The full reasoning can be found below the Sudoku.
Apr 09 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R1C2 can only be <6>
R8C1 can only be <1>
R1C8 can only be <1>
R3C3 can only be <1>
R1C1 can only be <8>
R1C9 can only be <7>
R9C8 can only be <6>
R2C9 can only be <6>
R3C7 can only be <9>
R2C1 can only be <4>
R8C9 can only be <3>
R8C5 can only be <9>
R5C8 can only be <8>
R2C5 can only be <1>
R7C5 can only be <8>
R7C6 can only be <2>
R1C5 can only be <5>
R1C6 can only be <9>
R1C4 can only be <2>
R4C1 is the only square in row 4 that can be <3>
R5C1 is the only square in row 5 that can be <9>
R6C4 is the only square in row 6 that can be <9>
Squares R9C5 and R9C6 in row 9 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R9C4 - removing <4> from <147> leaving <17>
Squares R3C6 and R9C6 in column 6 form a simple naked pair. These 2 squares both contain the 2 possibilities <34>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R4C6 - removing <4> from <458> leaving <58>
Squares R9C5, R9C6, R3C5 and R3C6 form a Type-1 Unique Rectangle on <34>.
R3C5 - removing <34> from <346> leaving <6>
R3C4 can only be <4>
R5C5 can only be <4>
R9C5 can only be <3>
R4C4 can only be <6>
R9C6 can only be <4>
R3C6 can only be <3>
R4C9 is the only square in row 4 that can be <4>
Squares R4C6, R6C6, R4C3 and R6C3 form a Type-4 Unique Rectangle on <58>.
R4C3 - removing <5> from <578> leaving <78>
R6C3 - removing <5> from <568> leaving <68>
Squares R5C9 (XY), R5C2 (XZ) and R4C7 (YZ) form an XY-Wing pattern on <7>. All squares that are buddies of both the XZ and YZ squares cannot be <7>.
R4C3 - removing <7> from <78> leaving <8>
R5C7 - removing <7> from <567> leaving <56>
R4C6 can only be <5>
R6C3 can only be <6>
R4C7 can only be <7>
R6C6 can only be <8>
R5C7 is the only square in row 5 that can be <6>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R6C9=<12>
These are called the BUG possibilities. In a BUG pattern, in each row, column and block, each unsolved possibility appears exactly twice. Such a pattern either has 0 or 2 solutions, so it cannot be part of a valid Sudoku
When a puzzle contains a BUG, and only one square in the puzzle has more than 2 possibilities, the only way to kill the BUG is to remove both of the BUG possibilities from the square, thus solving it
R6C9 - removing <12> from <125> leaving <5>
R6C1 can only be <2>
R6C7 can only be <1>
R5C9 can only be <2>
R9C9 can only be <1>
R9C4 can only be <7>
R7C7 can only be <5>
R5C2 can only be <7>
R9C1 can only be <5>
R7C3 can only be <7>
R9C2 can only be <2>
R7C4 can only be <1>
R5C3 can only be <5>
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