Aug 27 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R8C6 can only be <2>
R8C4 can only be <9>
R6C6 can only be <5>
R6C2 can only be <2>
R6C4 can only be <1>
R4C4 is the only square in row 4 that can be <2>
R5C5 is the only square in row 5 that can be <8>
R7C3 is the only square in row 7 that can be <9>
R1C5 is the only square in column 5 that can be <1>
R1C2 is the only square in row 1 that can be <4>
R6C5 is the only square in column 5 that can be <3>
R6C8 can only be <9>
R4C5 is the only square in row 4 that can be <9>
R3C5 is the only square in column 5 that can be <5>
Squares R2C4 and R2C6 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <47>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C1 - removing <7> from <678> leaving <68>
R2C2 - removing <7> from <378> leaving <38>
R2C8 - removing <7> from <1367> leaving <136>
R1C8 is the only square in column 8 that can be <7>
R1C3 can only be <2>
R1C7 can only be <5>
Squares R5C1 and R5C9 in row 5 and R7C1 and R7C9 in row 7 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in columns 1 and 9 can be removed.
R8C1 - removing <5> from <158> leaving <18>
R8C9 - removing <5> from <135> leaving <13>
Squares R2C4, R2C6, R5C4 and R5C6 form a Type-1 Unique Rectangle on <47>.
R5C6 - removing <47> from <467> leaving <6>
R5C7 can only be <3>
R4C6 can only be <7>
R5C9 can only be <5>
R7C9 can only be <2>
R4C8 can only be <6>
R7C7 can only be <4>
R4C2 can only be <5>
R2C6 can only be <4>
R5C4 can only be <4>
R2C4 can only be <7>
R7C5 can only be <7>
R9C7 can only be <6>
R3C7 can only be <2>
R7C1 can only be <5>
R9C5 can only be <4>
R8C8 is the only square in row 8 that can be <5>
R9C2 is the only square in column 2 that can be <7>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R2C9=<16>
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
R2C9 - removing <16> from <136> leaving <3>
R2C2 can only be <8>
R2C8 can only be <1>
R3C9 can only be <6>
R8C9 can only be <1>
R3C1 can only be <7>
R8C1 can only be <8>
R9C8 can only be <3>
R9C3 can only be <1>
R2C1 can only be <6>
R8C2 can only be <3>
R3C3 can only be <3>
R5C1 can only be <1>
R5C3 can only be <7>
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