Jul 16 - Super Hard

Puzzle Copyright © Kevin Stone

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## Reasoning

R1C8 can only be <1>

R4C6 can only be <4>

R6C6 can only be <8>

R1C2 is the only square in row 1 that can be <3>

R2C8 is the only square in row 2 that can be <2>

R3C6 is the only square in row 3 that can be <2>

R6C7 is the only square in row 6 that can be <2>

R7C3 is the only square in row 7 that can be <2>

R8C9 is the only square in row 8 that can be <1>

R7C7 can only be <3>

R7C4 can only be <8>

R4C7 can only be <1>

R7C5 can only be <7>

R7C6 can only be <1>

R1C5 can only be <5>

R9C6 can only be <6>

R1C6 can only be <7>

R1C4 can only be <6>

R3C5 is the only square in row 3 that can be <8>

R5C3 is the only square in row 5 that can be <1>

R9C5 is the only square in column 5 that can be <3>

R5C8 is the only square in column 8 that can be <3>

Squares R4C4, R6C4, R4C1 and R6C1 form a Type-3 Unique Rectangle on <35>. Upon close inspection, it is clear that:

(R4C1 or R6C1)<69>, R6C3<67> and R4C3<79> form a naked triplet on <679> in block 4. No other squares in the block can contain these possibilities

R5C1 - removing <6> from <568> leaving <58>

R5C2 - removing <6> from <68> leaving <8>

(R4C1 or R6C1)<69>, R6C3<67>, R5C2<68> and R4C3<79> form a naked quad on <6789> in block 4. No other squares in the block can contain these possibilities

R5C1 - removing <8> from <58> leaving <5>

(R4C1 or R6C1)<69> and R8C1<69> form a naked pair on <69> in column 1. No other squares in the column can contain these possibilities

R2C1 - removing <69> from <689> leaving <8>

Squares R2C9 and R5C9 in column 9 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.

R6C9 - removing <6> from <567> leaving <57>

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

R8C2=<67>

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

R8C2 - removing <67> from <679> leaving <9>

R8C1 can only be <6>

R8C5 can only be <4>

R2C2 can only be <6>

R9C2 can only be <7>

R8C8 can only be <7>

R2C5 can only be <9>

R9C4 can only be <9>

R9C8 can only be <4>

R3C4 can only be <4>

R2C9 can only be <4>

R3C3 can only be <9>

R5C9 can only be <6>

R3C7 can only be <6>

R4C3 can only be <7>

R5C7 can only be <4>

R4C9 can only be <5>

R6C3 can only be <6>

R4C4 can only be <3>

R6C9 can only be <7>

R6C1 can only be <3>

R4C1 can only be <9>

R6C4 can only be <5>

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