The full reasoning can be found below the Sudoku.
Mar 04 - Very Hard
Puzzle Copyright © Kevin Stone
Reasoning
R1C9 is the only square in row 1 that can be <6>
R1C7 is the only square in row 1 that can be <8>
R5C6 is the only square in row 5 that can be <6>
R2C6 can only be <3>
R2C5 is the only square in row 2 that can be <6>
Squares R2C7 and R2C8 in block 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 block.
R3C7 - removing <45> from <34579> leaving <379>
R3C8 - removing <45> from <234579> leaving <2379>
R3C9 - removing <5> from <257> leaving <27>
Intersection of row 6 with block 4. The value <3> only appears in one or more of squares R6C1, R6C2 and R6C3 of row 6. These squares are the ones that intersect with block 4. Thus, the other (non-intersecting) squares of block 4 cannot contain this value.
R4C1 - removing <3> from <2357> leaving <257>
Intersection of row 6 with block 5. The values <49> only appears in one or more of squares R6C4, R6C5 and R6C6 of row 6. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain these values.
R4C5 - removing <9> from <1279> leaving <127>
R4C6 - removing <9> from <1259> leaving <125>
Intersection of block 9 with row 7. The value <2> only appears in one or more of squares R7C7, R7C8 and R7C9 of block 9. These squares are the ones that intersect with row 7. Thus, the other (non-intersecting) squares of row 7 cannot contain this value.
R7C1 - removing <2> from <2567> leaving <567>
R7C2 - removing <2> from <2568> leaving <568>
R7C5 - removing <2> from <1279> leaving <179>
R7C6 - removing <2> from <1259> leaving <159>
Squares R4C9<578>, R5C8<57> and R6C9<578> in block 6 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <578>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C7 - removing <57> from <13579> leaving <139>
R4C8 - removing <57> from <3579> leaving <39>
R5C7 - removing <57> from <157> leaving <1>
Intersection of row 4 with block 5. The value <1> only appears in one or more of squares R4C4, R4C5 and R4C6 of row 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R6C4 - removing <1> from <14578> leaving <4578>
R6C5 - removing <1> from <1479> leaving <479>
R6C6 - removing <1> from <159> leaving <59>
Squares R2C7<45>, R7C7<457> and R9C7<457> in column 7 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <457>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C7 - removing <7> from <379> leaving <39>
Intersection of column 7 with block 9. The value <7> only appears in one or more of squares R7C7, R8C7 and R9C7 of column 7. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain this value.
R7C8 - removing <7> from <2457> leaving <245>
R7C9 - removing <7> from <257> leaving <25>
Squares R8C2<235>, R8C3<357>, R9C1<257> and R9C2<25> in block 7 form a comprehensive naked quad. These 4 squares can only contain the 4 possibilities <2357>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R7C1 - removing <57> from <567> leaving <6>
R7C2 - removing <5> from <568> leaving <68>
R7C3 - removing <57> from <45789> leaving <489>
R9C3 - removing <57> from <4579> leaving <49>
R7C2 can only be <8>
R3C2 is the only square in row 3 that can be <6>
R6C2 is the only square in column 2 that can be <1>
R8C2 is the only square in column 2 that can be <3>
Intersection of row 8 with block 8. The value <2> only appears in one or more of squares R8C4, R8C5 and R8C6 of row 8. These squares are the ones that intersect with block 8. Thus, the other (non-intersecting) squares of block 8 cannot contain this value.
R9C6 - removing <2> from <259> leaving <59>
R4C6 is the only square in column 6 that can be <2>
R5C2 is the only square in row 5 that can be <2>
R9C2 can only be <5>
R9C6 can only be <9>
R8C3 can only be <7>
R9C3 can only be <4>
R6C6 can only be <5>
R7C6 can only be <1>
R7C5 can only be <7>
R8C5 can only be <2>
R9C1 can only be <2>
R8C4 can only be <5>
R9C7 can only be <7>
R7C3 can only be <9>
R4C5 can only be <1>
R3C5 can only be <4>
R6C5 can only be <9>
R6C4 is the only square in row 6 that can be <4>
Squares R2C7, R2C8, R7C7 and R7C8 form a Type-1 Unique Rectangle on <45>.
R7C8 - removing <45> from <245> leaving <2>
R7C9 can only be <5>
R1C8 can only be <3>
R7C7 can only be <4>
R1C3 can only be <1>
R4C8 can only be <9>
R3C7 can only be <9>
R3C8 can only be <7>
R4C7 can only be <3>
R3C9 can only be <2>
R5C8 can only be <5>
R3C4 can only be <1>
R5C3 can only be <8>
R2C8 can only be <4>
R2C7 can only be <5>
R1C4 can only be <2>
R5C4 can only be <7>
R6C3 can only be <3>
R4C4 can only be <8>
R6C1 can only be <7>
R3C3 can only be <5>
R3C1 can only be <3>
R4C9 can only be <7>
R4C1 can only be <5>
R6C9 can only be <8>
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