Oct 15 - Super Hard
Puzzle Copyright © Kevin Stone
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Reasoning
R7C7 can only be <7>
R5C5 is the only square in row 5 that can be <9>
R5C4 is the only square in row 5 that can be <5>
R7C9 is the only square in row 7 that can be <5>
R9C5 is the only square in row 9 that can be <7>
R9C6 is the only square in row 9 that can be <1>
R8C1 is the only square in row 8 that can be <1>
R8C5 is the only square in column 5 that can be <2>
R6C5 is the only square in column 5 that can be <6>
R8C9 is the only square in column 9 that can be <3>
Squares R1C4 and R1C6 in row 1 form a simple naked pair. These 2 squares both contain the 2 possibilities <38>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R1C2 - removing <8> from <478> leaving <47>
R1C3 - removing <8> from <4589> leaving <459>
R1C5 - removing <38> from <13458> leaving <145>
R1C8 - removing <8> from <478> leaving <47>
Squares R1C2 and R1C8 in row 1 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.
R1C3 - removing <4> from <459> leaving <59>
R1C5 - removing <4> from <145> leaving <15>
R1C7 - removing <4> from <149> leaving <19>
Squares R4C5 and R7C5 in column 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <38>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C5 - removing <8> from <148> leaving <14>
R3C5 - removing <8> from <458> leaving <45>
Squares R3C9<478>, R4C9<78> and R6C9<48> in column 9 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <478>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R2C9 - removing <48> from <1248> leaving <12>
R5C9 - removing <478> from <12478> leaving <12>
Squares R4C1 and R4C5 in row 4 and R7C1 and R7C5 in row 7 form a Simple X-Wing pattern on possibility <3>. All other instances of this possibility in columns 1 and 5 can be removed.
R5C1 - removing <3> from <34678> leaving <4678>
Squares R1C2 and R1C8 in row 1 and R8C2 and R8C8 in row 8 form a Simple X-Wing pattern on possibility <4>. All other instances of this possibility in columns 2 and 8 can be removed.
R2C2 - removing <4> from <468> leaving <68>
R2C8 - removing <4> from <248> leaving <28>
R5C2 - removing <4> from <34678> leaving <3678>
R5C8 - removing <4> from <2478> leaving <278>
R9C2 - removing <4> from <3468> leaving <368>
R9C8 - removing <4> from <246> leaving <26>
Squares R3C1 and R3C9 in row 3 and R4C1 and R4C9 in row 4 form a Simple X-Wing pattern on possibility <7>. All other instances of this possibility in columns 1 and 9 can be removed.
R5C1 - removing <7> from <4678> leaving <468>
Squares R5C1<468>, R5C3<468> and R6C1<48> in block 4 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <468>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R4C1 - removing <8> from <378> leaving <37>
R5C2 - removing <68> from <3678> leaving <37>
Squares R2C1<468>, R5C1<468> and R6C1<48> in column 1 form a comprehensive naked triplet. These 3 squares can only contain the 3 possibilities <468>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C1 - removing <48> from <4789> leaving <79>
R7C1 - removing <8> from <389> leaving <39>
Squares R3C3 and R3C9 in row 3, R4C5 and R4C9 in row 4 and R7C3 and R7C5 in row 7 form a Swordfish pattern on possibility <8>. All other instances of this possibility in columns 3, 5 and 9 can be removed.
R5C3 - removing <8> from <468> leaving <46>
R6C9 - removing <8> from <48> leaving <4>
R9C3 - removing <8> from <468> leaving <46>
R6C1 can only be <8>
Squares R5C7 and R5C9 in row 5 form a simple naked pair. These 2 squares both contain the 2 possibilities <12>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R5C8 - removing <2> from <278> leaving <78>
Squares R5C3 and R9C3 in column 3 form a simple naked pair. These 2 squares both contain the 2 possibilities <46>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R3C3 - removing <4> from <4589> leaving <589>
Squares R8C2 and R9C3 in block 7 form a simple naked pair. These 2 squares both contain the 2 possibilities <46>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R9C2 - removing <6> from <368> leaving <38>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
R3C3=<58>
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
R3C3 - removing <58> from <589> leaving <9>
R3C1 can only be <7>
R3C7 can only be <4>
R1C3 can only be <5>
R7C3 can only be <8>
R3C5 can only be <5>
R9C7 can only be <2>
R1C8 can only be <7>
R7C5 can only be <3>
R9C2 can only be <3>
R7C1 can only be <9>
R4C5 can only be <8>
R9C4 can only be <8>
R5C2 can only be <7>
R1C4 can only be <3>
R9C8 can only be <6>
R5C7 can only be <1>
R9C3 can only be <4>
R8C8 can only be <4>
R1C5 can only be <1>
R1C6 can only be <8>
R1C7 can only be <9>
R2C5 can only be <4>
R5C6 can only be <3>
R1C2 can only be <4>
R5C8 can only be <8>
R3C9 can only be <8>
R2C1 can only be <6>
R4C1 can only be <3>
R4C9 can only be <7>
R2C8 can only be <2>
R5C9 can only be <2>
R2C9 can only be <1>
R8C2 can only be <6>
R5C3 can only be <6>
R2C2 can only be <8>
R5C1 can only be <4>
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