Aug 09 - Very Hard

Puzzle Copyright © Kevin Stone

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## Reasoning

R1C5 is the only square in row 1 that can be <4>

R2C7 is the only square in row 2 that can be <4>

R3C2 is the only square in row 3 that can be <2>

R3C5 is the only square in row 3 that can be <7>

R4C5 is the only square in row 4 that can be <2>

R5C3 is the only square in row 5 that can be <4>

R5C7 is the only square in row 5 that can be <5>

R5C9 is the only square in row 5 that can be <3>

R7C2 is the only square in row 7 that can be <4>

R8C3 is the only square in row 8 that can be <5>

Squares R3C4 and R3C6 in row 3 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.

R3C3 - removing <68> from <13689> leaving <139>

R3C7 - removing <68> from <13689> leaving <139>

R3C8 - removing <68> from <368> leaving <3>

R1C1 is the only square in row 1 that can be <3>

R8C7 is the only square in row 8 that can be <3>

R7C3 is the only square in row 7 that can be <3>

Squares R4C7 and R6C7 in column 7 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 column.

R7C7 - removing <68> from <1689> leaving <19>

R7C8 is the only square in row 7 that can be <8>

R1C8 can only be <6>

R9C8 can only be <5>

R7C5 is the only square in row 7 that can be <5>

R9C9 is the only square in column 9 that can be <6>

Intersection of column 3 with block 1. The value <1> only appears in one or more of squares R1C3, R2C3 and R3C3 of column 3. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.

R2C1 - removing <1> from <168> leaving <68>

Intersection of column 5 with block 5. The value <6> only appears in one or more of squares R4C5, R5C5 and R6C5 of column 5. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.

R5C4 - removing <6> from <168> leaving <18>

R5C6 - removing <6> from <689> leaving <89>

Intersection of block 5 with row 5. The values <18> only appears in one or more of squares R5C4, R5C5 and R5C6 of block 5. These squares are the ones that intersect with row 5. Thus, the other (non-intersecting) squares of row 5 cannot contain these values.

R5C1 - removing <8> from <689> leaving <69>

Intersection of block 4 with column 3. The value <8> only appears in one or more of squares R4C3, R5C3 and R6C3 of block 4. These squares are the ones that intersect with column 3. Thus, the other (non-intersecting) squares of column 3 cannot contain this value.

R2C3 - removing <8> from <168> leaving <16>

Squares R4C3, R4C7, R6C3 and R6C7 form a Type-1 Unique Rectangle on <68>.

R6C3 - removing <68> from <689> leaving <9>

R6C5 can only be <6>

R3C3 can only be <1>

R5C1 can only be <6>

R6C7 can only be <8>

R4C7 can only be <6>

R3C7 can only be <9>

R2C3 can only be <6>

R7C7 can only be <1>

R1C9 can only be <8>

R4C3 can only be <8>

R2C1 can only be <8>

R7C4 can only be <6>

R8C9 can only be <9>

R8C1 can only be <1>

R1C2 can only be <9>

R2C9 can only be <1>

R7C6 can only be <9>

R3C4 can only be <8>

R5C6 can only be <8>

R9C5 can only be <1>

R9C1 can only be <9>

R9C2 can only be <8>

R5C5 can only be <9>

R3C6 can only be <6>

R5C4 can only be <1>

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