 3 and R2C7 and R7C7 in column 7 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in rows 2 and 7 can be removed.
R2C2 - removing <5> from <1258> leaving <128>
R7C2 - removing <5> from <35> leaving <3>
R7C8 - removing <5> from <356> leaving <36>
R7C8 can only be <6>
R1C8 can only be <5>
R9C9 can only be <2>
R9C5 can only be <6>
R2C9 can only be <8>
R9C8 can only be <7>
R2C7 can only be <1>
R8C7 can only be <9>
R1C9 can only be <6>
R2C1 can only be <9>
R2C2 can only be <2>
R5C9 can only be <3>
R3C8 can only be <2>
R3C2 can only be <8>
R5C8 can only be <8>
R8C9 can only be <1>
R8C1 can only be <8>
R8C3 can only be <6>
R7C7 can only be <5>
R8C5 can only be <2>
R9C1 can only be <5>
R8C8 can only be <3>
R2C3 can only be <5>
R7C3 can only be <9>
R8C2 can only be <7>
R1C2 can only be <1>
R5C1 can only be <1>
R1C1 can only be <7>
R5C2 can only be <5>
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