Jun 25 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C4 can only be <1>
R2C1 can only be <6>
R4C9 can only be <1>
R5C3 can only be <8>
R5C6 can only be <5>
R2C9 can only be <3>
R5C5 can only be <1>
R4C1 can only be <2>
R6C1 can only be <1>
R1C3 is the only square in row 1 that can be <3>
R1C1 is the only square in row 1 that can be <7>
R1C9 is the only square in row 1 that can be <5>
R9C7 is the only square in row 9 that can be <1>
R9C6 is the only square in row 9 that can be <3>
R4C6 can only be <7>
R7C5 can only be <8>
R4C4 can only be <8>
R4C5 can only be <3>
R9C4 can only be <7>
R7C7 is the only square in row 7 that can be <3>
R9C3 is the only square in row 9 that can be <6>
Intersection of column 7 with block 3. The value <9> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C9 - removing <9> from <689> leaving <68>
Squares R3C1 and R3C9 in row 3 and R8C1 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 1 and 9 can be removed.
R9C1 - removing <8> from <489> leaving <49>
R9C9 - removing <8> from <2489> leaving <249>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R9C9=<29>
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
R9C9 - removing <29> from <249> leaving <4>
R9C1 can only be <9>
R9C8 can only be <8>
R6C9 can only be <6>
R7C9 can only be <2>
R6C4 can only be <4>
R3C9 can only be <8>
R5C7 can only be <4>
R7C3 can only be <5>
R8C1 can only be <8>
R9C2 can only be <2>
R1C8 can only be <4>
R8C9 can only be <9>
R1C7 can only be <9>
R3C1 can only be <5>
R5C4 can only be <6>
R7C1 can only be <4>
R3C3 can only be <2>
R1C2 can only be <8>
R1C6 can only be <2>
R3C7 can only be <6>
R3C5 can only be <9>
R6C5 can only be <2>
R6C6 can only be <9>
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