The full reasoning can be found below the Sudoku.
Apr 30 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C5 can only be <1>
R5C7 can only be <7>
R8C6 can only be <4>
R5C4 can only be <5>
R4C5 can only be <4>
R9C6 can only be <1>
R2C6 can only be <7>
R1C6 can only be <5>
R6C5 can only be <9>
R5C6 can only be <6>
R5C3 can only be <3>
R3C7 is the only square in row 3 that can be <2>
R4C1 is the only square in row 4 that can be <1>
R4C2 is the only square in row 4 that can be <7>
R1C2 can only be <6>
R6C2 can only be <5>
R6C1 can only be <6>
R9C2 can only be <4>
R3C9 is the only square in row 3 that can be <6>
R4C9 can only be <8>
R4C8 can only be <6>
R9C1 is the only square in row 9 that can be <2>
R9C3 is the only square in row 9 that can be <6>
R9C9 is the only square in row 9 that can be <5>
Intersection of block 3 with row 1. The values <48> only appears in one or more of squares R1C7, R1C8 and R1C9 of block 3. These squares are the ones that intersect with row 1. Thus, the other (non-intersecting) squares of row 1 cannot contain these values.
R1C1 - removing <8> from <3789> leaving <379>
R1C3 - removing <8> from <78> leaving <7>
R1C4 - removing <8> from <138> leaving <13>
Squares R2C1 and R2C4 in row 2 and R8C1 and R8C4 in row 8 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 1 and 4 can be removed.
R3C1 - removing <8> from <358> leaving <35>
R7C1 - removing <8> from <578> leaving <57>
R9C4 - removing <8> from <389> leaving <39>
Intersection of row 9 with block 9. The value <8> only appears in one or more of squares R9C7, R9C8 and R9C9 of row 9. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C7 - removing <8> from <48> leaving <4>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R1C9=<14>
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
R1C9 - removing <14> from <149> leaving <9>
R1C1 can only be <3>
R1C7 can only be <8>
R2C9 can only be <1>
R8C9 can only be <7>
R2C4 can only be <8>
R8C1 can only be <8>
R7C9 can only be <3>
R1C4 can only be <1>
R3C1 can only be <5>
R1C8 can only be <4>
R9C7 can only be <9>
R6C8 can only be <3>
R2C1 can only be <9>
R8C4 can only be <9>
R3C5 can only be <3>
R3C3 can only be <8>
R7C1 can only be <7>
R7C3 can only be <5>
R7C5 can only be <8>
R6C9 can only be <4>
R9C8 can only be <8>
R9C4 can only be <3>
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