May 14 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C9 can only be <8>
R3C2 can only be <9>
R5C5 can only be <5>
R8C7 can only be <5>
R2C7 can only be <9>
R3C8 can only be <2>
R1C1 can only be <5>
R3C6 can only be <7>
R4C7 can only be <8>
R2C3 can only be <8>
R2C4 can only be <5>
R3C4 can only be <8>
R6C7 can only be <1>
R6C2 can only be <2>
R4C9 is the only square in row 4 that can be <2>
R5C2 is the only square in row 5 that can be <1>
R6C9 is the only square in row 6 that can be <6>
R6C1 is the only square in row 6 that can be <8>
R7C6 is the only square in row 7 that can be <5>
R9C4 is the only square in row 9 that can be <6>
Squares R8C3 and R8C4 in row 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <37>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R8C5 - removing <3> from <239> leaving <29>
Intersection of row 5 with block 6. The value <3> only appears in one or more of squares R5C7, R5C8 and R5C9 of row 5. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain this value.
R4C8 - removing <3> from <359> leaving <59>
Squares R5C1 and R9C1 in column 1 and R5C9 and R9C9 in column 9 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in rows 5 and 9 can be removed.
R5C8 - removing <7> from <379> leaving <39>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R7C4=<47>
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
R7C4 - removing <47> from <347> leaving <3>
R7C2 can only be <4>
R7C8 can only be <7>
R1C4 can only be <4>
R8C4 can only be <7>
R9C5 can only be <1>
R6C8 can only be <5>
R9C9 can only be <3>
R8C3 can only be <3>
R9C6 can only be <4>
R2C5 can only be <2>
R9C1 can only be <7>
R1C6 can only be <9>
R5C9 can only be <7>
R1C5 can only be <3>
R8C6 can only be <2>
R2C6 can only be <1>
R8C5 can only be <9>
R5C1 can only be <9>
R6C3 can only be <7>
R4C8 can only be <9>
R4C2 can only be <3>
R4C3 can only be <5>
R4C1 can only be <4>
R5C8 can only be <3>
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