Sep 17 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C4 can only be <6>
R4C6 can only be <2>
R6C4 can only be <3>
R6C6 can only be <1>
R7C3 can only be <4>
R8C5 can only be <6>
R4C5 can only be <7>
R5C4 can only be <4>
R5C6 can only be <9>
R6C5 can only be <8>
R7C9 can only be <1>
R1C3 can only be <2>
R9C5 can only be <1>
R5C5 can only be <5>
R8C1 is the only square in row 8 that can be <3>
Squares R2C1 and R2C8 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <19>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C2 - removing <9> from <459> leaving <45>
Squares R5C2 and R8C2 in column 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <28>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R9C2 - removing <2> from <259> leaving <59>
Intersection of row 3 with block 3. The value <4> only appears in one or more of squares R3C7, R3C8 and R3C9 of row 3. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C7 - removing <4> from <4689> leaving <689>
R1C9 - removing <4> from <346> leaving <36>
R2C9 - removing <4> from <345> leaving <35>
Intersection of row 8 with block 9. The value <7> only appears in one or more of squares R8C7, R8C8 and R8C9 of row 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R9C8 - removing <7> from <679> leaving <69>
R9C9 - removing <7> from <2467> leaving <246>
Squares R1C9<36>, R2C9<35>, R3C7<46> and R3C9<456> in block 3 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <3456>. 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 <689> leaving <89>
R1C8 - removing <6> from <1689> leaving <189>
Squares R1C7 and R7C7 in column 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <89>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R9C7 - removing <9> from <469> leaving <46>
Squares R2C1, R2C8, R1C1 and R1C8 form a Type-4 Unique Rectangle on <19>.
R1C1 - removing <9> from <169> leaving <16>
R1C8 - removing <9> from <189> leaving <18>
Squares R3C7, R9C7, R3C9 and R9C9 form a Type-4 Unique Rectangle on <46>.
R3C9 - removing <6> from <456> leaving <45>
R9C9 - removing <6> from <246> leaving <24>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R9C1=<27>
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
R9C1 - removing <27> from <279> leaving <9>
R9C2 can only be <5>
R9C8 can only be <6>
R2C1 can only be <1>
R7C1 can only be <8>
R9C3 can only be <7>
R2C2 can only be <4>
R3C3 can only be <5>
R9C7 can only be <4>
R5C8 can only be <7>
R2C8 can only be <9>
R1C1 can only be <6>
R2C5 can only be <3>
R1C2 can only be <9>
R2C9 can only be <5>
R1C5 can only be <4>
R1C7 can only be <8>
R3C9 can only be <4>
R3C7 can only be <6>
R9C9 can only be <2>
R5C9 can only be <6>
R8C8 can only be <8>
R1C9 can only be <3>
R7C7 can only be <9>
R5C1 can only be <2>
R8C2 can only be <2>
R8C9 can only be <7>
R5C2 can only be <8>
R1C8 can only be <1>
R3C1 can only be <7>
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