Jul 09 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R8C9 can only be <9>
R9C2 can only be <9>
R9C7 can only be <4>
R1C5 is the only square in row 1 that can be <7>
R1C4 is the only square in row 1 that can be <3>
R9C4 can only be <5>
R5C4 can only be <2>
R8C5 can only be <3>
R8C1 can only be <5>
R1C1 is the only square in row 1 that can be <9>
R5C5 is the only square in row 5 that can be <9>
R9C5 is the only square in row 9 that can be <1>
Squares R1C3 and R1C8 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <24>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C9 - removing <24> from <12458> leaving <158>
Squares R5C6 and R5C7 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C9 - removing <58> from <34578> leaving <347>
Squares R1C8 and R3C9 in block 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <24>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R2C9 - removing <2> from <125> leaving <15>
Squares R5C3 and R6C1 in block 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C1 - removing <4> from <47> leaving <7>
R5C1 - removing <34> from <13467> leaving <167>
Squares R1C3 and R9C3 in column 3 and R1C8 and R9C8 in column 8 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in rows 1 and 9 can be removed.
R9C1 - removing <2> from <238> leaving <38>
R9C9 - removing <2> from <267> leaving <67>
Squares R1C3 and R5C3 in column 3 and R1C8 and R5C8 in column 8 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in rows 1 and 5 can be removed.
R5C9 - removing <4> from <347> leaving <37>
Squares R2C5 and R2C9 in row 2 and R6C5 and R6C9 in row 6 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in columns 5 and 9 can be removed.
R1C9 - removing <5> from <158> leaving <18>
Squares R7C5 (XY), R3C5 (XZ) and R7C1 (YZ) form an XY-Wing pattern on <2>. All squares that are buddies of both the XZ and YZ squares cannot be <2>.
R3C1 - removing <2> from <246> leaving <46>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R6C9=<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
R6C9 - removing <35> from <345> leaving <4>
R6C1 can only be <3>
R6C5 can only be <5>
R3C9 can only be <2>
R4C9 can only be <8>
R5C8 can only be <7>
R3C5 can only be <6>
R7C9 can only be <6>
R1C8 can only be <4>
R4C5 can only be <4>
R1C9 can only be <1>
R5C7 can only be <5>
R5C6 can only be <8>
R1C7 can only be <8>
R5C9 can only be <3>
R9C8 can only be <2>
R5C3 can only be <4>
R9C1 can only be <8>
R2C5 can only be <2>
R7C5 can only be <8>
R9C9 can only be <7>
R9C6 can only be <6>
R7C1 can only be <2>
R1C6 can only be <5>
R9C3 can only be <3>
R1C3 can only be <2>
R1C2 can only be <6>
R2C9 can only be <5>
R2C1 can only be <1>
R3C1 can only be <4>
R5C2 can only be <1>
R5C1 can only be <6>
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