The full reasoning can be found below the Sudoku.
Aug 08 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R3C4 can only be <3>
R5C5 can only be <3>
R5C7 can only be <1>
R7C5 can only be <8>
R1C4 can only be <2>
R5C3 can only be <6>
R3C5 can only be <7>
R9C6 can only be <1>
R7C6 can only be <6>
R9C2 is the only square in column 2 that can be <7>
R2C8 is the only square in column 8 that can be <7>
Intersection of column 2 with block 1. The values <46> only appears in one or more of squares R1C2, R2C2 and R3C2 of column 2. These squares are the ones that intersect with block 1. Thus, the other (non-intersecting) squares of block 1 cannot contain these values.
R2C1 - removing <4> from <2348> leaving <238>
R3C1 - removing <4> from <148> leaving <18>
Intersection of column 8 with block 3. The value <4> only appears in one or more of squares R1C8, R2C8 and R3C8 of column 8. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R1C7 - removing <4> from <34689> leaving <3689>
R2C7 - removing <4> from <3458> leaving <358>
R2C2 is the only square in row 2 that can be <4>
Intersection of column 8 with block 9. The values <15> only appears in one or more of squares R7C8, R8C8 and R9C8 of column 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain these values.
R7C9 - removing <5> from <2359> leaving <239>
R8C7 - removing <5> from <5689> leaving <689>
R8C9 - removing <5> from <2569> leaving <269>
R9C7 - removing <5> from <3589> leaving <389>
Squares R3C1<18>, R4C1<48> and R6C1<14> in column 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <148>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C1 - removing <8> from <238> leaving <23>
R7C1 - removing <1> from <1235> leaving <235>
R8C1 - removing <18> from <1258> leaving <25>
Squares R7C2<19>, R7C4<59> and R7C8<159> in row 7 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <159>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R7C1 - removing <5> from <235> leaving <23>
R7C9 - removing <9> from <239> leaving <23>
R8C1 is the only square in column 1 that can be <5>
Squares R1C3 and R1C7 in row 1 and R9C3 and R9C7 in row 9 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in columns 3 and 7 can be removed.
R2C3 - removing <3> from <238> leaving <28>
R2C7 - removing <3> from <358> leaving <58>
Squares R7C4, R9C4, R7C8 and R9C8 form a Type-4 Unique Rectangle on <59>.
R7C8 - removing <9> from <159> leaving <15>
R9C8 - removing <9> from <589> leaving <58>
Squares R1C6, R3C6, R1C8 and R3C8 form a Type-2 Unique Rectangle on <48>.
R1C7 - removing <9> from <3689> leaving <368>
R8C8 - removing <9> from <189> leaving <18>
R3C9 - removing <9> from <69> leaving <6>
R1C2 is the only square in row 1 that can be <6>
R8C7 is the only square in row 8 that can be <6>
Squares R1C7<38>, R2C7<58> and R2C9<35> in block 3 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <358>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R1C8 - removing <8> from <489> leaving <49>
R3C8 - removing <8> from <489> leaving <49>
Intersection of column 8 with block 9. The values <158> only appears in one or more of squares R7C8, R8C8 and R9C8 of column 8. These squares are the ones that intersect with block 9. Thus, the other (non-intersecting) squares of block 9 cannot contain these values.
R9C7 - removing <8> from <389> leaving <39>
Squares R8C2 (XYZ), R8C8 (XZ) and R7C2 (YZ) form an XYZ-Wing pattern on <1>. All squares that are buddies of all three squares cannot be <1>.
R8C3 - removing <1> from <1289> leaving <289>
R6C3 is the only square in column 3 that can be <1>
R6C1 can only be <4>
R6C7 can only be <9>
R4C1 can only be <8>
R6C9 can only be <7>
R9C7 can only be <3>
R4C9 can only be <5>
R1C7 can only be <8>
R7C9 can only be <2>
R1C6 can only be <4>
R2C7 can only be <5>
R2C9 can only be <3>
R4C7 can only be <4>
R2C1 can only be <2>
R4C3 can only be <7>
R3C1 can only be <1>
R7C1 can only be <3>
R8C9 can only be <9>
R1C8 can only be <9>
R3C6 can only be <8>
R1C3 can only be <3>
R3C8 can only be <4>
R2C3 can only be <8>
R8C3 can only be <2>
R9C3 can only be <9>
R3C2 can only be <9>
R7C2 can only be <1>
R7C8 can only be <5>
R8C2 can only be <8>
R7C4 can only be <9>
R9C8 can only be <8>
R8C8 can only be <1>
R9C4 can only be <5>
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