The full reasoning can be found below the Sudoku.
Apr 04 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R5C1 can only be <9>
R5C2 can only be <4>
R5C5 can only be <8>
R5C9 can only be <2>
R5C8 can only be <5>
R2C3 is the only square in row 2 that can be <3>
R4C6 is the only square in row 4 that can be <2>
R8C8 is the only square in row 8 that can be <8>
R1C9 is the only square in row 1 that can be <8>
R9C5 is the only square in row 9 that can be <1>
R9C8 is the only square in row 9 that can be <3>
R7C5 is the only square in row 7 that can be <3>
R9C2 is the only square in row 9 that can be <8>
R9C3 is the only square in column 3 that can be <4>
R7C9 is the only square in column 9 that can be <6>
R7C7 is the only square in row 7 that can be <4>
R6C8 is the only square in block 6 that can be <1>
R7C8 is the only square in block 9 that can be <7>
Squares R8C7 and R9C7 in column 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <29>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R1C7 - removing <29> from <1259> leaving <15>
R2C7 - removing <9> from <159> leaving <15>
Squares R4C9 and R6C9 in column 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <39>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C9 - removing <9> from <479> leaving <47>
R3C9 - removing <9> from <479> leaving <47>
Intersection of row 3 with block 1. The value <1> only appears in one or more of squares R3C1, R3C2 and R3C3 of row 3. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain this value.
R1C2 - removing <1> from <12679> leaving <2679>
R2C2 - removing <1> from <179> leaving <79>
Intersection of row 7 with block 7. The value <2> only appears in one or more of squares R7C1, R7C2 and R7C3 of row 7. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R8C1 - removing <2> from <2567> leaving <567>
R9C1 - removing <2> from <26> leaving <6>
Intersection of column 5 with block 2. The value <5> only appears in one or more of squares R1C5, R2C5 and R3C5 of column 5. These squares are the ones that intersect with block 2. Thus, the other (non-intersecting) squares of block 2 cannot contain this value.
R1C6 - removing <5> from <1569> leaving <169>
R2C6 - removing <5> from <159> leaving <19>
R2C7 is the only square in row 2 that can be <5>
R1C7 can only be <1>
R2C6 is the only square in row 2 that can be <1>
Squares R1C3<579>, R2C2<79> and R3C3<579> in block 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <579>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C2 - removing <79> from <2679> leaving <26>
R3C1 - removing <57> from <1257> leaving <12>
R3C2 - removing <79> from <12679> leaving <126>
Intersection of column 1 with block 7. The value <5> only appears in one or more of squares R7C1, R8C1 and R9C1 of column 1. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R8C3 - removing <5> from <579> leaving <79>
Squares R1C2<26>, R1C6<69> and R1C8<29> in row 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <269>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C3 - removing <9> from <579> leaving <57>
R1C5 - removing <69> from <5679> leaving <57>
Squares R1C3, R1C5, R3C3 and R3C5 form a Type-3 Unique Rectangle on <57>. Upon close inspection, it is clear that:
(R3C3 or R3C5)<69>, R3C8<29>, R3C2<126> and R3C1<12> form a naked quad on <1269> in row 3. No other squares in the row can contain these possibilities
R3C4 - removing <69> from <4679> leaving <47>
Squares R3C4 and R3C9 in row 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <47>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R3C3 - removing <7> from <579> leaving <59>
R3C5 - removing <7> from <5679> leaving <569>
Squares R3C4, R3C9, R2C4 and R2C9 form a Type-1 Unique Rectangle on <47>.
R2C4 - removing <47> from <479> leaving <9>
R2C2 can only be <7>
R9C4 can only be <2>
R1C6 can only be <6>
R9C7 can only be <9>
R8C4 can only be <6>
R8C7 can only be <2>
R1C2 can only be <2>
R3C5 can only be <5>
R2C9 can only be <4>
R4C2 can only be <1>
R1C3 can only be <5>
R3C9 can only be <7>
R3C3 can only be <9>
R1C5 can only be <7>
R3C4 can only be <4>
R6C4 can only be <7>
R1C8 can only be <9>
R3C2 can only be <6>
R7C2 can only be <9>
R3C1 can only be <1>
R4C5 can only be <9>
R3C8 can only be <2>
R8C3 can only be <7>
R4C9 can only be <3>
R6C5 can only be <6>
R4C1 can only be <7>
R6C9 can only be <9>
R6C1 can only be <3>
R7C6 can only be <5>
R7C1 can only be <2>
R8C6 can only be <9>
R8C1 can only be <5>
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