The Definitive Guide to Sudoku Nishio Forcing Chains Strategy

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Nishio

Nishio is a strategy for examining the ramifications of a hypothesis and concluding that the hypothesis is wrong if a contradiction develops. (Ideally, the exploration should take place in one’s mind.)

It smells like crap as soon as different sheets of paper or an eraser are involved.)

Terminology Nishio is commonly used in two ways:

I pick a single cell’s value and examine the results. If a contradiction is discovered, the chosen value is ruled out as an option. (ii) Select a d value for a single cell and check whether all other digits d may still be placed with this value—this second version, which may or may not be the correct version.

An example that may be used in a variety of ways

Ruby lips suggested the following starting position.

This is finished with the standard arguments.

Nishio Forcing Chains

Nishio has a long history in the Sudoku community as a semi-trial-and-error method to difficult Sudoku bottlenecks. Still, this website’s implementation fits quite neatly within the family of forcing chains and is a formal ‘pattern’ based technique that leverages AICs. In reality, because Nishio operates on a single candidate, it’s extremely similar to Digit Forcing Chains.

However, whereas Digit Forcing Chains considers the implications of a candidate being both ON and OFF, Nishio looks for a contradiction when the candidate is just ON. Then, two chains emerge from the candidate in opposite directions, attempting to link up with another candidate later. The rule is that if the start candidate is ON and another candidate is ON and OFF simultaneously, it indicates an impossible situation.

As a result, the original applicant is ineligible.

A Nishio forcing chain is sometimes the polar opposite of a Digit Forcing Chain.

Example of a Nishio Forcing Chain:

In this case, the first possibility in J4 is 6. The shorter blue chain states that when J4 is ON, the 6s in box 7 are removed, leaving 6 in G2 as the sole surviving 6 and hence the answer.

On the other hand, the larger purple chain indicates that 6 cannot be the answer in G2. Therefore, a 6 in J4 indicates that 6s have been deleted from the rest of box 8, which activates the six in H9. This forces a 2 in J9, eliminating a 2 from J1, resulting in G2 being two rather than 6.

As a result, the 6 in J4 isn’t a solution and may be eliminated.

Type 1

Nishio Forcing Digit (same candidate in both “ON” and “OFF”): the two chains meet in a Cell with the same candidate in both “ON” and “OFF” states.

At the start of the chains, the candidate cannot be the solution for that beginning Cell since the same basic assumption cannot lead to contradicting results.

Assuming candidate 6 is the solution in A4 results in a contradiction: the blue chain says that candidate 6 cannot be the solution in C7, yet the pink chain implies that candidate 6 must be the solution in C7.

As a result, the initial assumption is incorrect: candidate 6 cannot answer A4 and must be removed.

Type 2

(emptying a cell) Nishio Forcing Digit: the two chains meet in a “Bi-Value” Cell with both candidates “OFF,” leaving this Cell “empty.”

As a result, the contender at the beginning of the chains cannot be the starting Cell’s answer.

Assuming that candidate 3 is the answer in B6 results in a contradiction.

If candidate 3 is the solution in B6, candidate 3 may be eliminated in B8 (blue chain), yet the pink chain suggests that candidate 8 cannot be the solution in B8.

Therefore, B8 would be unable to find a solution.

As a result, the original assumption is incorrect: candidate 3 cannot be the answer in B6 and must be removed.

Type 3

Nishio Forcing Digit (two candidates “ON” in a region): the two chains turn on the same candidate in two distinct Cells of the same area (Row, Column, or Square).

The original assumption is incorrect since a candidate can only be the solution in one Cell of an area.

As a result, the contender at the beginning of the chains cannot be the starting Cell’s answer.

Assuming that candidate 7 is the solution in H6 results in a contradiction: the blue chain suggests that candidate 3 must be the answer in either G1 or H1, yet the pink chain says that candidate 3 must be the solution in H2.

Candidate 3 is the answer in two separate Square 7 cells, which is impossible.

As a result, the original assumption is incorrect: candidate 7 cannot be the answer in H6 and must be removed.

Type 4

(emptying an area) Nishio Forcing Digit: the two chains turn all occurrences of a candidate “OFF” in a region (Row, Column, or Square). The original assumption is incorrect since a candidate must be the solution in just one Cell of an area.

As a result, the contender at the beginning of the chains cannot be the starting Cell’s answer.

Assuming that candidate 4 is the answer in C2 results in an impossibility: the blue chain says that candidate 8 must be removed from E3, while the pink chain implies that candidate 8 must be removed from E7, leaving Row “E” with no viable solution for candidate 8, which is impossible.

As a result, the original assumption is incorrect: candidate 4 cannot be the answer in C2 and must be removed.

Resuming the series on forcing chains, this piece explains that Andrew Stuart’s sudokuwiki site’s Nishio forcing chain is yet another re-interpretation of the AIC poisonous set, also known as the deleting nearly pleasant loop (ANL).

We further contend that approaches called “Nishio,” including this one, are incompatible with sysudokie solution aims among the Sudoku community.

Stuart’s current sudowikiNishio forcing chain example turns out to be an intriguing ANL with grouped hazardous set members.

He chooses the 6-candidate in r9c4 as the emanating candidate for his Nishio forcing chain, maybe because it removes the 6-group in SWr9, and then looks for an AIC from the 6-group in Sr8 back to the initial group.

Would you do something like that? I am certain that followers of this site will build the AIC from r7c2 to r8c9, then realize the expansions to the 6-groups, resulting in the effective hazardous set.

In the post-X-panel investigation, I was ecstatic to see an X-chain example with clustered pincers. Unfortunately, attributing the removal to a Nishio approach minimizes the logical achievement.

The arbitrary selection of a candidate and the determination of the implications of it being true is a recurring subject in Nishio’s techniques.

Aside from that, there is significant ambiguity among the Sudoku community regarding what the phrase means.

For some, Nishio is as simple as this:

choose any candidate.
Assume it to be correct.
Determine whether the problem solution or a contradiction.

If the latter is the case, it should be removed. Others require a Nishio bifurcation as well. That is, there is a choice between two options.

Finally, others describe Nishio as a restricted type of trial and error in which the chosen candidate in the second stage is deemed false if “it stops the candidate’s number from being placed in the remaining placements.”

What is involved in setting this constraint is shown by our recent research into pattern enumeration. By performing a single freeform without the Nishio candidate and the candidates it observes, one may test for patterns of the remaining candidates.

Consider how restrictive this criterion is in terms of Nishio candidates. Isn’t it a little underwhelming?

I believe that human solvers may increase their Sudoku enjoyment by simply forbidding any Nishio strategies, regardless of definition.

Followers of this blog are aware of how much Nishio conceals, substituting a mechanical hunt for a solution for the task of deciphering the puzzle’s logic as if the final solution were the most important.

If you’re “lucky,” you’ll come up with a fast contradiction, attempt the opposite, and see how quickly it collapses. However, all you get is the solution. If you’re unfortunate, you’ll have to go through the exhausting exercise of computer logic, testing path after path until the last unit contradicts you.

As the puzzle falls apart, you feel relieved, but you have nothing to share with a buddy.

Do two pals compare their sequence of guesses if they Nishio their way to the solution?

So, where does Stuart’s relatively new Nishio forcing chain fit into the Nishio zoo?

It’s “a formal ‘pattern-based method that leverages AICs,” he says. That is entirely correct. As we’ll see below, it’s an ANL removal or, if you prefer, a poisonous set removal.

This method begins with (1) and (2) above and qualifies as a Nishio technique. That, however, is not “pattern-based.” An AIC begins with a slink and alternates to a slink at the finish. There’s a trend here. The statement “it is false” solely explains why this pattern yields a poisonous set.

“Two (forcing) chains originate from the candidate in separate directions and try to link up later on another candidate,” according to sudowiki.org in the Nishio forcing chain version of (3) above.

This medication invariably results in an ANL that is eliminated. We must inquire, “How do the chains connect on that candidate?”

By the AIC of each chain’s terminating slinks convergent on a candidate. The loop thus has an even number of connections, two defects, and no conclusion. The narrative is the same when the “join up” candidate sees the two AIC end candidates, i.e., in converging winks.

As a second re-interpretation of the removing ALN, Stuart’s Nishio forcing chain joins his “digit forcing chain.” The only difference is that he begins his routine with two winks at the contender to be dismissed. Is this a good way to solve a problem? Forget about it. Too many applicants are blind to the dangers of an AIC. Not suitable for humans or computers.