Resuming the series on forcing chains, this post reveals that the Nishio forcing chain introduced on Andrew Stuart’s sudokuwiki site is yet another re-interpretation of the AIC toxic set, a.k.a. the eliminating almost nice loop(ANL). Here we also assert that techniques labeled “Nishio” in the Sudoku community, including this one, are contrary to sysudokie solving objectives .

Stuart’s current sudowiki example of a *Nishio forcing chain* turns out to be an interesting ANL with grouped toxic set members. For his Nishio forcing chain, he selects the 6-candidate in r9c4 as the emanating candidate, perhaps because it eliminates the 6-group in SWr9, then seeks an AIC from the 6-group in Sr8 back to the first group.

Would you do that? I trust that readers of this blog would construct the AIC from r7c2 to r8c9, then realize the extensions to the 6-groups, forming the effective toxic set. I was elated to find an X-chain example of grouped pincers back in the post X-panel analysis. Crediting the removal to a Nishio technique is to belittle the logical accomplishment here.

A common theme of Nishio methods is the arbitrary selection of a candidate, and the determination of the consequences of it being true. Outside of that, there is some confusion in the Sudoku community about the meaning of the term.

For some, Nishio is simply (1) choose any candidate, (2) take it to be true and (3) solve to a solution or a contradiction. Remove it if the latter. Others require also a bifurcation in a Nishio. That is to say, a choice between two. Other writers define Nishio as a limited form of trial and error in which the chosen candidate in the second step is assumed false if “it prevents the remaining placements of the candidate’s number”. Our recent exploration of pattern enumeration tells us what is involved in imposing this limit. One can test for patterns of the remaining candidates by attempting a single freeform excluding the Nishio candidate and the candidates it sees. Now imagine how limiting on the choice of Nishio candidates this requirement is. Underwhelming, wouldn’t you say?

I maintain that human solvers could double their Sudoku pleasure by just swearing off all Nishio methods, regardless of definition. Followers of this blog realize how much Nishio covers up, replacing the challenge of decoding the logic of the puzzle with a mechanical search for a solution, as if it were the final solution that is important. If you are “lucky”, you’ll hit on a quick contradiction, try the opposite, and mark the quick collapse. But the solution is all that you gain. If you are unlucky, you will play through the tiring exercise of computer logic, trying path after path, sometimes going on and on until the final unit contradicts you. When the puzzle collapses, you gain relief, but nothing to share with a friend. If two friends Nishio their way to the solution, do they compare the series of guesses they used?

So exactly where in the Nishio menagerie is Stuart’s relatively new Nishio forcing chain? He describes it as “ a formal ‘pattern’ based strategy that uses AICs”. That is all too true. It’s an eliminating ANL, or if you prefer, a toxic set removal, as we will prove below. Qualifying as a Nishio technique, this one starts with (1) and (2) above. But that’s not “pattern based”. An AIC starts with a slink, and alternates to an ending slink. That’s a pattern. We only use the “if _ is false” phrase in describing why this pattern produces a toxic set.

In the Nishio forcing chain version of (3) above, to quote from sudowiki.org, “Two (forcing) chains emanate from the candidate in different directions and try to join up later on another candidate”. This prescription always produces an eliminating ANL. We have to ask: How do the chains join up on that candidate? By the ending slinks of each chain’s AIC converging on a candidate? Then we have an nice loop with an even number of links, two faults and no conclusion. Same story if the “join up” candidate sees the two AIC end candidates, i.e. in converging winks. All that remains is the only effective way to join the two emanating AIC on a candidate, to merge them into one AIC, producing an ANL eliminating the starting digit. Just as in Andrew’s example above.

Stuart’s Nishio forcing chain joins his “digit forcing chain” as a second re-interpretation of the eliminating ALN. The only difference is that he starts his pattern at the candidate to be removed, with two winks. An efficient solving approach? Forget it. Too many candidates not seeing the toxic ends of an AIC. Not recommended for human or computer.

I know there is this debate about “trial and error” of one flavor or another being something to be shunned by a sudoku “purist” – kind of like the debate between pure mathematics and practical mathematics. I would like to see that schism better described. It seems to me that many of the advanced techniques like “coloring” are doing just that – in fact any elimination approach that looks ahead for a contradiction and finds that contradiction and that goes back to where one started is similar to backtracking in the Guess and Check approach. Are not all those techniques philosophically just a more rigorous or structured backtracking of a sort?

Richard, its not about purity vs practicality. I write about methods which uncover the logical content buried in the puzzle. T&E does not do that. It eventually reveals the solution, or a solution, without doing that. Sudoku fans are interested in the logical content or not. If not, they lose nothing by using T&E.

Coloring builds a logical structure which eliminates candidates that do not fit. Eventually, it collapses into a host of clues that usually finishes the puzzle. There is no guesswork involved. Systematic Sudoku is about science and engineering, not philosophy. The science is the logical patterns. The engineering is about procedures and devices which enable humans to find these patterns.