Nov 26 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C6 can only be <6>
R5C1 can only be <1>
R5C5 can only be <2>
R8C4 can only be <6>
R2C7 can only be <4>
R1C5 can only be <3>
R5C6 can only be <5>
R9C5 can only be <9>
R8C6 can only be <7>
R4C5 can only be <6>
R6C5 can only be <7>
R2C4 can only be <2>
R2C3 can only be <3>
R6C6 can only be <9>
R4C6 can only be <1>
R3C8 is the only square in row 3 that can be <3>
R4C8 is the only square in row 4 that can be <9>
R7C8 can only be <2>
R7C3 can only be <8>
Squares R4C2 and R6C2 in column 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <38>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R8C2 - removing <3> from <12349> leaving <1249>
R9C2 - removing <3> from <1346> leaving <146>
Squares R7C2<69>, R8C1<39> and R9C1<36> in block 7 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <369>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C2 - removing <9> from <1249> leaving <124>
R9C2 - removing <6> from <146> leaving <14>
Squares R9C2 and R9C8 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <14>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C9 - removing <4> from <346> leaving <36>
Squares R1C2 and R1C8 in row 1 and R9C2 and R9C8 in row 9 form a Simple X-Wing pattern on possibility <1>. All other instances of this possibility in columns 2 and 8 can be removed.
R3C2 - removing <1> from <126> leaving <26>
R8C2 - removing <1> from <124> leaving <24>
R8C8 - removing <1> from <145> leaving <45>
Squares R3C2 and R3C7 in row 3 and R7C2 and R7C7 in row 7 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 2 and 7 can be removed.
R1C2 - removing <6> from <156> leaving <15>
Squares R4C2, R4C4, R6C2 and R6C4 form a Type-1 Unique Rectangle on <38>.
R6C4 - removing <38> from <348> leaving <4>
R6C8 can only be <8>
R5C4 can only be <8>
R6C2 can only be <3>
R5C9 can only be <4>
R4C4 can only be <3>
R4C2 can only be <8>
R2C9 is the only square in row 2 that can be <8>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R1C8=<17>
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
R1C8 - removing <17> from <157> leaving <5>
R1C2 can only be <1>
R1C9 can only be <6>
R2C8 can only be <7>
R8C8 can only be <4>
R1C1 can only be <7>
R9C9 can only be <3>
R3C7 can only be <1>
R2C1 can only be <9>
R3C3 can only be <2>
R8C7 can only be <9>
R8C1 can only be <3>
R7C7 can only be <6>
R8C2 can only be <2>
R9C8 can only be <1>
R9C2 can only be <4>
R9C1 can only be <6>
R8C9 can only be <5>
R2C2 can only be <5>
R3C2 can only be <6>
R8C3 can only be <1>
R7C2 can only be <9>
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