Oct 08 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C8 can only be <8>
R6C7 can only be <5>
R4C7 can only be <9>
R4C4 can only be <5>
R5C9 can only be <4>
R3C4 can only be <3>
R7C4 can only be <9>
R3C7 can only be <2>
R2C2 is the only square in row 2 that can be <9>
R3C9 is the only square in row 3 that can be <8>
R4C5 is the only square in row 4 that can be <1>
R5C6 is the only square in row 5 that can be <3>
R5C5 is the only square in row 5 that can be <9>
R9C5 is the only square in row 9 that can be <8>
R6C6 is the only square in row 6 that can be <8>
R9C9 is the only square in row 9 that can be <9>
R8C5 is the only square in column 5 that can be <2>
R7C6 can only be <5>
R3C6 can only be <4>
R3C3 can only be <1>
R4C6 can only be <2>
R4C3 can only be <4>
R3C1 can only be <5>
R6C5 is the only square in row 6 that can be <4>
R7C9 is the only square in column 9 that can be <2>
R7C1 is the only square in row 7 that can be <1>
R9C8 is the only square in row 9 that can be <1>
R1C9 is the only square in row 1 that can be <1>
R9C2 is the only square in row 9 that can be <5>
R8C2 can only be <4>
R1C1 is the only square in row 1 that can be <4>
Squares R2C5 and R2C9 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <57>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C8 - removing <5> from <356> leaving <36>
Squares R6C3 and R7C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <67>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C3 - removing <6> from <236> leaving <23>
R9C3 - removing <6> from <236> leaving <23>
Intersection of column 8 with block 3. The value <6> only appears in one or more of squares R1C8, R2C8 and R3C8 of column 8. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C7 - removing <6> from <367> leaving <37>
Squares R2C1 and R2C8 in row 2 and R8C1 and R8C8 in row 8 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in columns 1 and 8 can be removed.
R1C8 - removing <3> from <356> leaving <56>
R9C1 - removing <3> from <236> leaving <26>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R5C1=<27>
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
R5C1 - removing <27> from <267> leaving <6>
R5C2 can only be <2>
R5C4 can only be <7>
R2C1 can only be <3>
R9C1 can only be <2>
R6C3 can only be <7>
R1C2 can only be <6>
R6C4 can only be <6>
R7C3 can only be <6>
R7C7 can only be <7>
R1C7 can only be <3>
R8C9 can only be <5>
R8C8 can only be <3>
R2C9 can only be <7>
R9C3 can only be <3>
R9C7 can only be <6>
R1C3 can only be <2>
R8C1 can only be <7>
R1C8 can only be <5>
R2C8 can only be <6>
R1C5 can only be <7>
R2C5 can only be <5>
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