May 31 - Very Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R6C6 can only be <2>
R4C1 is the only square in row 4 that can be <1>
R2C3 is the only square in row 2 that can be <1>
R1C6 is the only square in row 1 that can be <1>
R5C7 is the only square in row 5 that can be <9>
R6C5 is the only square in row 6 that can be <9>
R7C2 is the only square in row 7 that can be <1>
R8C1 is the only square in column 1 that can be <2>
Intersection of column 4 with block 5. The value <8> 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 this value.
R4C5 - removing <8> from <3578> leaving <357>
Intersection of column 5 with block 8. The values <68> only appears in one or more of squares R7C5, R8C5 and R9C5 of column 5. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain these values.
R9C4 - removing <6> from <456> leaving <45>
Squares R9C4 and R9C6 in row 9 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 row.
R9C8 - removing <5> from <357> leaving <37>
R9C9 - removing <5> from <3567> leaving <367>
Squares R9C4 and R9C6 in block 8 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.
R7C5 - removing <5> from <568> leaving <68>
R8C5 - removing <5> from <568> leaving <68>
Squares R8C5 and R8C7 in row 8 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.
R8C3 - removing <6> from <356> leaving <35>
R8C9 - removing <68> from <3568> leaving <35>
Intersection of column 6 with block 5. The value <3> only appears in one or more of squares R4C6, R5C6 and R6C6 of column 6. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R4C5 - removing <3> from <357> leaving <57>
Intersection of column 7 with block 3. The value <7> 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 this value.
R1C8 - removing <7> from <578> leaving <58>
R1C9 - removing <7> from <56789> leaving <5689>
R2C9 - removing <7> from <789> leaving <89>
R3C8 - removing <7> from <257> leaving <25>
Intersection of block 4 with row 5. The value <7> only appears in one or more of squares R5C1, R5C2 and R5C3 of block 4. These squares are the ones that intersect with row 5. Thus, the other (non-intersecting) squares of row 5 cannot contain this value.
R5C4 - removing <7> from <4678> leaving <468>
R5C8 - removing <7> from <378> leaving <38>
R9C8 is the only square in column 8 that can be <7>
R5C8 is the only square in column 8 that can be <3>
R5C6 can only be <4>
R9C6 can only be <5>
R9C4 can only be <4>
R4C6 can only be <3>
Squares R4C9 and R6C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <78>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C9 - removing <8> from <5689> leaving <569>
R2C9 - removing <8> from <89> leaving <9>
Squares R8C5, R8C7, R7C5 and R7C7 form a Type-1 Unique Rectangle on <68>.
R7C7 - removing <68> from <268> leaving <2>
R3C8 is the only square in row 3 that can be <2>
R3C5 is the only square in row 3 that can be <5>
R4C5 can only be <7>
R1C4 can only be <7>
R4C9 can only be <8>
R2C5 can only be <3>
R4C4 can only be <5>
R6C9 can only be <7>
R2C1 can only be <8>
R2C7 can only be <7>
R6C1 can only be <6>
R3C7 can only be <6>
R3C2 can only be <7>
R8C7 can only be <8>
R1C9 can only be <5>
R6C4 can only be <8>
R1C1 can only be <9>
R5C3 can only be <7>
R5C4 can only be <6>
R8C5 can only be <6>
R7C8 can only be <5>
R1C2 can only be <6>
R9C1 can only be <3>
R9C2 can only be <9>
R1C8 can only be <8>
R8C9 can only be <3>
R3C3 can only be <3>
R5C2 can only be <8>
R8C3 can only be <5>
R7C3 can only be <6>
R7C5 can only be <8>
R9C9 can only be <6>
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