The full reasoning can be found below the Sudoku.
Jan 31 - Very Hard
Puzzle Copyright © Kevin Stone
www.brainbashers.com
Reasoning
R1C7 is the only square in row 1 that can be <7>
R2C7 is the only square in row 2 that can be <5>
R5C6 is the only square in row 5 that can be <6>
R5C3 is the only square in column 3 that can be <5>
R5C4 is the only square in column 4 that can be <8>
R5C8 is the only square in column 8 that can be <7>
Intersection of column 6 with block 5. The value <1> only appears in one or more of squares R4C6, R5C6 and R6C6 of column 6. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R4C5 - removing <1> from <1345> leaving <345>
R6C5 - removing <1> from <1479> leaving <479>
Intersection of block 2 with row 1. The value <2> 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.
R1C1 - removing <2> from <2349> leaving <349>
R1C3 - removing <2> from <1234> leaving <134>
R1C9 - removing <2> from <1234> leaving <134>
Intersection of block 6 with column 9. The value <1> only appears in one or more of squares R4C9, R5C9 and R6C9 of block 6. These squares are the ones that intersect with column 9. Thus, the other (non-intersecting) squares of column 9 cannot contain this value.
R1C9 - removing <1> from <134> leaving <34>
R3C9 - removing <1> from <1234> leaving <234>
R7C9 - removing <1> from <12358> leaving <2358>
R1C3 is the only square in row 1 that can be <1>
Squares R6C1<24>, R6C6<12> and R6C9<124> in row 6 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <124>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R6C4 - removing <2> from <279> leaving <79>
R6C5 - removing <4> from <479> leaving <79>
Squares R1C9<34>, R3C9<234>, R4C9<124> and R6C9<124> in column 9 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1234>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R7C9 - removing <23> from <2358> leaving <58>
R9C9 - removing <3> from <358> leaving <58>
Intersection of column 9 with block 3. The value <3> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R2C8 - removing <3> from <239> leaving <29>
R3C8 - removing <3> from <1239> leaving <129>
Squares R1C1 and R1C9 in row 1 and R6C1 and R6C9 in row 6 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 1 and 9 can be removed.
R3C1 - removing <4> from <234689> leaving <23689>
R3C9 - removing <4> from <234> leaving <23>
R4C1 - removing <4> from <234> leaving <23>
R4C9 - removing <4> from <124> leaving <12>
R4C5 is the only square in row 4 that can be <4>
R5C5 can only be <3>
R8C5 can only be <1>
R4C4 can only be <2>
R4C1 can only be <3>
R4C9 can only be <1>
R6C6 can only be <1>
R4C6 can only be <5>
R9C6 can only be <3>
R9C4 can only be <7>
R1C6 can only be <2>
R9C3 can only be <6>
R6C4 can only be <9>
R7C5 can only be <5>
R6C5 can only be <7>
R1C4 can only be <3>
R7C9 can only be <8>
R9C9 can only be <5>
R9C7 can only be <9>
R7C1 can only be <2>
R9C1 can only be <8>
R8C7 can only be <2>
R1C9 can only be <4>
R1C1 can only be <9>
R6C9 can only be <2>
R6C1 can only be <4>
R3C9 can only be <3>
R5C7 can only be <4>
R7C2 can only be <3>
R7C3 can only be <7>
R7C8 can only be <1>
R8C3 can only be <4>
R7C7 can only be <6>
R8C2 can only be <9>
R8C8 can only be <3>
R3C7 can only be <1>
R3C1 can only be <6>
R3C3 can only be <2>
R5C2 can only be <2>
R3C8 can only be <9>
R2C3 can only be <3>
R2C2 can only be <8>
R3C5 can only be <8>
R2C8 can only be <2>
R2C5 can only be <9>
R3C2 can only be <4>
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