Sep 17 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R2C7 can only be <4>
R5C5 can only be <9>
R6C7 is the only square in row 6 that can be <3>
R9C4 is the only square in row 9 that can be <3>
R7C2 is the only square in row 7 that can be <3>
R9C6 is the only square in row 9 that can be <6>
Squares R2C3 and R6C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <56>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R8C3 - removing <5> from <158> leaving <18>
Intersection of column 1 with block 4. The values <79> only appears in one or more of squares R4C1, R5C1 and R6C1 of column 1. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain these values.
R4C2 - removing <7> from <2478> leaving <248>
R5C2 - removing <7> from <678> leaving <68>
Squares R7C4<19>, R7C6<89> and R9C5<18> in block 8 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <189>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C4 - removing <19> from <1479> leaving <47>
R8C5 - removing <18> from <1248> leaving <24>
R8C6 - removing <89> from <2789> leaving <27>
Intersection of row 8 with block 9. The value <9> only appears in one or more of squares R8C7, R8C8 and R8C9 of row 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C8 - removing <9> from <189> leaving <18>
Squares R1C4<46>, R3C4<67> and R8C4<47> in column 4 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <467>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C4 - removing <67> from <1679> leaving <19>
Squares R2C2<567>, R3C2<5678>, R5C2<68> and R8C2<58> in column 2 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <5678>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C2 - removing <8> from <248> leaving <24>
R6C2 - removing <56> from <2456> leaving <24>
Squares R4C3 and R8C3 in column 3 and R4C7 and R8C7 in column 7 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in rows 4 and 8 can be removed.
R4C1 - removing <8> from <1789> leaving <179>
R8C2 - removing <8> from <58> leaving <5>
R4C8 - removing <8> from <12489> leaving <1249>
R8C8 - removing <8> from <1589> leaving <159>
R9C9 is the only square in row 9 that can be <5>
Squares R1C9 and R6C9 in column 9 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 column.
R4C9 - removing <2> from <127> leaving <17>
R5C9 - removing <6> from <167> leaving <17>
Squares R4C9 and R5C9 in block 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C8 - removing <1> from <1249> leaving <249>
R5C8 - removing <1> from <168> leaving <68>
Squares R5C2 and R5C8 in row 5 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 row.
R5C1 - removing <8> from <178> leaving <17>
Squares R4C1<179>, R4C3<18>, R4C7<89> and R4C9<17> in row 4 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1789>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R4C8 - removing <9> from <249> leaving <24>
Squares R1C1 and R9C1 in column 1 and R1C5 and R9C5 in column 5 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in rows 1 and 9 can be removed.
R1C6 - removing <8> from <258> leaving <25>
Squares R4C2, R4C8, R6C2 and R6C8 form a Type-1 Unique Rectangle on <24>.
R6C8 - removing <24> from <2469> leaving <69>
R6C2 is the only square in row 6 that can be <4>
R4C2 can only be <2>
R4C8 can only be <4>
R6C9 is the only square in row 6 that can be <2>
R1C9 can only be <6>
R1C4 can only be <4>
R3C8 can only be <5>
R2C8 can only be <2>
R8C4 can only be <7>
R2C5 can only be <1>
R8C6 can only be <2>
R3C4 can only be <6>
R8C5 can only be <4>
R1C6 can only be <5>
R1C1 can only be <8>
R2C4 can only be <9>
R9C5 can only be <8>
R9C1 can only be <1>
R1C5 can only be <2>
R7C6 can only be <9>
R3C2 can only be <7>
R2C6 can only be <7>
R7C4 can only be <1>
R2C2 can only be <6>
R3C6 can only be <8>
R7C8 can only be <8>
R5C8 can only be <6>
R8C7 can only be <9>
R8C8 can only be <1>
R4C7 can only be <8>
R8C3 can only be <8>
R5C1 can only be <7>
R2C3 can only be <5>
R5C2 can only be <8>
R6C3 can only be <6>
R4C3 can only be <1>
R5C9 can only be <1>
R4C1 can only be <9>
R6C8 can only be <9>
R4C9 can only be <7>
R6C1 can only be <5>
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