Jul 09 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C5 is the only square in row 2 that can be <6>
R3C5 is the only square in row 3 that can be <9>
R8C5 is the only square in row 8 that can be <3>
R1C6 is the only square in row 1 that can be <3>
R1C5 is the only square in block 2 that can be <2>
R1C1 is the only square in row 1 that can be <7>
Squares R1C8 and R4C8 in column 8 form a simple naked pair. These 2 squares both contain the 2 possibilities <45>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R6C8 - removing <5> from <568> leaving <68>
R8C8 - removing <5> from <568> leaving <68>
Squares R8C1 and R9C2 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <26>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C3 - removing <2> from <278> leaving <78>
R9C1 - removing <26> from <269> leaving <9>
R9C3 - removing <2> from <2789> leaving <789>
Intersection of row 1 with block 3. The values <145> only appears in one or more of squares R1C7, R1C8 and R1C9 of row 1. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain these values.
R2C7 - removing <1> from <189> leaving <89>
R2C9 - removing <14> from <1489> leaving <89>
Intersection of column 1 with block 1. The value <1> only appears in one or more of squares R1C1, R2C1 and R3C1 of column 1. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R2C2 - removing <1> from <123> leaving <23>
Intersection of column 9 with block 9. The value <2> only appears in one or more of squares R7C9, R8C9 and R9C9 of column 9. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C7 - removing <2> from <2678> leaving <678>
R8C7 - removing <2> from <125678> leaving <15678>
Squares R4C4<14>, R4C5<145> and R4C8<45> in row 4 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <145>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C2 - removing <1> from <123> leaving <23>
R4C3 - removing <5> from <2359> leaving <239>
R4C7 - removing <5> from <259> leaving <29>
R6C2 is the only square in column 2 that can be <1>
R9C2 is the only square in column 2 that can be <6>
R8C1 can only be <2>
R2C1 can only be <1>
R2C6 can only be <4>
R3C1 can only be <5>
R3C3 can only be <4>
R5C1 can only be <6>
R3C6 is the only square in column 6 that can be <1>
R3C4 can only be <7>
Squares R2C2, R2C3, R4C2 and R4C3 form a Type-1 Unique Rectangle on <23>.
R4C3 - removing <23> from <239> leaving <9>
R4C7 can only be <2>
R5C3 can only be <5>
R4C2 can only be <3>
R6C3 can only be <2>
R2C3 can only be <3>
R2C2 can only be <2>
Squares R2C7 and R5C7 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.
R6C7 - removing <8> from <568> leaving <56>
R7C7 - removing <8> from <678> leaving <67>
R8C7 - removing <8> from <15678> leaving <1567>
Squares R2C7, R2C9, R5C7 and R5C9 form a Type-1 Unique Rectangle on <89>.
R5C9 - removing <89> from <489> leaving <4>
R5C5 can only be <8>
R4C8 can only be <5>
R1C8 can only be <4>
R6C7 can only be <6>
R5C7 can only be <9>
R6C6 can only be <7>
R2C7 can only be <8>
R6C5 can only be <5>
R7C6 can only be <8>
R6C8 can only be <8>
R7C7 can only be <7>
R8C8 can only be <6>
R7C9 can only be <2>
R7C5 can only be <4>
R8C4 can only be <1>
R2C9 can only be <9>
R7C4 can only be <6>
R4C5 can only be <1>
R8C7 can only be <5>
R4C4 can only be <4>
R9C4 can only be <2>
R9C5 can only be <7>
R8C9 can only be <8>
R1C7 can only be <1>
R8C3 can only be <7>
R9C9 can only be <1>
R9C3 can only be <8>
R1C9 can only be <5>
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