Jul 21 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C7 can only be <5>
R5C6 can only be <4>
R3C8 is the only square in row 3 that can be <5>
R3C3 is the only square in row 3 that can be <7>
R6C6 is the only square in row 6 that can be <7>
R8C1 is the only square in row 8 that can be <5>
R8C2 is the only square in row 8 that can be <1>
R7C5 is the only square in row 7 that can be <1>
R5C4 is the only square in row 5 that can be <1>
R9C8 is the only square in row 9 that can be <7>
R7C8 is the only square in column 8 that can be <4>
R9C9 is the only square in block 9 that can be <3>
Squares R9C4 and R9C6 in row 9 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 row.
R9C1 - removing <6> from <468> leaving <48>
R9C2 - removing <6> from <468> leaving <48>
Squares R4C9 and R6C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <29>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C9 - removing <29> from <12689> leaving <168>
R2C9 - removing <9> from <1689> leaving <168>
R8C9 - removing <29> from <2689> leaving <68>
Squares R9C1 and R9C2 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R8C3 - removing <8> from <689> leaving <69>
Intersection of row 2 with block 3. The value <9> only appears in one or more of squares R2C7, R2C8 and R2C9 of row 2. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C8 - removing <9> from <239> leaving <23>
Intersection of block 4 with column 1. The value <6> only appears in one or more of squares R4C1, R5C1 and R6C1 of block 4. These squares are the ones that intersect with column 1. Thus, the other (non-intersecting) squares of column 1 cannot contain this value.
R1C1 - removing <6> from <1368> leaving <138>
R2C1 - removing <6> from <13468> leaving <1348>
Squares R3C2 and R3C5 in row 3, R5C3 and R5C5 in row 5 and R7C2 and R7C3 in row 7 form a Swordfish pattern on possibility <3>. All other instances of this possibility in columns 2, 3 and 5 can be removed.
R1C2 - removing <3> from <2368> leaving <268>
R2C2 - removing <3> from <3468> leaving <468>
R2C3 - removing <3> from <368> leaving <68>
R4C5 - removing <3> from <236> leaving <26>
Squares R9C1, R9C2, R2C1 and R2C2 form a Type-4 Unique Rectangle on <48>.
R2C1 - removing <8> from <1348> leaving <134>
R2C2 - removing <8> from <468> leaving <46>
Squares R9C4, R9C6, R4C4 and R4C6 form a Type-4 Unique Rectangle on <56>.
R4C4 - removing <6> from <3569> leaving <359>
R4C6 - removing <6> from <569> leaving <59>
Squares R2C2 (XY), R9C2 (XZ) and R2C3 (YZ) form an XY-Wing pattern on <8>. All squares that are buddies of both the XZ and YZ squares cannot be <8>.
R1C2 - removing <8> from <268> leaving <26>
R9C2 is the only square in column 2 that can be <8>
R9C1 can only be <4>
R2C2 is the only square in row 2 that can be <4>
Squares R1C2<26>, R1C4<369>, R1C6<69> and R1C8<23> in row 1 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2369>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C1 - removing <3> from <138> leaving <18>
R1C9 - removing <6> from <168> leaving <18>
Squares R3C7 (XY), R1C8 (XZ) and R3C5 (YZ) form an XY-Wing pattern on <3>. All squares that are buddies of both the XZ and YZ squares cannot be <3>.
R1C4 - removing <3> from <369> leaving <69>
R1C8 is the only square in row 1 that can be <3>
R2C8 can only be <9>
R8C8 can only be <2>
R1C2 is the only square in row 1 that can be <2>
R2C1 is the only square in row 2 that can be <3>
R4C1 can only be <6>
R3C2 can only be <6>
R3C5 can only be <3>
R3C7 can only be <2>
R7C2 can only be <3>
R2C3 can only be <8>
R5C5 can only be <8>
R4C5 can only be <2>
R6C1 can only be <8>
R4C9 can only be <9>
R4C6 can only be <5>
R6C9 can only be <2>
R5C3 can only be <3>
R6C5 can only be <6>
R1C1 can only be <1>
R6C4 can only be <9>
R1C9 can only be <8>
R8C9 can only be <6>
R2C7 can only be <6>
R2C9 can only be <1>
R7C7 can only be <9>
R4C4 can only be <3>
R9C6 can only be <6>
R1C4 can only be <6>
R7C3 can only be <6>
R8C7 can only be <8>
R8C3 can only be <9>
R9C4 can only be <5>
R1C6 can only be <9>
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