Jun 24 - Super Hard

Puzzle Copyright © Kevin Stone

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## Reasoning

R1C4 can only be <2>

R5C1 is the only square in row 5 that can be <5>

R8C5 is the only square in row 8 that can be <5>

R2C7 is the only square in row 2 that can be <5>

R2C3 is the only square in row 2 that can be <2>

R3C7 is the only square in row 3 that can be <2>

R3C4 is the only square in row 3 that can be <5>

R5C2 is the only square in column 2 that can be <6>

R7C6 is the only square in column 6 that can be <6>

R7C7 is the only square in column 7 that can be <4>

Squares R6C1 and R6C9 in row 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <19>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.

R6C3 - removing <19> from <1479> leaving <47>

R6C5 - removing <19> from <1469> leaving <46>

R6C7 - removing <9> from <679> leaving <67>

Intersection of row 9 with block 8. The values <47> only appears in one or more of squares R9C4, R9C5 and R9C6 of row 9. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain these values.

R7C5 - removing <7> from <1379> leaving <139>

Intersection of column 2 with block 7. The value <7> only appears in one or more of squares R7C2, R8C2 and R9C2 of column 2. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.

R7C3 - removing <7> from <13789> leaving <1389>

R8C3 - removing <7> from <1379> leaving <139>

Squares R4C1<189>, R4C3<189> and R6C1<19> in block 4 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.

R5C3 - removing <189> from <14789> leaving <47>

Intersection of row 5 with block 6. The values <38> only appears in one or more of squares R5C7, R5C8 and R5C9 of row 5. These squares are the ones that intersect with block 6. Thus, the other (non-intersecting) squares of block 6 cannot contain these values.

R4C9 - removing <8> from <1289> leaving <129>

Squares R1C1 and R1C9 in row 1 and R6C1 and R6C9 in row 6 form a Simple X-Wing pattern on possibility <9>. All other instances of this possibility in columns 1 and 9 can be removed.

R4C1 - removing <9> from <189> leaving <18>

R4C9 - removing <9> from <129> leaving <12>

R5C9 - removing <9> from <12389> leaving <1238>

R9C1 - removing <9> from <139> leaving <13>

R9C9 - removing <9> from <139> leaving <13>

Squares R9C1 and R9C9 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <13>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.

R9C4 - removing <1> from <149> leaving <49>

R9C5 - removing <13> from <13479> leaving <479>

R9C6 - removing <13> from <1347> leaving <47>

R7C5 is the only square in block 8 that can be <3>

R2C8 is the only square in row 2 that can be <3>

R7C4 is the only square in block 8 that can be <1>

Squares R6C1 and R6C9 in row 6 and R9C1 and R9C9 in row 9 form a Simple X-Wing pattern on possibility <1>. All other instances of this possibility in columns 1 and 9 can be removed.

R4C1 - removing <1> from <18> leaving <8>

R4C9 - removing <1> from <12> leaving <2>

R5C9 - removing <1> from <1238> leaving <238>

R5C5 is the only square in row 5 that can be <2>

Squares R3C2 (XYZ), R3C8 (XZ) and R2C2 (YZ) form an XYZ-Wing pattern on <8>. All squares that are buddies of all three squares cannot be <8>.

R3C3 - removing <8> from <1389> leaving <139>

R7C3 is the only square in column 3 that can be <8>

Squares R7C2, R7C8, R8C2 and R8C8 form a Type-2 Unique Rectangle on <79>.

R8C3 - removing <1> from <139> leaving <39>

Squares R8C3 (XY), R9C1 (XZ) and R4C3 (YZ) form an XY-Wing pattern on <1>. All squares that are buddies of both the XZ and YZ squares cannot be <1>.

R6C1 - removing <1> from <19> leaving <9>

R6C9 can only be <1>

R1C1 can only be <3>

R4C3 can only be <1>

R9C9 can only be <3>

R9C1 can only be <1>

R5C9 can only be <8>

R1C6 can only be <7>

R1C5 can only be <8>

R9C6 can only be <4>

R3C3 can only be <9>

R1C9 can only be <9>

R9C4 can only be <9>

R5C6 can only be <1>

R2C5 can only be <1>

R3C8 can only be <8>

R2C2 can only be <8>

R3C5 can only be <4>

R3C6 can only be <3>

R8C3 can only be <3>

R6C5 can only be <6>

R3C2 can only be <1>

R6C7 can only be <7>

R4C5 can only be <9>

R6C3 can only be <4>

R8C7 can only be <9>

R5C8 can only be <9>

R8C2 can only be <7>

R4C7 can only be <6>

R5C7 can only be <3>

R7C8 can only be <7>

R9C5 can only be <7>

R5C4 can only be <4>

R5C3 can only be <7>

R7C2 can only be <9>

R8C8 can only be <1>

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