Jan 22 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C3 can only be <9>
R7C5 can only be <2>
R8C3 can only be <2>
R4C6 can only be <3>
R7C9 can only be <1>
R7C1 can only be <8>
R9C8 can only be <2>
R1C8 can only be <1>
R1C2 can only be <5>
R5C8 can only be <6>
R9C2 can only be <1>
R5C2 can only be <8>
R1C3 is the only square in row 1 that can be <4>
R1C7 is the only square in row 1 that can be <8>
R1C9 is the only square in row 1 that can be <2>
R3C1 is the only square in row 3 that can be <1>
R5C1 can only be <3>
R6C6 is the only square in row 6 that can be <6>
R1C6 can only be <9>
R5C6 can only be <4>
R9C6 can only be <8>
R9C5 is the only square in row 9 that can be <4>
Intersection of column 4 with block 5. The values <25> 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 these values.
R5C5 - removing <5> from <579> leaving <79>
Squares R5C5 and R5C9 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <79>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C4 - removing <79> from <2579> leaving <25>
R5C7 - removing <79> from <1279> leaving <12>
Squares R3C5 and R3C9 in row 3 and R5C5 and R5C9 in row 5 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in columns 5 and 9 can be removed.
R1C5 - removing <7> from <367> leaving <36>
R2C5 - removing <7> from <567> leaving <56>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R9C7=<35>
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
R9C7 - removing <35> from <359> leaving <9>
R9C4 can only be <3>
R9C9 can only be <5>
R6C7 can only be <1>
R8C7 can only be <3>
R3C9 can only be <7>
R3C5 can only be <5>
R5C9 can only be <9>
R2C7 can only be <5>
R5C5 can only be <7>
R6C3 can only be <5>
R5C7 can only be <2>
R8C5 can only be <9>
R1C4 can only be <7>
R1C1 can only be <6>
R4C4 can only be <2>
R2C5 can only be <6>
R4C7 can only be <7>
R5C4 can only be <5>
R5C3 can only be <1>
R6C4 can only be <9>
R1C5 can only be <3>
R9C1 can only be <7>
R2C3 can only be <7>
R9C3 can only be <6>
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