The full reasoning can be found below the Sudoku.
Jan 12 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R7C5 can only be <2>
R8C5 can only be <9>
R3C5 can only be <6>
R2C5 can only be <3>
R1C7 is the only square in row 1 that can be <3>
R9C1 is the only square in row 9 that can be <3>
R4C2 is the only square in row 4 that can be <3>
R8C8 is the only square in row 8 that can be <3>
R9C3 is the only square in row 9 that can be <9>
Squares R9C4 and R9C6 in row 9 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.
R9C7 - removing <7> from <267> leaving <26>
R9C9 - removing <7> from <267> leaving <26>
Squares R3C2 and R8C2 in column 2 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.
R2C2 - removing <8> from <248> leaving <24>
R5C2 - removing <1> from <1249> leaving <249>
R6C2 - removing <18> from <1289> leaving <29>
Squares R6C2 and R6C4 in row 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <29>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R6C3 - removing <2> from <1278> leaving <178>
R6C8 - removing <29> from <2679> leaving <67>
R6C9 - removing <29> from <126789> leaving <1678>
Squares R9C4 and R9C6 in block 8 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 block.
R7C6 - removing <47> from <4578> leaving <58>
R8C4 - removing <7> from <578> leaving <58>
Intersection of row 4 with block 6. The value <2> only appears in one or more of squares R4C7, R4C8 and R4C9 of row 4. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R5C7 - removing <2> from <1257> leaving <157>
R5C8 - removing <2> from <2579> leaving <579>
Intersection of column 8 with block 6. The value <9> only appears in one or more of squares R4C8, R5C8 and R6C8 of column 8. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R4C9 - removing <9> from <1289> leaving <128>
Squares R1C1<58>, R3C1<158> and R3C2<18> in block 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <158>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C3 - removing <8> from <268> leaving <26>
R2C3 - removing <8> from <2468> leaving <246>
Squares R1C1<58>, R1C4<589> and R1C6<589> in row 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <589>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C9 - removing <89> from <2689> leaving <26>
R3C9 is the only square in column 9 that can be <9>
Squares R1C9 and R9C9 in column 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 column.
R4C9 - removing <2> from <128> leaving <18>
R6C9 - removing <6> from <1678> leaving <178>
R6C8 is the only square in row 6 that can be <6>
Intersection of column 9 with block 6. The value <8> only appears in one or more of squares R4C9, R5C9 and R6C9 of column 9. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R4C7 - removing <8> from <128> leaving <12>
Squares R6C1 and R7C1 in column 1 and R6C9 and R7C9 in column 9 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in rows 6 and 7 can be removed.
R6C3 - removing <7> from <178> leaving <18>
R7C3 - removing <7> from <1478> leaving <148>
R7C8 - removing <7> from <57> leaving <5>
R7C6 can only be <8>
R2C6 can only be <7>
R8C4 can only be <5>
R2C8 can only be <2>
R9C6 can only be <4>
R3C4 can only be <8>
R2C2 can only be <4>
R4C8 can only be <9>
R1C9 can only be <6>
R3C2 can only be <1>
R3C7 can only be <7>
R1C4 can only be <9>
R8C7 can only be <1>
R5C8 can only be <7>
R8C2 can only be <8>
R4C7 can only be <2>
R5C7 can only be <5>
R7C9 can only be <7>
R9C4 can only be <7>
R4C6 can only be <1>
R1C6 can only be <5>
R6C4 can only be <2>
R1C1 can only be <8>
R1C3 can only be <2>
R9C9 can only be <2>
R2C7 can only be <8>
R2C3 can only be <6>
R3C1 can only be <5>
R4C9 can only be <8>
R5C6 can only be <9>
R9C7 can only be <6>
R4C1 can only be <4>
R6C9 can only be <1>
R5C2 can only be <2>
R6C2 can only be <9>
R5C4 can only be <4>
R6C3 can only be <8>
R8C3 can only be <7>
R6C1 can only be <7>
R7C1 can only be <1>
R5C3 can only be <1>
R7C3 can only be <4>
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