Dec 31 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R9C6 can only be <3>
R1C3 is the only square in row 1 that can be <4>
R1C4 is the only square in row 1 that can be <3>
R2C5 is the only square in row 2 that can be <8>
R4C6 is the only square in row 4 that can be <2>
R4C1 is the only square in row 4 that can be <9>
R5C5 is the only square in row 5 that can be <4>
R5C6 is the only square in row 5 that can be <9>
R6C5 is the only square in row 6 that can be <3>
R7C2 is the only square in row 7 that can be <3>
R3C8 is the only square in row 3 that can be <3>
R2C3 is the only square in row 2 that can be <3>
R3C9 is the only square in row 3 that can be <9>
R8C2 is the only square in row 8 that can be <2>
R8C8 is the only square in row 8 that can be <9>
R9C7 is the only square in row 9 that can be <4>
R3C5 is the only square in column 5 that can be <6>
R3C2 can only be <5>
R1C6 can only be <1>
R6C6 can only be <6>
R3C1 can only be <1>
R5C7 is the only square in column 7 that can be <8>
R4C4 is the only square in row 4 that can be <8>
Squares R7C5 and R7C8 in row 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C1 - removing <5> from <568> leaving <68>
R7C9 - removing <15> from <1568> leaving <68>
R6C9 is the only square in column 9 that can be <1>
R5C4 is the only square in row 5 that can be <1>
Squares R7C8 and R8C7 in block 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <15>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R9C9 - removing <5> from <568> leaving <68>
Squares R7C9 and R9C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <68>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C9 - removing <6> from <567> leaving <57>
Squares R6C1 and R9C1 in column 1 and R6C4 and R9C4 in column 4 form a Simple X-Wing pattern on possibility <5>. All other instances of this possibility in rows 6 and 9 can be removed.
R9C3 - removing <5> from <567> leaving <67>
Squares R2C2 and R5C2 in column 2 and R2C8 and R5C8 in column 8 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in rows 2 and 5 can be removed.
R5C3 - removing <7> from <567> leaving <56>
Intersection of column 3 with block 7. The value <7> only appears in one or more of squares R7C3, R8C3 and R9C3 of column 3. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C1 - removing <7> from <5678> leaving <568>
Squares R7C1, R7C9, R9C1 and R9C9 form a Type-1 Unique Rectangle on <68>.
R9C1 - removing <68> from <568> leaving <5>
R9C4 can only be <7>
R6C1 can only be <7>
R8C3 can only be <7>
R9C3 can only be <6>
R6C4 can only be <5>
R1C1 can only be <6>
R5C2 can only be <6>
R4C5 can only be <7>
R9C9 can only be <8>
R5C3 can only be <5>
R7C1 can only be <8>
R7C9 can only be <6>
R1C7 can only be <5>
R2C2 can only be <7>
R1C9 can only be <7>
R8C7 can only be <1>
R4C9 can only be <5>
R2C8 can only be <1>
R2C7 can only be <6>
R7C8 can only be <5>
R5C8 can only be <7>
R7C5 can only be <1>
R8C5 can only be <5>
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