Jul 08 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C7 can only be <8>
R6C5 can only be <4>
R4C5 can only be <8>
R1C3 is the only square in row 1 that can be <5>
R2C6 is the only square in row 2 that can be <8>
R5C1 is the only square in row 5 that can be <7>
R2C2 is the only square in row 2 that can be <7>
R6C8 is the only square in row 6 that can be <7>
Intersection of row 9 with block 9. The values <46> only appears in one or more of squares R9C7, R9C8 and R9C9 of row 9. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain these values.
R8C8 - removing <6> from <12356> leaving <1235>
R8C9 - removing <6> from <1568> leaving <158>
Squares R1C1<269>, R1C2<2469>, R2C1<26> and R2C3<46> in block 1 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2469>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R3C1 - removing <269> from <12369> leaving <13>
R3C2 - removing <2469> from <123469> leaving <13>
Intersection of row 3 with block 2. The value <9> 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.
R1C4 - removing <9> from <2469> leaving <246>
R1C6 - removing <9> from <2679> leaving <267>
Squares R5C9<19>, R6C9<59>, R7C9<1589> and R8C9<158> in column 9 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1589>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C9 - removing <5> from <456> leaving <46>
R9C9 - removing <1> from <146> leaving <46>
R2C8 is the only square in row 2 that can be <5>
R2C1 is the only square in row 2 that can be <2>
R4C6 is the only square in row 4 that can be <5>
Intersection of row 1 with block 2. The values <27> only appears in one or more of squares R1C4, R1C5 and R1C6 of row 1. These squares are the ones that intersect with block 2. Thus, the other (non-intersecting) squares of block 2 cannot contain these values.
R3C4 - removing <2> from <2469> leaving <469>
R3C6 - removing <2> from <269> leaving <69>
Squares R7C6<23>, R9C5<27> and R9C6<237> in block 8 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 block.
R7C4 - removing <2> from <1258> leaving <158>
R8C4 - removing <2> from <1258> leaving <158>
R9C4 - removing <2> from <12> leaving <1>
Squares R7C2<123>, R7C3<13> and R7C6<23> in row 7 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <123>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C8 - removing <123> from <1239> leaving <9>
R7C9 - removing <1> from <1589> leaving <589>
Intersection of row 4 with block 4. The values <49> only appears in one or more of squares R4C1, R4C2 and R4C3 of row 4. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain these values.
R6C2 - removing <9> from <369> leaving <36>
Intersection of row 7 with block 7. The value <1> only appears in one or more of squares R7C1, R7C2 and R7C3 of row 7. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R8C1 - removing <1> from <136> leaving <36>
R8C2 - removing <1> from <1236> leaving <236>
Squares R3C2<13>, R6C2<36>, R7C2<123> and R8C2<236> in column 2 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1236>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C2 - removing <6> from <469> leaving <49>
R4C2 - removing <13> from <1349> leaving <49>
Squares R7C4, R7C9, R8C4 and R8C9 form a Type-1 Unique Rectangle on <58>.
R8C9 - removing <58> from <158> leaving <1>
R5C9 can only be <9>
R6C9 can only be <5>
R6C7 can only be <3>
R7C9 can only be <8>
R7C4 can only be <5>
R6C2 can only be <6>
R4C8 can only be <1>
R8C4 can only be <8>
R6C4 can only be <9>
R5C3 can only be <1>
R7C3 can only be <3>
R7C6 can only be <2>
R4C3 can only be <4>
R8C1 can only be <6>
R8C2 can only be <2>
R7C2 can only be <1>
R5C6 can only be <6>
R9C5 can only be <7>
R1C1 can only be <9>
R8C7 can only be <5>
R8C8 can only be <3>
R9C6 can only be <3>
R1C5 can only be <2>
R1C2 can only be <4>
R4C1 can only be <3>
R4C2 can only be <9>
R2C3 can only be <6>
R1C4 can only be <6>
R2C9 can only be <4>
R9C9 can only be <6>
R3C7 can only be <2>
R3C8 can only be <6>
R9C7 can only be <4>
R3C4 can only be <4>
R3C6 can only be <9>
R9C8 can only be <2>
R3C1 can only be <1>
R5C4 can only be <2>
R1C6 can only be <7>
R3C2 can only be <3>
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