## Hodoku’s Continuous Nice Loops

This post considers the terminology of nice loops prevalent when Hodoku was put up. It also demonstrates nice loop coloring, an important reason to reject another Hodoku assertion. And it also transcribes two Hodoku nice loop examples for better access by sysudokie readers.

You have to wonder why the Sudoku community, and therefore Hodoku, ever distinguished nice loops from almost nice loops by calling them “Continuous” versus “Discontinuous”. In the real world, loops are continuous. Discontinuous loops are strings. “Nice” and “almost nice” have better connection to the idea of alternating strong and weak links, the “almost nice” loop having one exceptional, naughty link.

Sadly, at the beginning of the Hodoku section on Continuous Nice Loops, Hodoku is unable to define what he calls a Continuous Nice Loop. Instead he cites the bygone clutter of propogation rules, contradictions and start/end cells. It’s just not that hard.

My “abc” picture of X-loop types shows why I prefer Andrew Stewart’s “almost nice” terminology.

Loop c is “nice” because it has perfect (continuous) alternation. Loop a, an eliminating almost nice loop(ANL) has two winks together just once. The b loop, a confirming ANL, has two slinks together just once.

As to the affect of the nice loop, Hodoku makes this statement:

“What makes Continuous Nice Loops so effective is that all weak links in the loop are converted into strong links. That means that all additional candidates in the houses or cells providing the weak links can be eliminated.”

Another unforced error.

Slinks can be used as winks where needed, yes. The terminals of a wink in a nice loop are a toxic set, but that doesn’t make a nice loop wink into a slink. It happens because, as the “c” loop demonstrates above, the wink terminals are also the slink terminals of an AIC.

No such wink to slink conversion occurs. In fact, the position of the winks needs to be preserved. Unless all candidates of the loop are somehow false, the true ones alternate with the false ones. This is a basis for a coloring cluster. We don’t know which set is true, but the alternation defines a clockwise and a counterclockwise set of candidates, one true and the other, false. The corresponding coloring can be merged or bridged with other clusters of any origin. Also the nice loop clusters can be extended along any alternating chains leaving the loop, giving additional opportunities for a color to reach a contradiction and declare a color false.

Nice loop coloring with extensions was first explained here in Nice Loop Coloring of Insane 465 of 8/27/13.

Here is another beautiful example I got for my birthday. My sysudokie friend Gordon Fick is working through Michael Mepham’s Diabolical Sudoku collection, and passing the toughest ones to me for a filtered review next year.

The Puzzle 15 bv field supports three small unconnected clusters. With a new bv in hand, I was searching for an iXYZ wing, but instead, found this AIC nice loop.

The 1 removals imply 1r6c9 and the loop then merges the three clusters into one. In the clockwise direction, orange, peach and green are true.  Counter-clockwise, it’s red, blue and violet. In an easy trial, counter-clockwise puts two 7’s in r8, or two 2’s in c9, and clockwise solves it.

No, Bernhard. We’ll keep nice loop coloring, and nice loop winks are not slinks.

Hodoku does display two fine examples of nice loop prowess in the Continuous Nice Loops section. At the risk of being fined for cluttering, I have color coded the respective toxic sets and victims of the wink ends in the first one. Using this diagram to accompany Hodoku’s above quoted description of the nice loop effect, you can see that what he is saying is right, as far as it goes.

But it’s not where they are, it’s what they see.

The second example, also ungrouped, is a true AIC, depending heavily on the AIC hinge winks. To fight off clutter, I abandoned my diamonds for Hodoku pink to mark victims, but had to compromise my candidate lists to get the digit coloring in. Can’t have everything.

Neither of these examples require nice loop coloring extensions, of course. They are composed to be solved with the technique ordered. My question is, how did Bernhard Hobiger do that?

Next we use the focused puzzle compositions of Hodoku to explore the difficult problem of finding grouped AIC, like the Insane nice loop linked above.