Aug 13 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C3 can only be <4>
R1C7 can only be <6>
R2C2 can only be <8>
R3C1 can only be <3>
R9C7 can only be <5>
R5C3 can only be <9>
R9C3 can only be <3>
R1C1 can only be <1>
R5C7 can only be <3>
R5C5 is the only square in row 5 that can be <1>
R5C9 is the only square in row 5 that can be <6>
R5C6 is the only square in row 5 that can be <2>
R6C6 can only be <8>
R1C5 is the only square in row 1 that can be <2>
R1C4 is the only square in row 1 that can be <8>
R5C1 is the only square in row 5 that can be <8>
R6C9 is the only square in row 6 that can be <2>
R6C5 is the only square in row 6 that can be <6>
R7C5 is the only square in row 7 that can be <3>
R4C6 is the only square in row 4 that can be <3>
R8C5 is the only square in row 8 that can be <8>
R9C1 is the only square in row 9 that can be <2>
R9C9 is the only square in row 9 that can be <8>
Intersection of column 4 with block 5. The value <5> only appears in one or more of squares R4C4, R5C4 and R6C4 of column 4. 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 <5> from <4579> leaving <479>
Squares R5C2 and R8C2 in column 2 and R5C8 and R8C8 in column 8 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in rows 5 and 8 can be removed.
R5C4 - removing <4> from <457> leaving <57>
Squares R5C2 (XY), R6C1 (XZ) and R5C4 (YZ) form an XY-Wing pattern on <5>. All squares that are buddies of both the XZ and YZ squares cannot be <5>.
R6C4 - removing <5> from <45> leaving <4>
R6C1 can only be <5>
R9C5 is the only square in row 9 that can be <4>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R4C4=<59>
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
R4C4 - removing <59> from <579> leaving <7>
R4C1 can only be <4>
R4C5 can only be <9>
R5C4 can only be <5>
R9C4 can only be <9>
R3C5 can only be <5>
R5C8 can only be <4>
R5C2 can only be <7>
R8C8 can only be <7>
R4C9 can only be <5>
R8C2 can only be <4>
R2C8 can only be <5>
R7C9 can only be <4>
R9C6 can only be <7>
R1C6 can only be <9>
R1C9 can only be <7>
R2C5 can only be <7>
R3C9 can only be <9>
R7C1 can only be <7>
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