Oct 01 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C8 is the only square in row 4 that can be <2>
R4C2 is the only square in row 4 that can be <5>
R5C4 is the only square in row 5 that can be <4>
R5C3 is the only square in row 5 that can be <9>
R6C8 is the only square in row 6 that can be <5>
R6C2 is the only square in row 6 that can be <6>
R6C5 is the only square in row 6 that can be <9>
R3C4 is the only square in row 3 that can be <9>
R7C3 is the only square in row 7 that can be <4>
R8C8 is the only square in row 8 that can be <4>
R8C9 is the only square in row 8 that can be <3>
R9C1 is the only square in row 9 that can be <6>
R6C3 is the only square in column 3 that can be <8>
R6C7 can only be <3>
R3C7 is the only square in column 7 that can be <6>
R9C9 is the only square in column 9 that can be <7>
R8C2 is the only square in column 2 that can be <7>
Squares R7C4 and R7C7 in row 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <18>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C5 - removing <1> from <127> leaving <27>
Squares R5C8 and R9C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <18>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C8 - removing <1> from <139> leaving <39>
R2C8 - removing <1> from <139> leaving <39>
Intersection of row 3 with block 2. The value <2> only appears in one or more of squares R3C4, R3C5 and R3C6 of row 3. These squares are the ones that intersect with block 2. Thus, the other (non-intersecting) squares of block 2 cannot contain this value.
R2C5 - removing <2> from <1236> leaving <136>
R2C6 - removing <2> from <2356> leaving <356>
Squares R3C6<23>, R5C6<37> and R7C6<27> in column 6 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <237>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C6 - removing <3> from <356> leaving <56>
Squares R1C9, R2C9, R1C2 and R2C2 form a Type-4 Unique Rectangle on <12>.
R1C2 - removing <1> from <123> leaving <23>
R2C2 - removing <1> from <1238> leaving <238>
Squares R5C2 and R9C2 in column 2 and R5C8 and R9C8 in column 8 form a Simple X-Wing pattern on possibility <1>. All other instances of this possibility in rows 5 and 9 can be removed.
R5C7 - removing <1> from <178> leaving <78>
Squares R1C1 (XY), R3C3 (XZ) and R1C8 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R1C2 - removing <3> from <23> leaving <2>
R1C9 can only be <1>
R1C1 can only be <9>
R2C9 can only be <2>
R1C8 can only be <3>
R2C8 can only be <9>
Squares R2C1 and R7C4 form a remote naked pair. <18> can be removed from any square that is common to their groups.
R2C4 - removing <1> from <15> leaving <5>
R2C6 can only be <6>
R8C6 can only be <5>
R8C5 is the only square in row 8 that can be <6>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R3C5=<12>
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
R3C5 - removing <12> from <123> leaving <3>
R3C3 can only be <1>
R3C6 can only be <2>
R2C5 can only be <1>
R4C5 can only be <7>
R7C6 can only be <7>
R4C7 can only be <1>
R7C5 can only be <2>
R5C6 can only be <3>
R4C3 can only be <3>
R7C7 can only be <8>
R5C8 can only be <8>
R5C2 can only be <1>
R5C7 can only be <7>
R9C8 can only be <1>
R7C4 can only be <1>
R9C2 can only be <8>
R2C1 can only be <8>
R8C4 can only be <8>
R8C1 can only be <1>
R2C2 can only be <3>
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