dailysudoku.com

[keith]

Another good and challenging one:

[someone_somewhere]

Hi,

After applying standard techniques and finding:

Hidden Pair 4 6 in r8c1 and r8c3
Nacked Pair 3 5 in r2c1 and r7c1

I get to:

[David Bryant]

This puzzle is very tough. I did find a way to break through the logjam by considering the forcing chains originating in r6c5, but I wonder if there might not be a shorter path to the solution.

After a series of fairly routine moves I arrived at this position.

[keith]

Because of the forcing chains, I think it is unlikely that two people will solve this puzzle the same way. Here is a solution that involves a Swordfish, and at the end, another BUG.

I don't know how common this BUG pattern is, but it is curious that three of the most recent puzzles in this discussion topic have them.

(This is not how I solved the puzzle. It is a solution generated by a computer program that mimics how a human might solve the puzzle.)

R1C4 must be <3>.
R3C6 must be <5>.
R1C5 must be <7>.
R3C4 must be <1>.
R4C4 must be <6>.
R2C5 must be <6>.
R8C5 is the only square in column 5 that can be <2>.
R9C6 is the only square in column 6 that can be <6>.
R5C9 is the only square in column 9 that can be <6>.
R5C8 is the only square in row 5 that can be <8>.
R1C8 must be <9>.
R5C1 is the only square in row 5 that can be <7>.
R7C9 is the only square in column 9 that can be <9>.
R7C8 is the only square in row 7 that can be <2>.
R8C8 is the only square in column 8 that can be <3>.

Squares R2C1 and R7C1 in column 1 form a simple naked 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 column.
R6C1 - removing <5> from <569> leaving <69>.
R8C1 - removing <5> from <456> leaving <46>.
A set of 2 squares form a simple hidden pair. R8C1 and R8C3 all contain the 2 possibilities <46>. No other squares in row 8 have those possibilities. Since the 2 squares are the only possible locations for 2 possible values, any additional possibilities these squares have (if any) can be eliminated. These squares now become a simple naked pair.
R8C3 - removing <58> from <4568> leaving <46>.

Found a 4-link Comprehensive Forcing Chain. If we assume that square R2C2 is <8> then we can make the following chain of conclusions:
R8C2 must be <5> (force), which means that
R7C1 must be <3> (force), which means that
R2C1 must be <5> (force), which means that
R2C2 must be <3> (R2 pin).
Since this is logically inconsistent, R2C2 cannot be <8>.
Intersection of column 2 with block 7. The value <8> only appears in one or more of squares R7C2, R8C2 and R9C2 of column 2. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R9C3 - removing <8> from <589> leaving <59>.

Found a 5-link Comprehensive Forcing Chain. If we assume that square R9C2 is <9> then we can make the following chain of conclusions:
R3C2 must be <2> (force), which means that
R3C9 must be <4> (force), which means that
R1C7 must be <8> (force), which means that
R9C7 must be <5> (force), which means that
R9C2 must be <8> (R9 pin).
Since this is logically inconsistent, R9C2 cannot be <9>.
R9C3 is the only square in row 9 that can be <9>.
R4C3 must be <2>.
R6C7 is the only square in row 6 that can be <2>.

Squares R5C2 and R5C5 in row 5, R8C2 and R8C7 in row 8 and R9C2, R9C5 and R9C7 in row 9 form a Swordfish pattern on possibility <5>. All other instances of this possibility in columns 2, 5 and 7 can be removed.
R2C2 - removing <5> from <235> leaving <23>.
R6C5 - removing <5> from <159> leaving <19>.
Found a 7-link Simple Forcing Loop. If we assume that square R5C2 is <5> then we can make the following chain of conclusions:
R8C2 must be <8>, which means that
R8C9 must be <7>, which means that
R4C9 must be <4>, which means that
R3C9 must be <2>, which means that
R3C2 must be <9>, which means that
R4C2 must be <1>, which means that
R5C2 must be <5>.
This forms a inherently bi-directional loop through the puzzle that permits reductions. On each edge of the loop, one of the two squares must have a particular value, so their common buddies cannot contain that value, as follows:
One of R5C2 and R8C2 must be <5>.
One of R8C2 and R8C9 must be <8>.
One of R8C9 and R4C9 must be <7>.
One of R4C9 and R3C9 must be <4>.
One of R3C9 and R3C2 must be <2>.
One of R3C2 and R4C2 must be <9>.
One of R4C2 and R5C2 must be <1>.
Thus we can deduce that:
R9C2 - cannot contain <5> because of R5C2 and R8C2.
R8C7 - cannot contain <8> because of R8C2 and R8C9.
R2C9 - cannot contain <7> because of R8C9 and R4C9.
A set of 2 squares form a simple hidden pair. R2C7 and R2C8 all contain the 2 possibilities <17>. No other squares in row 2 have those possibilities. Since the 2 squares are the only possible locations for 2 possible values, any additional possibilities these squares have (if any) can be eliminated. These squares now become a simple naked pair.
R2C7 - removing <8> from <178> leaving <17>.
Found a 8-link Simple Forcing Loop. If we assume that square R6C8 is <7> then we can make the following chain of conclusions:
R4C9 must be <4>, which means that
R3C9 must be <2>, which means that
R2C9 must be <8>, which means that
R2C3 must be <5>, which means that
R6C3 must be <6>, which means that
R6C1 must be <9>, which means that
R6C5 must be <1>, which means that
R6C8 must be <7>.
This forms a inherently bi-directional loop through the puzzle that permits reductions. On each edge of the loop, one of the two squares must have a particular value, so their common buddies cannot contain that value, as follows:
One of R6C8 and R4C9 must be <7>.
One of R4C9 and R3C9 must be <4>.
One of R3C9 and R2C9 must be <2>.
One of R2C9 and R2C3 must be <8>.
One of R2C3 and R6C3 must be <5>.
One of R6C3 and R6C1 must be <6>.
One of R6C1 and R6C5 must be <9>.
One of R6C5 and R6C8 must be <1>.
Thus we can deduce that:
R4C7 - cannot contain <7> because of R6C8 and R4C9.

The puzzle is now: