May 14 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C1 is the only square in row 5 that can be <9>
R4C7 is the only square in row 4 that can be <9>
R9C1 is the only square in column 1 that can be <4>
R6C1 is the only square in column 1 that can be <8>
R1C4 is the only square in column 4 that can be <8>
R1C6 is the only square in row 1 that can be <9>
R9C6 is the only square in column 6 that can be <3>
R3C8 is the only square in column 8 that can be <3>
Intersection of column 4 with block 5. The values <45> 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.
R4C6 - removing <5> from <2567> leaving <267>
R5C6 - removing <5> from <256> leaving <26>
Intersection of column 7 with block 3. The values <45> only appears in one or more of squares R1C7, R2C7 and R3C7 of column 7. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain these values.
R1C9 - removing <4> from <12467> leaving <1267>
R3C9 - removing <4> from <1468> leaving <168>
Squares R4C6<267>, R5C6<26>, R6C4<126> and R6C6<1267> in block 5 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1267>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C4 - removing <26> from <2456> leaving <45>
R5C4 - removing <26> from <2456> leaving <45>
Squares R1C2, R7C2 and R9C2 in column 2, R7C5 and R9C5 in column 5 and R1C8 and R9C8 in column 8 form a Swordfish pattern on possibility <7>. All other instances of this possibility in rows 1, 7 and 9 can be removed.
R7C1 - removing <7> from <1567> leaving <156>
R1C3 - removing <7> from <457> leaving <45>
R7C3 - removing <7> from <23567> leaving <2356>
R9C3 - removing <7> from <2679> leaving <269>
R1C9 - removing <7> from <1267> leaving <126>
Squares R1C3 and R3C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <45>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C3 - removing <5> from <2567> leaving <267>
R7C3 - removing <5> from <2356> leaving <236>
Squares R1C3 and R3C3 in block 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <45>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C2 - removing <5> from <157> leaving <17>
R2C1 - removing <5> from <157> leaving <17>
Squares R2C1 and R8C1 in column 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <17>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C1 - removing <7> from <567> leaving <56>
R7C1 - removing <1> from <156> leaving <56>
Intersection of block 4 with column 3. The value <7> 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.
R8C3 - removing <7> from <379> leaving <39>
Squares R1C2<17>, R1C5<126>, R1C8<67> and R1C9<126> in row 1 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <1267>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C7 - removing <12> from <1245> leaving <45>
Squares R1C3, R1C7, R3C3 and R3C7 form a Type-1 Unique Rectangle on <45>.
R3C7 - removing <45> from <1458> leaving <18>
R3C3 is the only square in row 3 that can be <4>
R1C3 can only be <5>
R1C7 can only be <4>
R3C6 is the only square in row 3 that can be <5>
R2C7 is the only square in row 2 that can be <5>
R6C7 is the only square in column 7 that can be <2>
R6C9 is the only square in row 6 that can be <3>
R8C3 is the only square in row 8 that can be <3>
R7C7 is the only square in row 7 that can be <3>
R8C4 is the only square in row 8 that can be <9>
R9C3 is the only square in row 9 that can be <9>
Intersection of row 9 with block 8. The value <6> 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 this value.
R7C4 - removing <6> from <126> leaving <12>
R7C5 - removing <6> from <1267> leaving <127>
Intersection of column 4 with block 8. The value <2> only appears in one or more of squares R7C4, R8C4 and R9C4 of column 4. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R7C5 - removing <2> from <127> leaving <17>
R9C5 - removing <2> from <1267> leaving <167>
R1C5 is the only square in column 5 that can be <2>
R2C6 can only be <1>
R2C1 can only be <7>
R3C5 can only be <6>
R2C9 can only be <2>
R8C1 can only be <1>
R1C2 can only be <1>
R8C9 can only be <7>
R9C8 can only be <8>
R9C7 can only be <1>
R5C8 can only be <6>
R1C9 can only be <6>
R1C8 can only be <7>
R4C9 can only be <4>
R4C4 can only be <5>
R5C9 can only be <8>
R5C6 can only be <2>
R3C9 can only be <1>
R9C5 can only be <7>
R3C7 can only be <8>
R4C1 can only be <6>
R5C4 can only be <4>
R5C2 can only be <5>
R9C2 can only be <2>
R7C5 can only be <1>
R4C6 can only be <7>
R7C1 can only be <5>
R6C3 can only be <7>
R4C3 can only be <2>
R6C6 can only be <6>
R6C4 can only be <1>
R7C4 can only be <2>
R9C4 can only be <6>
R7C2 can only be <7>
R7C3 can only be <6>
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