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Sudoku Solution Path

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R5C6 is the only square in row 5 that can be <2>
R5C1 is the only square in column 1 that can be <9>
R3C8 is the only square in column 8 that can be <3>
R5C9 is the only square in column 9 that can be <1>
R2C9 is the only square in column 9 that can be <7>
Squares R4C3 and R4C4 in row 4 form a simple locked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
   R4C5 - removing <58> from <13578> leaving <137>
   R4C6 - removing <8> from <138> leaving <13>
Squares R4C4 and R7C4 in column 4 form a simple locked pair. These 2 squares both contain the 2 possibilities <58>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
   R3C4 - removing <8> from <489> leaving <49>
   R5C4 - removing <8> from <468> leaving <46>
   R6C4 - removing <5> from <4569> leaving <469>
Intersection of row 3 with block 1. The values <15> 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 these values.
   R1C1 - removing <5> from <258> leaving <28>
   R1C2 - removing <5> from <4568> leaving <468>
Squares R1C1 and R2C1 in column 1 form a simple locked pair. These 2 squares both contain the 2 possibilities <28>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
   R8C1 - removing <28> from <2358> leaving <35>
   R9C1 - removing <8> from <358> leaving <35>
Squares R1C1 and R2C1 in block 1 form a simple locked pair. These 2 squares both contain the 2 possibilities <28>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
   R1C2 - removing <8> from <468> leaving <46>
   R2C3 - removing <28> from <2468> leaving <46>
   R3C2 - removing <8> from <1458> leaving <145>
   R3C3 - removing <8> from <1458> leaving <145>
Squares R1C2 and R2C3 in block 1 form a simple locked 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.
   R3C2 - removing <4> from <145> leaving <15>
   R3C3 - removing <4> from <145> leaving <15>
Squares R8C1 and R9C1 in block 7 form a simple locked pair. These 2 squares both contain the 2 possibilities <35>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
   R7C2 - removing <5> from <1568> leaving <168>
   R7C3 - removing <5> from <12568> leaving <1268>
   R8C3 - removing <5> from <258> leaving <28>
   R9C2 - removing <5> from <1568> leaving <168>
R3C2 is the only square in column 2 that can be <5>
R3C3 can only be <1>
Intersection of column 4 with block 5. The value <6> only appears in one or more of squares R4C4, R5C4 and R6C4 of column 4. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
   R5C5 - removing <6> from <4678> leaving <478>
   R6C5 - removing <6> from <3456> leaving <345>
Intersection of column 6 with block 5. The value <3> only appears in one or more of squares R4C6, R5C6 and R6C6 of column 6. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
   R4C5 - removing <3> from <137> leaving <17>
   R6C5 - removing <3> from <345> leaving <45>
Squares R6C3 and R6C5 in row 6 form a simple locked pair. These 2 squares both contain the 2 possibilities <45>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
   R6C4 - removing <4> from <469> leaving <69>
Squares R2C7<248>, R3C7<48> and R8C7<248> in column 7 form a comprehensive locked triplet. These 3 squares can only contain the 3 possibilities <248>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
   R7C7 - removing <28> from <2678> leaving <67>
Squares R8C1, R9C1, R8C5 and R9C5 form a Type-3 Unique Rectangle on <35>. Upon close inspection, it is clear that:
(R8C5 or R9C5)<18> and R7C6<18> form a locked pair on <18> in block 8. No other squares in the block can contain these possibilities
   R7C4 - removing <8> from <58> leaving <5>
(R8C5 or R9C5)<18>, R5C5<478>, R4C5<17>, R2C5<468> and R1C5<468> form a locked set on <14678> in column 5. No other squares in the column can contain these possibilities
   R6C5 - removing <4> from <45> leaving <5>
R6C3 can only be <4>
R4C4 can only be <8>
R4C3 can only be <5>
R2C3 can only be <6>
R5C2 can only be <8>
R1C2 can only be <4>
R1C5 is the only square in row 1 that can be <6>
R8C9 is the only square in column 9 that can be <4>
R8C1 is the only square in row 8 that can be <5>
R9C1 can only be <3>
R8C5 is the only square in row 8 that can be <3>
Squares R3C6 and R3C7 in row 3, R7C3 and R7C6 in row 7 and R8C3 and R8C7 in row 8 form a Swordfish pattern on possibility <8>. All other instances of this possibility in columns 3, 6 and 7 can be removed.
   R2C7 - removing <8> from <248> leaving <24>
The puzzle can be reduced to a Bivalue Universal Grave (BUG) pattern, by making this reduction:
   R7C8=<27>
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
   R7C8 - removing <27> from <267> leaving <6>
R7C2 can only be <1>
R7C7 can only be <7>
R5C8 can only be <7>
R9C8 can only be <5>
R9C9 can only be <8>
R1C8 can only be <2>
R9C5 can only be <1>
R1C9 can only be <5>
R8C7 can only be <2>
R1C1 can only be <8>
R2C7 can only be <4>
R2C5 can only be <8>
R3C7 can only be <8>
R3C6 can only be <9>
R5C5 can only be <4>
R4C7 can only be <3>
R7C6 can only be <8>
R9C2 can only be <6>
R7C3 can only be <2>
R8C3 can only be <8>
R4C5 can only be <7>
R2C1 can only be <2>
R3C4 can only be <4>
R6C6 can only be <3>
R4C6 can only be <1>
R6C7 can only be <6>
R5C4 can only be <6>
R6C4 can only be <9>


 

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