Dec 03 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R4C4 can only be <8>
R5C2 can only be <7>
R5C4 can only be <4>
R6C3 can only be <8>
R3C4 can only be <7>
R6C6 can only be <7>
R6C7 can only be <4>
R6C5 can only be <5>
R7C4 can only be <5>
R2C5 can only be <8>
R6C4 can only be <1>
R3C5 can only be <2>
R3C6 can only be <4>
R1C1 is the only square in row 1 that can be <4>
R7C9 is the only square in row 7 that can be <4>
R3C1 is the only square in column 1 that can be <8>
R7C6 is the only square in column 6 that can be <8>
R9C8 is the only square in column 8 that can be <9>
R7C1 is the only square in column 1 that can be <9>
R8C1 is the only square in column 1 that can be <2>
R7C7 is the only square in row 7 that can be <2>
R1C9 is the only square in row 1 that can be <2>
R9C9 is the only square in column 9 that can be <8>
Squares R7C5 and R8C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <37>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C5 - removing <3> from <369> leaving <69>
R5C5 - removing <3> from <369> leaving <69>
Intersection of column 9 with block 3. The value <6> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R3C7 - removing <6> from <356> leaving <35>
Squares R1C8, R5C8, R1C7 and R5C7 form a Type-4 Unique Rectangle on <38>.
R1C7 - removing <3> from <3578> leaving <578>
R5C7 - removing <3> from <368> leaving <68>
Squares R1C3 (XYZ), R1C2 (XZ) and R7C3 (YZ) form an XYZ-Wing pattern on <3>. All squares that are buddies of all three squares cannot be <3>.
R3C3 - removing <3> from <356> leaving <56>
Intersection of row 3 with block 3. The value <3> 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.
R1C8 - removing <3> from <38> leaving <8>
R5C8 can only be <3>
R5C6 can only be <2>
R4C7 can only be <6>
R4C5 can only be <9>
R5C7 can only be <8>
R5C3 can only be <9>
R4C6 can only be <3>
R4C3 can only be <2>
R5C5 can only be <6>
Squares R1C7 (XY), R1C2 (XZ) and R9C7 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R9C2 - removing <3> from <35> leaving <5>
R1C2 can only be <3>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R9C3=<36>
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
R9C3 - removing <36> from <367> leaving <7>
R9C1 can only be <6>
R9C7 can only be <3>
R1C3 can only be <5>
R7C3 can only be <3>
R3C7 can only be <5>
R8C9 can only be <7>
R1C7 can only be <7>
R3C3 can only be <6>
R2C9 can only be <6>
R2C1 can only be <7>
R3C9 can only be <3>
R7C5 can only be <7>
R8C5 can only be <3>
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