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Jun 04 - Super Hard

## Reasoning

R2C2 can only be <6>

R2C5 can only be <3>

R3C3 can only be <3>

R8C2 can only be <5>

R5C2 can only be <4>

R2C8 can only be <8>

R1C3 can only be <8>

R7C3 can only be <9>

R4C1 can only be <6>

R9C3 can only be <6>

R8C5 can only be <7>

R8C8 can only be <3>

R5C8 can only be <5>

R5C3 can only be <7>

R7C1 is the only square in row 7 that can be <3>

R9C1 can only be <4>

R9C9 is the only square in row 9 that can be <9>

Squares R5C4 and R5C7 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <23>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.

R5C5 - removing <2> from <12689> leaving <1689>

R5C6 - removing <3> from <139> leaving <19>

R5C9 - removing <23> from <236> leaving <6>

R6C9 can only be <7>

R6C4 can only be <5>

R6C6 can only be <9>

R6C1 can only be <8>

R5C6 can only be <1>

R5C5 can only be <8>

R6C5 can only be <6>

R5C1 can only be <9>

R1C5 is the only square in row 1 that can be <9>

R3C7 is the only square in row 3 that can be <7>

R9C5 is the only square in row 9 that can be <1>

Squares R5C4 and R9C4 in column 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <23>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.

R4C4 - removing <23> from <2347> leaving <47>

Squares R5C4 and R5C7 in row 5 and R9C4 and R9C7 in row 9 form a Simple X-Wing pattern on possibility <2>. All other instances of this possibility in columns 4 and 7 can be removed.

R7C7 - removing <2> from <245> leaving <45>

The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:

R1C9=<13>

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

R1C9 - removing <13> from <134> leaving <4>

R1C4 can only be <7>

R1C7 can only be <3>

R3C9 can only be <1>

R7C9 can only be <2>

R3C1 can only be <5>

R7C5 can only be <5>

R4C9 can only be <3>

R9C7 can only be <5>

R9C6 can only be <3>

R7C7 can only be <4>

R1C6 can only be <5>

R4C4 can only be <4>

R1C1 can only be <1>

R3C5 can only be <4>

R5C7 can only be <2>

R4C5 can only be <2>

R5C4 can only be <3>

R4C6 can only be <7>

R9C4 can only be <2>

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