Jan 08 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C5 can only be <2>
R7C3 can only be <3>
R7C2 can only be <6>
R7C7 can only be <2>
R7C8 can only be <5>
R3C3 is the only square in row 3 that can be <5>
R4C4 is the only square in row 4 that can be <1>
R6C6 is the only square in row 6 that can be <4>
R6C2 is the only square in row 6 that can be <8>
R9C9 is the only square in row 9 that can be <4>
Squares R2C7, R3C7 and R3C8 in block 3 form a simple naked triplet. These 3 squares all 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.
R1C8 - removing <369> from <12369> leaving <12>
R1C9 - removing <36> from <12368> leaving <128>
R2C9 - removing <36> from <1368> leaving <18>
Intersection of row 3 with block 3. The value <6> 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.
R2C7 - removing <6> from <369> leaving <39>
Intersection of column 7 with block 3. The value <9> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C8 - removing <9> from <369> leaving <36>
Squares R2C1 and R2C9 in row 2 and R8C1 and R8C9 in row 8 form a Simple X-Wing pattern on possibility <1>. All other instances of this possibility in columns 1 and 9 can be removed.
R1C1 - removing <1> from <13679> leaving <3679>
R1C9 - removing <1> from <128> leaving <28>
R9C1 - removing <1> from <129> leaving <29>
Squares R8C6 (XY), R8C7 (XZ) and R9C5 (YZ) form an XY-Wing pattern on <6>. All squares that are buddies of both the XZ and YZ squares cannot be <6>.
R9C8 - removing <6> from <136> leaving <13>
R8C4 - removing <6> from <236> leaving <23>
Squares R9C2 (XY), R9C8 (XZ) and R3C2 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R3C8 - removing <3> from <36> leaving <6>
R4C9 is the only square in row 4 that can be <6>
R8C7 is the only square in row 8 that can be <6>
Squares R9C8 (XY), R9C2 (XZ) and R4C8 (YZ) form an XY-Wing pattern on <9>. All squares that are buddies of both the XZ and YZ squares cannot be <9>.
R4C2 - removing <9> from <379> leaving <37>
Intersection of block 4 with column 1. The values <59> only appears in one or more of squares R4C1, R5C1 and R6C1 of block 4. These squares are the ones that intersect with column 1. Thus, the other (non-intersecting) squares of column 1 cannot contain these values.
R1C1 - removing <9> from <3679> leaving <367>
R2C1 - removing <9> from <13679> leaving <1367>
R8C1 - removing <9> from <1279> leaving <127>
R9C1 - removing <9> from <29> leaving <2>
R8C4 is the only square in row 8 that can be <2>
Squares R3C2 (XY), R4C2 (XZ) and R2C3 (YZ) form an XY-Wing pattern on <7>. All squares that are buddies of both the XZ and YZ squares cannot be <7>.
R1C2 - removing <7> from <1379> leaving <139>
R4C2 is the only square in column 2 that can be <7>
R4C6 can only be <5>
R6C4 can only be <7>
R9C4 is the only square in row 9 that can be <5>
R9C5 is the only square in row 9 that can be <6>
R1C5 can only be <9>
Squares R8C6 and R9C6 in column 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <39>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C6 - removing <3> from <378> leaving <78>
R2C6 - removing <3> from <378> leaving <78>
Intersection of column 2 with block 1. The value <3> only appears in one or more of squares R1C2, R2C2 and R3C2 of column 2. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R1C1 - removing <3> from <367> leaving <67>
R2C1 - removing <3> from <1367> leaving <167>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R2C1=<16>
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
R2C1 - removing <16> from <167> leaving <7>
R2C3 can only be <9>
R2C6 can only be <8>
R1C1 can only be <6>
R8C1 can only be <1>
R2C7 can only be <3>
R8C3 can only be <7>
R3C2 can only be <3>
R2C9 can only be <1>
R1C6 can only be <7>
R2C4 can only be <6>
R3C7 can only be <9>
R8C9 can only be <3>
R1C8 can only be <2>
R1C2 can only be <1>
R9C2 can only be <9>
R8C6 can only be <9>
R5C9 can only be <5>
R9C8 can only be <1>
R9C6 can only be <3>
R1C4 can only be <3>
R1C9 can only be <8>
R6C8 can only be <9>
R5C1 can only be <3>
R6C9 can only be <2>
R6C1 can only be <5>
R4C8 can only be <3>
R4C1 can only be <9>
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