May 12 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C2 can only be <6>
R5C8 can only be <3>
R3C7 is the only square in row 3 that can be <3>
R3C6 is the only square in row 3 that can be <4>
R3C3 is the only square in row 3 that can be <8>
R3C8 is the only square in row 3 that can be <9>
R7C8 can only be <7>
R7C6 can only be <9>
R6C6 can only be <8>
R8C5 is the only square in row 8 that can be <7>
R2C5 can only be <5>
R1C5 can only be <1>
R3C4 can only be <7>
R5C4 can only be <1>
R3C2 is the only square in row 3 that can be <1>
R7C2 can only be <5>
R9C9 is the only square in row 9 that can be <8>
Squares R8C3 and R9C1 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <69>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R7C1 - removing <6> from <26> leaving <2>
R7C3 - removing <6> from <1236> leaving <123>
R9C3 - removing <69> from <1369> leaving <13>
Intersection of column 9 with block 3. The value <5> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C7 - removing <5> from <2567> leaving <267>
Squares R2C3, R2C7, R1C3 and R1C7 form a Type-2 Unique Rectangle on <27>.
R1C1 - removing <6> from <567> leaving <57>
R1C9 - removing <6> from <256> leaving <25>
Squares R1C3 and R1C7 in row 1 and R8C3 and R8C7 in row 8 form a Simple X-Wing pattern on possibility <6>. All other instances of this possibility in columns 3 and 7 can be removed.
R7C7 - removing <6> from <146> leaving <14>
R9C7 - removing <6> from <169> leaving <19>
Squares R1C1 (XY), R1C9 (XZ) and R2C3 (YZ) form an XY-Wing pattern on <2>. All squares that are buddies of both the XZ and YZ squares cannot be <2>.
R2C7 - removing <2> from <27> leaving <7>
R1C3 - removing <2> from <267> leaving <67>
R2C3 can only be <2>
Squares R5C1 (XY), R4C3 (XZ) and R5C5 (YZ) form an XY-Wing pattern on <4>. All squares that are buddies of both the XZ and YZ squares cannot be <4>.
R4C5 - removing <4> from <46> leaving <6>
R4C4 can only be <5>
R9C5 can only be <3>
R9C3 can only be <1>
R7C4 can only be <6>
R6C4 can only be <3>
R7C9 can only be <4>
R7C7 can only be <1>
R5C9 can only be <2>
R9C7 can only be <9>
R7C3 can only be <3>
R9C1 can only be <6>
R8C7 can only be <6>
R5C6 can only be <7>
R1C9 can only be <5>
R4C7 can only be <4>
R8C3 can only be <9>
R1C7 can only be <2>
R3C1 can only be <5>
R1C1 can only be <7>
R3C9 can only be <6>
R4C3 can only be <7>
R6C7 can only be <5>
R5C1 can only be <9>
R4C6 can only be <2>
R6C3 can only be <4>
R1C3 can only be <6>
R5C5 can only be <4>
R6C5 can only be <9>
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