Jan 15 - Super Hard

Puzzle Copyright © Kevin Stone

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## Reasoning

R4C3 can only be <8>

R5C8 can only be <8>

R6C6 can only be <8>

R8C5 can only be <7>

R4C6 can only be <7>

R4C7 can only be <2>

R6C3 can only be <6>

R5C2 can only be <9>

R4C4 can only be <3>

R3C6 can only be <1>

R5C5 can only be <5>

R6C7 can only be <1>

R6C4 can only be <9>

R7C6 can only be <4>

R2C5 can only be <9>

R7C4 can only be <8>

R3C4 can only be <7>

R7C7 can only be <9>

R2C8 is the only square in row 2 that can be <3>

R2C2 is the only square in row 2 that can be <4>

R3C3 can only be <5>

R3C7 can only be <4>

R3C9 can only be <9>

R3C1 can only be <6>

R1C1 is the only square in row 1 that can be <9>

R7C1 is the only square in row 7 that can be <3>

R8C8 is the only square in row 8 that can be <4>

R8C2 is the only square in row 8 that can be <6>

R9C3 is the only square in row 9 that can be <4>

Squares R2C1 and R2C9 in row 2 and R8C1 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in columns 1 and 9 can be removed.

R1C9 - removing <8> from <1578> leaving <157>

R9C1 - removing <8> from <178> leaving <17>

R9C9 - removing <8> from <12578> leaving <1257>

Squares R9C1 and R9C8 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.

R9C9 - removing <17> from <1257> leaving <25>

The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:

R1C9=<15>

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

R1C9 - removing <15> from <157> leaving <7>

R1C3 can only be <2>

R1C8 can only be <1>

R2C9 can only be <8>

R7C9 can only be <2>

R2C1 can only be <7>

R8C9 can only be <1>

R1C7 can only be <5>

R7C3 can only be <7>

R9C9 can only be <5>

R8C1 can only be <8>

R9C8 can only be <7>

R9C1 can only be <1>

R9C7 can only be <8>

R1C2 can only be <8>

R9C2 can only be <2>

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