Jul 14 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C1 is the only square in row 1 that can be <6>
R2C9 is the only square in row 2 that can be <7>
R8C5 is the only square in row 8 that can be <7>
R6C6 is the only square in row 6 that can be <7>
R9C1 is the only square in row 9 that can be <4>
R9C4 is the only square in row 9 that can be <6>
R4C5 is the only square in row 4 that can be <6>
R3C1 is the only square in column 1 that can be <1>
Squares R9C2<27>, R9C3<237>, R9C7<1237> and R9C8<123> in row 9 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1237>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C9 - removing <123> from <12359> leaving <59>
Squares R5C3 and R9C3 in column 3 and R5C8 and R9C8 in column 8 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in rows 5 and 9 can be removed.
R5C4 - removing <3> from <2359> leaving <259>
R5C7 - removing <3> from <1238> leaving <128>
R9C7 - removing <3> from <1237> leaving <127>
R3C7 is the only square in column 7 that can be <3>
Squares R1C2 and R5C2 in column 2 and R1C7 and R5C7 in column 7 form a Simple X-Wing pattern on possibility <8>. All other instances of this possibility in rows 1 and 5 can be removed.
R1C3 - removing <8> from <2789> leaving <279>
R5C3 - removing <8> from <2389> leaving <239>
R5C6 - removing <8> from <4589> leaving <459>
R1C9 - removing <8> from <1248> leaving <124>
R4C6 is the only square in column 6 that can be <8>
Squares R4C9<23>, R7C9<259>, R8C9<35> and R9C9<59> in column 9 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2359>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C9 - removing <2> from <124> leaving <14>
R3C9 - removing <2> from <248> leaving <48>
R6C9 - removing <23> from <1238> leaving <18>
Intersection of block 3 with row 1. The values <12> only appears in one or more of squares R1C7, R1C8 and R1C9 of block 3. These squares are the ones that intersect with row 1. Thus, the other (non-intersecting) squares of row 1 cannot contain these values.
R1C2 - removing <2> from <278> leaving <78>
R1C3 - removing <2> from <279> leaving <79>
R1C4 - removing <2> from <259> leaving <59>
Intersection of column 4 with block 5. The values <23> only appears in one or more of squares R4C4, R5C4 and R6C4 of column 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain these values.
R5C5 - removing <2> from <12459> leaving <1459>
R6C5 - removing <2> from <12> leaving <1>
R6C9 can only be <8>
R3C9 can only be <4>
R3C5 can only be <2>
R1C9 can only be <1>
R1C8 can only be <2>
R3C3 can only be <8>
R2C5 can only be <9>
R1C7 can only be <8>
R2C1 can only be <2>
R7C5 can only be <5>
R1C4 can only be <5>
R1C2 can only be <7>
R5C5 can only be <4>
R9C6 can only be <9>
R9C9 can only be <5>
R8C9 can only be <3>
R1C3 can only be <9>
R9C2 can only be <2>
R1C6 can only be <4>
R5C6 can only be <5>
R6C1 can only be <3>
R7C1 can only be <8>
R6C4 can only be <2>
R4C1 can only be <9>
R8C1 can only be <5>
R5C3 can only be <2>
R5C4 can only be <9>
R4C9 can only be <2>
R9C8 can only be <1>
R5C2 can only be <8>
R7C3 can only be <7>
R9C7 can only be <7>
R5C8 can only be <3>
R4C4 can only be <3>
R7C9 can only be <9>
R5C7 can only be <1>
R7C7 can only be <2>
R9C3 can only be <3>
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