Feb 05 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C1 is the only square in row 1 that can be <3>
R1C3 is the only square in row 1 that can be <5>
R2C1 is the only square in row 2 that can be <7>
R4C9 is the only square in row 4 that can be <3>
R5C1 is the only square in row 5 that can be <5>
R5C5 is the only square in row 5 that can be <9>
R4C5 can only be <8>
R4C1 can only be <9>
R8C1 is the only square in row 8 that can be <1>
R9C8 is the only square in row 9 that can be <3>
R9C9 is the only square in row 9 that can be <5>
R1C4 is the only square in column 4 that can be <8>
R2C8 is the only square in column 8 that can be <8>
R7C9 is the only square in column 9 that can be <7>
Squares R1C8 and R2C9 in block 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <26>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C7 - removing <6> from <169> leaving <19>
R1C9 - removing <26> from <12469> leaving <149>
R3C9 - removing <26> from <246> leaving <4>
R1C2 is the only square in row 1 that can be <4>
Intersection of block 9 with row 8. The value <2> only appears in one or more of squares R8C7, R8C8 and R8C9 of block 9. These squares are the ones that intersect with row 8. Thus, the other (non-intersecting) squares of row 8 cannot contain this value.
R8C2 - removing <2> from <269> leaving <69>
Squares R3C1 and R3C5 in row 3 and R7C1 and R7C5 in row 7 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 1 and 5 can be removed.
R1C5 - removing <2> from <267> leaving <67>
R9C1 - removing <2> from <2468> leaving <468>
R9C5 - removing <2> from <1247> leaving <147>
Squares R6C1 and R6C5 in row 6 and R7C1 and R7C5 in row 7 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 1 and 5 can be removed.
R9C1 - removing <4> from <468> leaving <68>
R9C5 - removing <4> from <147> leaving <17>
Squares R2C2 and R8C8 form a remote naked pair. <26> can be removed from any square that is common to their groups.
R8C2 - removing <6> from <69> leaving <9>
R9C7 is the only square in row 9 that can be <9>
R1C7 can only be <1>
R1C9 can only be <9>
R5C7 can only be <6>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R9C6=<47>
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
R9C6 - removing <47> from <247> leaving <2>
R9C2 can only be <6>
R9C4 can only be <1>
R1C6 can only be <7>
R5C6 can only be <4>
R7C5 can only be <4>
R1C5 can only be <6>
R5C3 can only be <8>
R6C5 can only be <1>
R6C9 can only be <8>
R9C5 can only be <7>
R5C4 can only be <2>
R6C1 can only be <4>
R5C9 can only be <1>
R7C1 can only be <2>
R9C1 can only be <8>
R2C2 can only be <2>
R1C8 can only be <2>
R3C5 can only be <2>
R8C8 can only be <6>
R2C9 can only be <6>
R3C1 can only be <6>
R8C9 can only be <2>
R9C3 can only be <4>
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