This post identifies the conjugacy chain, the last fundamental chain type to be introduced in *The Hidden Logic of Sudoku*, as a weakened X-chain. Since the last few chapters introduce extensions for XY chains, it becomes clear that *THLS* is incomplete on the fundamentals of links and chains. Such deficiencies explain why symmetry spaces are deemed necessary for SudoRules.

Before examining c-chains, here’s a checkpoint on Gordon Fick’s AIC on the line marked grid of Royle 17-1002. If you placed your AIC hinges, it was easy. There were only the same three, and the AIC went through all three. Following the SOB, I didn’t put on my AIC hinge glasses before the X-panel scan of the last post, and this chain sneaked by undetected in front of my clutter distracted eyes.

Sysudokies can live with Denis Berthier’s description of XY chains. His idea that it is cells, rather than candidates, that are linked seems a little quaint, but the mechanism of left candidates and right candidates linking the bv cells does account for the very simple AIC action of the XY chain. Besides, many early Sudoku authors shared this view.

But reading through *THLS* Chapter XVI on “conjugacy chains” and realizing that the remaining chapters are about extended XY chains (xyt and xyzt chains), a sysudokie experiences a disquieting sense of loss. The conjugacy chain, the last major category of chains in *THLS*, is just an X-chain, and a weakened one at that.

Earlier in THLS we read that two cells are *c-linked*, when they are in the same unit and share a number __exclusively__ in that unit. That makes the c-link is a unit induced strong link, a slink.

Beginning Chapter XVI, the THLS definition of the *conjugacy chain* reads:

That is to say, a chain of identical values, with every other link, starting with the first, a slink, and ending with a slink. No nice loops, no confirming ANL and no grouping..

How peculiar. In *THLS* there is no reference to the general concept of an alternate inference chain. Come to think of it, throughout *THLS* there is no mention of the logical definition of strong and weak links between candidates. The slink and the wink are confined in narrow, limiting definitions based on cells sharing a unit. Mixed type AIC do not exist. Grouped inference chains do not exist. The AIC anatomy of an XY chain and the bv reversed AIC chain is undefinable. Nice loop coloring and extension is impossible.

Actually, these missing fundamental elements may account for the whole edifice of hidden symmetries. *THLS* can be interpreted as proposing computer generated hidden logic symmetry spaces as a remedy for a huge blind spot in supposedly expert human solvers, requiring the starting line generation and constant maintenance of extra grid spaces. You can believe this necessary if you lack the candidate linking fundamentals required to identify in nrc space the reverse XY non-hidden counterparts of XY chains in the hidden logic spaces. Berthier invites that very conclusion with this:

“For advanced examples, see chapters XV, XVII and XVIII, where hidden chains of various types are introduced and shown to be irreducible to non-hidden chains.”

In this c-chain chapter, there is little incentive follow the “proof” that X-chains should never be closed. It plays out somehow in super-hidden subset rules, with subsuming of longer chains with loops. On that point, how can one see that the loop can be bypassed, and not see the logical consequences of the loop itself? *THLS* misses the nice loop again, here in its simplest form.

Ironically, Berthier includes an observation that regular fish are special cases of the c-chain. Or as *THLS* states it “the Swordfish(row) and Swordfish(col) rules are subsumed by the c6-chain rule”.

Here is an instructive example from the Absolutely Nasty IV review. The c6-chain on the same candidates as a swordfish is a nice loop! Not only that, but this nice loop c-chain has a victim the swordfish cannot touch. So much for subsumation.

But sometimes, the fish wins. If row 2 had an 8 candidate in r2c2, the fish would be fine, but the c-chain would lose a vital c-link.

Sysudokie friends, the nice loop is not to be disparaged. You put one together before you even think about all the removals it may cause.

With the many decisive X-chains appearing in the earlier Sysudoku review puzzles, I expected the same in the chapter on conjugacy chains. The first c-chain example, Royle 17-57, illustrates one reason there are few.

On the freshly line marked grid, the bv field is so rich, there is a 6-cell remote pair that overrides the X-chain (red). With it’s identical bv, the remote pair is a string of flashing lights.

After the removal, why mess around? The first coloring is an easy wrap.

Of his second c-chain example, Royle 17-118, Berthier states: “its L4_0+C4 elaboration is equal to its L1_0 elaboration.”

The Sysudoku basic equivalent to L1_0 follows conventions designed to enable readers to generate traces that match the blog checkpoints. Apparently, the Sudorules resolution path does reach a state in which a four cell c-chain exists. It doesn’t matter, when immediately after the simple line marking, there is a coloring wrap(r6c9).

To finish the puzzle from this point *THLS* suggests either a 4-cell XY chain in nrc or a 4-cell XY chain in crn space. This may explain why there are no pencil marks on* THLS* grids. It can be embarrassing at times.

Berthier follows with two more examples of disappearing conjugacy chains, before reaching one of relevance to Sysudoku. It is Royle 17-147, showing alternative resolution paths via c4-chain and XY4-chain. In a bv field like this there may be more, but these should be among the shortest.

The reason the bv are not marked with green squares? It’s because the grid was filled in the bypass. Are you following this, Wayne?

The grid above is also a near BUG (Bi-value Universal Grave). Coloring. Here the arrows show the order of a coloring that resolves it with the 4 and 8 in r3c2 being the same color. That color is thereby wrapped, and 9r3c2 carries through to the solution.

In *The Logic of Sudoku,* Andrew Stuart suggested an unproven rule for near BUGs. It was to confirm the 3-cell candidate that appears three times in a line. Unfortunately, in his book, he applied it to a puzzle with three solutions. In THLS, Berthier points this out, and faults Andrew’s rule for assuming uniqueness, i.e. that the puzzle possesses a single solution. Denis claims that using the rule hides the non-uniqueness of the puzzle.

Andrew gave no rationale for the rule, and certainly no explanation of why it depends on absolute uniqueness throughout the puzzle. The incident may have inspired Berthier to make a similar claim against unique rectangles, confusing uniqueness among cells on the four corners of a rectangle spanning two boxes, with absolute uniqueness of the whole puzzle.

In coloring, as illustrated above, you are continuing to follow the implications of the givens without assuming the value of any cell. It such a logical train arrives at a solution, it is unique. Coloring is a car on this train, a collective property of a slink net. Coloring can be used to uncover multiplicity of solutions only by assuming one of a cluster’s colors to be true.

Andrew’s unproven rule had no such standing, and he made no claim that it did.

Let’s do one more, another X-chain steam roller like Royle 17-1020 back in the 1/16/17 post. One that makes up for these logically least complex c-chain softies. Introducing Royle 17-16774, a line marking monster where bv are few. But also, where the X-chain is the buzz saw that cuts through the trees and moves out the wood.

Then, when line marking is complete, there’s a 4-chain and an XY nice loop that SudoRules never saw. Also, a decisive 6-ANL that SudoRules did see.

And we’ll get to see what fisherman Fick caught from the beach bordering the tropical forest.

This seems like as good a time as any to ask. How can a BUG have *three* solutions? Don’t all the bi-value squares form a color network, which then reaches the two candidates in the tri-value square that aren’t also trilocal? So then either those two candidates are the same color (which wraps, leaving the trilocal candidate standing alone and solving the puzzle) or they are different colors (and the trilocal candidate is caught in a color trap, leaving exactly two solutions). I don’t see how it is possible to have three solutions.

In your post titled “Mashing the BUG” you mentioned something about a mix of blue and green candidates in a solution. But the definition of coloring should make that impossible, shouldn’t it? If colors can be mixed in a solution, how can coloring make any eliminations at all?

Another excellent question, dov. Andrew Stuart’s rule was supposed to resolve a “near BUG”, not a BUG. The rule can fail when there are multiple solutions. This is what happened when the rule produced conflicting slinks (mixed coloring) and no solution. When it works, it produces a color wrap. One color, one solution. Coloring wins every time. Coloring trials found the three solutions Denis refers to. I have a case somewhere in the reviews, where an apparent BUG produces a color conflict, and no solution.

So Stuart’s rule does assume uniqueness. But unique rectangle methods do not. They assume rectangle uniqueness, a far different animal.