## Hodoku Nice Loop Mythology

This post dismisses the Stephens/Hobiger theory of cell links, propogation rules, and loop closing rules as unnecessary and overcomplicated. It uses the first two Hodoku AIC ANL examples to demonstrate the alternative Sysudoku chain construction process.

As a blogger, one thing I like about Bernard Hobiger’s site, is that an entire Sysudoku post can emerge from actually thinking about just one Hodoku declaration. This time it is the Hodoku definition of a nice loop. To sysudokies, a nice loop is a perfect alternating inference loop of candidates, not cells. An almost nice loop is the same, except for one extra link that creates a pair of slinks or winks, marking an elimination or the confirmation of new clue.

The Hodoku declaration du jour says “Nice loops are chains that link back to the cell where they started.” That sounds innocent enough, but with their intended meaning of “cell where they started”, Bernhard Hobiger and predecessor Paul Stephens are saying:

1. Chain nodes are cells, not candidates or groups, and links are between cells. Internal links within cells and ALS do not exist.
2. A loop has a starting cell. No joke, they mean it’s a real thing.
3. “Nice loops” include nice loops(“continuous nice loops”) and almost nice loops(“discontinuous nice loops”.

Point 3 is inconsequential, just a matter of awkward nomenclature. Points 1 and 2 are bad news. Point 1 would mire us in the distracting indirectness of cells being linked and not linked, and the problem of internal links being invisible.

Point 2 is the disaster. Instead of linking up candidates into chains and observing the consequences, we have to start by trying out cells as starting cells to see if any chains come back. To help me do that Hodoku provides a set of propagation rules. To understand the propagation rules, I have to read set of tutorials from PaulsPages. At the end, when my chain comes back. I get to determine if the chain creates a contradiction within the starting cell and the assumption I started the chain with. What was it? Did I forget to write it down? A contradiction means it’s a discontinuous nice loop.

Paul’s propogation rules and loop closing decision rules are robot rules for a computer to follow.

A good example for a demonstration is Hodoku’s first under Discontinuous Nice Loops. The starting cell is nothing more than where Hodoku starts. Here, for an undisclosed reason, Hodoku starts a chain with the premise that 7r1c8 is true. It so happens that 5r4c8 is then false by XY-chain, and a slink to 5 would make 5r1c8 true, contradicting the premise. Nothing is said here about how many such premises were made before this one paid off. And no advice is given anywhere about making effective premises. Just try them all? Any particular order?

There has to be a constructive way to avoid this lottery.

In Sysudoku, constructive is the right word. We are at the AIC stage, so first, to the bv we have already mined for XY chains, we add a set of sysudokie AIC hinges. These are cells having two or more slinks to candidates in other cells. They permit a slink/wink/slink inference path of candidates through cells, bv and non-bv alike. What is left to find is AIC using these paths!

So I try to construct chains from the AIC hinges to nearby bv, AIC hinges and ALS nodes. I’ll go from left to right.

The r1c1 hinge links directly to the r3c1 hinge. Note that the other link from 4r1c1 goes to 4r1c4, stopping there but adding 4 as a possible ANL terminal

Using 4r1c1 as a terminal, I get to the 3r3c8 terminal. Restarting from this terminal is AIC 33531, with a dead toxic set of 3’s.

But another branch is AIC 34755487. Oops. I have constructed an ANL removing 7r1c8, and WHOLLY MOLEY – it’s the Hodoku discontinuous nice loop! No premise involved.

In fact, this amounts to surveying the jungle, as opposed to flailing away at it with a machete. Just facts on the grid, uncovered by a systematic construction of every AIC from the AIC hinges, the exactly right grown-up thing to be doing at this stage. It is the direct way to uncover AIC without wading through a lot of false starts based on arbitrary starting premises.

Just to show that all roads do not lead to Rome, here are the AIC generated by the remaining hinges. They connect with each other, but both ends fizzle out. This chain fails to connect to the chains we had. But we knew that already.

The Hodoku result looked unlikely to find, but it’s a human scale Sysudoku operation, building chains from a very limited number of starting points and quickly exhausting the hiding places. And the finding is inevitable. Tell ‘em where you got it.

Was that a fluke? No, but you’re not going to find these by guessing.

In the second AIC ANL, we start with five hinges and three terminals. The AIC from terminal 3r3c6 is isolated and dead.

Black is the Hodoku confirming ANL. It can be traversed by rail only from terminal 7r8c2 through the hinge in that cell. The branch into red at 4r9c9 traverses a different confirming ANL for the same clue. The green track from the 5r4c7 terminal cannot merge with the 4 slink in r8 to complete the ANL.

Compared to this systematic logical process, Hodoku’s starting with a “premise” that 4r8c2 is not 4 is just too convenient, and too improbable, for belief. No, it’s not going to work that way for you, pilgrim.

Hodoku graces this stab in the dark with an account of how it leads to a “contradiction that proves that r8c2 has indeed to be 4” and that the example can also be viewed as a “verify” in which “(r8c2=4 and r8c2<>4) lead to the same result”.

The contrast between the Hobiger/Stephens “guess and follow theory” and the construction process illustrated above is stark.  Which path are you going to follow?

My next post continues in Hodoku Chains and Loops, with the good news that AIC Discontinuous Nice Loops Type 1 and Type 2 are not something to bother with. They are simply more examples of the AIC ANL. You might enjoy trying out your AIC construction skills before looking at the post. I’m going to throw in Medusa coloring where appropriate in the Hodoku examples, because it is generally present in Sysudoku AIC building. In Hodoku, nothing builds on anything else, another departure from the real baseball.

By the way, Hodoku covers a crippled form of coloring, X-slink net (single number) coloring, not Medusa coloring. Hobiger explains color traps and wraps, but not bridging logic. In his examples, coloring is represented by cell shading, in keeping with the Paul Stephens theory of links between cells instead of candidates. It’s awkward.