Sep 24 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C3 can only be <3>
R3C9 can only be <7>
R5C5 can only be <9>
R7C1 can only be <4>
R3C1 can only be <1>
R3C2 can only be <2>
R1C3 can only be <8>
R6C1 can only be <3>
R4C1 can only be <8>
R1C6 is the only square in row 1 that can be <1>
R5C8 is the only square in row 5 that can be <6>
R4C3 is the only square in row 4 that can be <6>
R5C2 is the only square in row 5 that can be <1>
R6C8 is the only square in row 6 that can be <1>
R6C7 is the only square in row 6 that can be <7>
R6C5 is the only square in row 6 that can be <2>
R4C5 can only be <5>
R2C5 can only be <7>
R8C5 can only be <1>
R7C6 is the only square in row 7 that can be <8>
R8C4 is the only square in row 8 that can be <7>
R9C3 is the only square in row 9 that can be <1>
R9C4 is the only square in row 9 that can be <6>
R7C9 is the only square in row 7 that can be <6>
Squares R4C2 and R4C9 in row 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <49>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C7 - removing <49> from <2349> leaving <23>
Squares R1C4 and R5C4 in column 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 column.
R2C4 - removing <4> from <245> leaving <25>
R3C4 - removing <3> from <359> leaving <59>
Squares R4C2, R4C9, R6C2 and R6C9 form a Type-1 Unique Rectangle on <49>.
R6C2 - removing <49> from <459> leaving <5>
R6C3 can only be <9>
R7C2 can only be <9>
R6C9 can only be <4>
R8C3 can only be <5>
R4C2 can only be <4>
R4C9 can only be <9>
R3C4 is the only square in column 4 that can be <9>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R2C6=<24>
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
R2C6 - removing <24> from <245> leaving <5>
R2C4 can only be <2>
R2C7 can only be <4>
R3C6 can only be <3>
R9C6 can only be <9>
R1C7 can only be <3>
R3C8 can only be <5>
R5C6 can only be <4>
R1C4 can only be <4>
R7C8 can only be <2>
R5C4 can only be <3>
R7C4 can only be <5>
R4C8 can only be <3>
R8C7 can only be <9>
R8C6 can only be <2>
R9C7 can only be <5>
R4C7 can only be <2>
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