Dec 10 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C1 can only be <2>
R7C2 can only be <4>
R1C7 is the only square in row 1 that can be <1>
R3C1 is the only square in row 3 that can be <4>
R3C2 is the only square in row 3 that can be <8>
R5C2 can only be <9>
R8C2 can only be <6>
R2C2 can only be <3>
R2C3 is the only square in row 2 that can be <6>
R1C3 can only be <7>
R1C5 can only be <3>
R1C9 can only be <6>
R3C9 is the only square in row 3 that can be <3>
R4C5 is the only square in row 4 that can be <9>
R4C3 is the only square in row 4 that can be <4>
R6C3 can only be <5>
R5C1 can only be <8>
R4C4 is the only square in row 4 that can be <8>
R5C6 is the only square in row 5 that can be <6>
R6C5 is the only square in row 6 that can be <4>
R6C4 is the only square in row 6 that can be <1>
R5C8 is the only square in row 5 that can be <1>
R7C1 is the only square in row 7 that can be <1>
R9C1 can only be <5>
R8C8 is the only square in row 8 that can be <9>
R9C5 is the only square in row 9 that can be <1>
R9C9 is the only square in row 9 that can be <4>
R7C4 is the only square in column 4 that can be <3>
Squares R9C3, R9C7, R8C3 and R8C7 form a Type-1 Unique Rectangle on <28>.
R8C7 - removing <28> from <258> leaving <5>
R8C5 can only be <2>
R8C3 can only be <8>
R7C6 can only be <5>
R9C3 can only be <2>
R9C7 can only be <8>
Intersection of row 2 with block 3. The value <2> only appears in one or more of squares R2C7, R2C8 and R2C9 of row 2. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C8 - removing <2> from <257> leaving <57>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R2C8=<25>
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
R2C8 - removing <25> from <257> leaving <7>
R2C5 can only be <5>
R2C7 can only be <2>
R3C8 can only be <5>
R7C8 can only be <2>
R3C4 can only be <2>
R7C9 can only be <7>
R5C9 can only be <2>
R5C5 can only be <7>
R4C7 can only be <3>
R3C6 can only be <7>
R5C4 can only be <5>
R6C6 can only be <3>
R4C6 can only be <2>
R6C7 can only be <7>
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