The full reasoning can be found below the Sudoku.
Apr 09 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C8 is the only square in row 4 that can be <2>
R5C7 is the only square in row 5 that can be <3>
R3C9 is the only square in row 3 that can be <3>
R2C6 is the only square in row 2 that can be <3>
R4C2 is the only square in row 4 that can be <3>
R6C5 is the only square in row 6 that can be <9>
R8C9 is the only square in row 8 that can be <6>
R9C3 is the only square in row 9 that can be <2>
R5C5 is the only square in column 5 that can be <6>
R5C4 is the only square in row 5 that can be <5>
R4C1 is the only square in row 4 that can be <5>
R6C8 is the only square in row 6 that can be <5>
R1C7 is the only square in row 1 that can be <5>
R7C5 is the only square in row 7 that can be <5>
R8C2 is the only square in row 8 that can be <5>
Squares R8C4 and R9C4 in column 4 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.
R1C4 - removing <18> from <1678> leaving <67>
R2C4 - removing <1> from <167> leaving <67>
Squares R1C6 and R5C6 in column 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 column.
R8C6 - removing <18> from <1489> leaving <49>
R9C6 - removing <18> from <1489> leaving <49>
Squares R4C9 and R6C9 in column 9 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.
R2C9 - removing <1> from <179> leaving <79>
R7C9 - removing <1> from <179> leaving <79>
Squares R1C3<189>, R1C6<18> and R1C8<19> in row 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <189>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C2 - removing <1> from <167> leaving <67>
Squares R1C3 and R1C6 in row 1 and R5C3 and R5C6 in row 5 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 3 and 6 can be removed.
R3C3 - removing <8> from <148> leaving <14>
Squares R3C5 and R5C3 form a remote naked pair. <18> can be removed from any square that is common to their groups.
R3C3 - removing <1> from <14> leaving <4>
R2C8 is the only square in row 2 that can be <4>
Intersection of row 2 with block 1. The value <1> only appears in one or more of squares R2C1, R2C2 and R2C3 of row 2. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R1C3 - removing <1> from <189> leaving <89>
R3C1 - removing <1> from <178> leaving <78>
Squares R1C2, R1C4, R2C2 and R2C4 form a Type-1 Unique Rectangle on <67>.
R2C2 - removing <67> from <167> leaving <1>
R6C2 can only be <4>
R9C2 can only be <7>
R1C2 can only be <6>
R1C4 can only be <7>
R2C4 can only be <6>
Squares R8C4, R9C4, R8C8 and R9C8 form a Type-2 Unique Rectangle on <18>.
R1C8 - removing <9> from <19> leaving <1>
R7C9 - removing <9> from <79> leaving <7>
R1C6 can only be <8>
R3C7 can only be <7>
R3C1 can only be <8>
R2C9 can only be <9>
R1C3 can only be <9>
R5C6 can only be <1>
R3C5 can only be <1>
R2C1 can only be <7>
R6C1 can only be <1>
R4C5 can only be <8>
R4C9 can only be <1>
R6C9 can only be <8>
R5C3 can only be <8>
R7C3 can only be <1>
R7C7 can only be <4>
R7C1 can only be <9>
R9C7 can only be <1>
R9C4 can only be <8>
R8C1 can only be <4>
R8C6 can only be <9>
R8C8 can only be <8>
R9C6 can only be <4>
R8C4 can only be <1>
R9C8 can only be <9>
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