The full reasoning can be found below the Sudoku.
Nov 05 - Super Hard
Puzzle Copyright © Kevin Stone
Reasoning
R5C5 can only be <9>
R9C4 can only be <7>
R2C5 is the only square in row 2 that can be <7>
R3C5 can only be <5>
R8C5 can only be <3>
R7C5 can only be <8>
R2C4 is the only square in row 2 that can be <8>
R3C3 is the only square in row 3 that can be <7>
R4C2 is the only square in row 4 that can be <8>
R8C8 is the only square in row 8 that can be <7>
R6C7 is the only square in row 6 that can be <7>
R6C2 is the only square in column 2 that can be <9>
R8C9 is the only square in block 9 that can be <6>
R8C4 can only be <9>
Squares R1C2 and R1C4 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <12>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C6 - removing <2> from <239> leaving <39>
Squares R6C3 and R6C9 in row 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <13>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R6C1 - removing <1> from <156> leaving <56>
R6C8 - removing <3> from <356> leaving <56>
Squares R7C3 and R7C7 in row 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C6 - removing <4> from <246> leaving <26>
Squares R5C7 and R7C7 in column 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C7 - removing <4> from <149> leaving <19>
Squares R2C8 and R9C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C8 - removing <3> from <39> leaving <9>
R4C8 - removing <4> from <2459> leaving <259>
R5C8 - removing <34> from <2346> leaving <26>
R1C6 can only be <3>
R3C7 can only be <1>
R3C4 can only be <6>
R4C7 can only be <9>
R3C6 can only be <9>
R7C4 can only be <2>
R2C6 can only be <2>
R7C6 can only be <6>
R1C4 can only be <1>
R1C2 can only be <2>
Intersection of column 2 with block 7. The value <5> only appears in one or more of squares R7C2, R8C2 and R9C2 of column 2. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R8C1 - removing <5> from <145> leaving <14>
Intersection of column 3 with block 4. The values <12> only appears in one or more of squares R4C3, R5C3 and R6C3 of column 3. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain these values.
R4C1 - removing <1> from <145> leaving <45>
Squares R5C3 and R5C7 in row 5 and R7C3 and R7C7 in row 7 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 3 and 7 can be removed.
R4C3 - removing <4> from <124> leaving <12>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R5C3=<24>
These are called the BUG possibilities. In a BUG pattern, in each row, column and block, each unsolved possibility appears exactly twice. Such a pattern either has 0 or 2 solutions, so it cannot be part of a valid Sudoku
When a puzzle contains a BUG, and only one square in the puzzle has more than 2 possibilities, the only way to kill the BUG is to remove both of the BUG possibilities from the square, thus solving it
R5C3 - removing <24> from <234> leaving <3>
R5C2 can only be <6>
R5C7 can only be <4>
R6C3 can only be <1>
R7C3 can only be <4>
R7C7 can only be <3>
R4C9 can only be <1>
R6C9 can only be <3>
R4C3 can only be <2>
R2C9 can only be <4>
R8C1 can only be <1>
R9C8 can only be <4>
R8C2 can only be <5>
R2C1 can only be <6>
R8C6 can only be <4>
R9C2 can only be <3>
R9C6 can only be <5>
R2C8 can only be <3>
R2C2 can only be <1>
R6C1 can only be <5>
R4C8 can only be <5>
R4C1 can only be <4>
R6C8 can only be <6>
R5C8 can only be <2>
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