Hodoku Grouped AIC II

This post continues with two more Hodoku examples, which illustrate how a human solver finds grouped AIC. As usual, Hobinger attributes them to making just the right premise at just the right place.

Hodoku group AIC 3In the Hodoku Grouped AIC section, the third example is a group connected XY chain. The suggested starting premise is that 5r7c2 is false, and with that, Hodoku sets off to reach a contradiction.

My contention of this series is that there is no reason to start this way, and nobody should do it. It would be reasonable to notice the “almost remote pair” 56-chain and attempt to extend it.

Hodoku group AIC 3xThis could lead to an exploration such as shown here, working from the three ends of the 56-chain, and reversing polarities of the strong links. Along the way, the 7-group paths in the C box might be tried, one of them leading to the ANL of the example.

More likely than this effort, and more consistent with the Sysudoku order of battle, would be . . .


Hodoku group AIC 3c to notice the reach of the two easy Medusa coloring clusters that lie beyond the 3579 fog bank.

The two clusters collide in c2 where red and blue claim the true 6, implying green or orange. Thus

blue => orange.

So if we could not find our way to the grouped ANL, we could try blue and orange. With the blue 6 removing the middle 7 in the N box, the orange 7 forces two 7’s in c6. So green wins, soon forcing red.

With the collapse, we can leave satisfied, but wondering how Hobinger’s Hodoku comes up with these examples.

Hodoku grp AIC 4 LM wingUnlike the other examples I worked through, Hodoku’s last Grouped AIC example did not come straight out of line marking. First you have to apply this 7-wing.

Now on the example grid, Hodoku claims to start with the premise of r2c5 <>4. That’s an arbitrary guess.

Hodoku grp AIC 4 ANLThe forcing chain looking South makes 1r1c4 true, which makes 4r2c5 true, a Hodoku contradiction.

But get real. On the 1- panel you have a very noticeable grouped chain.

If you didn’t find the i471-wing earlier, you would probably be tracing the 14 bv connecting to the short 4-chain.


We rest our case. Hodoku’s starting premise is actually what a computer does t

housands of times to find a particular instance of a successful technique. It is not a practical or pleasurable way for a human solver to start looking for an AIC.

Hodoku grp AIC 4 471wingOh yes, the i471-wing? It collapses the example puzzle well before AIC in the SOOB. Hodoku didn’t catch it because the solver just doesn’t do “seeing” by forcing chain. Nor do most solvers, for that matter. With the XYZ map, sysudokies do.

Next week, I begin sifting through the very advanced fishing theory of Hodoku’s day, hoping to refine my posts of April/May 2012 on that topic.

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Hodoku Grouped AIC I

This post uses two grouped AIC ANL from Hodoku to illustrate how such chains are inspired and constructed. After the first, readers are challenged to try this approach on the following examples, with checkpoint to follow.

I was prepared to skip the next four Hodoku examples of grouped AIC chains, thinking that, beyond the addition of grouped chains, there would be nothing new in them. But of course I had to try one. That led to demonstrations that grouping expands the possibilities for AIC ANL, and suggestions on finding them.

Hodoku group AIC 1So here we are with Hodoku’s first Grouped Nice Loop/AIC. We’re going to call them Grouped AIC 1, 2, 3 and 4. After looking at Grouped AIC 1, you might want to try some of the others, before reading my version.

As expected, line marking was a chore. That completed, I added the AIC hinges and set out to transcribe the Hodoku grouped AIC chain into Sysudoku chain graphics, for “the better to see you with, my dear ” (Wolf to Little Red Riding Hood).

Hodoku knows exactly what premise to start a successful grouped AIC , but let’s think seriously on how a human could find this confirming ANL.

Hodoku group AIC 1 2-chain aHodoku group AIC 1 2-chain bFor example, on the 2- panel, you might see the 2-group above as part of the 2-chain on the left. Frustrated at extending from 2 or 9 in r2c1, you switch the c1 link back to a slink, to use an r2 wink and look for the confirming ANL, as on the right. Moving to r2c5 to extend it, the North 8-group becomes a way to do that. It provides the wink and slink out.

That much done, the chain completes easily, giving the Hodoku loop.

But there is another way to complete it.

Hodoku group AIC 1 altContinuing from the 8-group, you can use two more 8-groups and an AIC hinge to enter the 2-group, completing the confirming ANL. Or you can drop the 2-group and the confirmation slink and use the three 8-group ANL to eliminate two rival 2-candidates in the SW box to get the same result.



What would you think if you came up with this alternative, and Hodoku check pointed you with the solution it gives. I’d be sure I’d made a mistake.

Hodoku grp AIC 2aMoving on, in Grouped AIC 2 sysudokies seeing the bizarre mass of candidates would probably invest in some coloring in the bv bands around it. This nets a trap 4r9c5, but more tellingly, the three link 1-chain in the blue/green cluster could prompt a search for an AIC connection.




Hodoku grp AIC 2panelThe 2-panel helps with a grouped slink to escape from r9c9, and makes it easy to see the c4 grouped slink into r6c4. Grouped chains are usually found when the search is focused in this way. The entire example is below.

As to finding this AIC nice loop, the unlikely Hodoku premise that  r9c9<>2  was undoubtedly made after the nice loop was found, not before.


Hodoku grp AIC 2bNext time we continue looking for plausible reasons for constructing grouped AIC with Hodoku examples 3 and 4.

Want to get there first? It would be great to hear from you before the post comes out.

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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.

Mepham 15Here 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 nice loop 1Hodoku 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.

Hodoku nice loop 2The 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.


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Simplifying Hodoku AIC

This post examines the AIC examples from the Discontinuous Nice Loop section of Hodoku Chain and Loop techniques page. Hodoku claims to demonstrate a new type of elimination technique based on AIC loops, called AIC Discontinuous Type 2. AIC ANL are to be called AIC Discontinuous Type 1. AIC hinges are demonstrated as a primary aid for AIC construction. Type 2 is shown to be a particular form of eliminating ANL, The “Type 1” and “Type 2” designations can be dropped.

We have some truly instructive examples of AIC loops to puzzle over, but first, we have to dispose of the customary fog Hodoku throws around them. Alternate Inference Chain Type 1 starts with

“Any AIC can be seen as a combination of one or more Discontinuous Nice Loops . . .”. What can this possibly mean?

Hodoku mult ANLMy guess is that “combination” means “superposition” and “AIC” means “AIC Type 1”, an eliminating ANL. The AIC is the terminal to terminal AIC having multiple victims seeing both terminals. Each victim forms an ANL.

Hodoku defines the AIC Type 1 this way:

“An AIC Type 1 starts and ends on a strong link for the same digit, thus proving that that digit has to be placed in one of the end cells of the AIC.”

Did Hobiger mean to say “on strong links to the same digit in two cells”, or not?

The corresponding definition of AIC Type 2 reads:

“An AIC Type 2 starts and ends on a strong link for two different digits in two cells, that see each other. This proves that the end digit can’t be in the start cell and the start digit cannot be in the end cell.”

Looks new, and makes you wonder why he left out “in two cells” in Type 1. Oh, well. The example diagrams will show what he means, if we can interpret them.

Hodoku AIC 3The AIC hinges guide you to this first Type 1. There were enough bv to color, and the toxic pair are both blue, but that is not a color wrap. Both chain ends can be true, but both cannot be false. Now if one of the ANL victims were blue, that would be a wrap.





Hodoku AIC 4Hodoku’s second Type 1 is two X-chains with an AIC hinge between them. Two AIC hinges guide the construction, leading to two overlapping ANL with the same result. From the r6c1 hinge, if you complete the southern chain first, then three eliminations confirm 6r3c8. If you follow the northern leg from 7 first, then a single elimination 6r2c7 confirms the clue, which eliminates 6r3c45.

These Type 1 ANL are ordinary AIC ANL, so we conclude that the only reason for Type 1 is the difference with Type 2. We can conclude that Hodoku meant for the Type 1 definition to be ambiguous about one or two cells, to include confirming ANL along with eliminating ANL.

Hodoku AIC 5aAll right, let’s derive the Sysudoku puzzle state for Hodoku’s first Type 2, and see what we can make of it.

The AIC connects 4r6c2 and 8r6c4. The cells see each other via 4 or 8, so 8r6c2 and 4r6c4 are removed by the Type 2 rule.

Hodoku AIC 5b



In this case, however, why not just complete the confirming ANL? It’s wicked, but it works






Hodoku AIC 6Hodoku’s second “Type 2” example is a cannibalistic AIC ANL. The two digits in cells that see each other are 2r7c8 and 6r6c7. Remove 2r6c7.

But it also works just as well as an AIC ANL.

So we have two Type 2 examples, that are also eliminating ANL. Is that always the case?

Hodoku Disco 2Let’s think outside Bernhard’s box and draw a schematic diagram of Type 2. X and Y are the terminals of an AIC, ending in cells that see each other, via a third number Z. We don’t want to know anything else. The “seeing” is Stephens’ cell seeing, i.e. containing candidates of the same number that see each other.

Hodoku Disco ANLNow if X is in the right box, the X in the left box sees it. This allows the right X to see both ends of the terminals of the AIC slink, so it becomes the X that wasn’t there. Do the same for the Y in the right box if you’re not entirely satisfied.

Is this the only known application of cell seeing? Well, not exactly. As the right diagram confirms, if the Type 2 has any effect at all, it is also one or two superimposed ANL a.k.a Discontinuous Nice Loops. We can just forget about AIC Type 1 and Type 2. I’d rather, if you don’t mind.

Next post, we move on to nice loops. Or in Hodoku speak, Continuous Nice Loops.

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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.

Hodoku AIC 1 LMA 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!

Hodoku AIC 1So 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.

Hodoku AIC 1 missesJust 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.

Hodoku AIC 2In 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.

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Hodoku X-chains on the X-panel

On this post, X-panel chain analysis is illustrated on a Hodoku example. Hobiger’s coverage of X-chains is shallow on fundamentals and distracted with named X-chains. Several X-chain highlights of the blog are recalled.

Hodoku starts the chain types with X-chains. Hobiger over-defines the type in the same manner as XY-chains, saying they must end in slinks. This overlooks, for one thing, their use as forcing chains.

X-chains may look simple, but that is deceptive. They are the basis of grouping and ER forcing chains, demanding very thorough analysis at times. X-chain loops can also get complicated. Unlike XY-chains, the links do not carry strong and weak badges on their sleeves. Sometimes X-chains work because they can cut through whatever is going on in the grid around them. With all that, it is absolutely not useful to name different lengths and shapes of X-chains as if different methods are involved.

Hodoku Xchain 1Here is Hodoku’s first X-chain ANL, in Sysudoku notation.

Have your Sudoku student list and locate the slinks not noted in the Hodoku grid. The fill strings of line marking are left on. Two naked singles turned up in line marking, but the going was tough.

When I discovered some of the complexities of X-chains, I made a decision that analysis of fish and X-chains require that candidates of a number be examined absent the noise of clues and other candidates. So I constructed a template for that, the X-panel. Puzzles that survive the bv scan have their surviving candidates copied to an X-panel.  Each panel is scanned for X-chains and fish. Later, the X-panel becomes an aid in coloring and and pattern analysis.

Hodoku 7-chainAt left is the 7-panel containing the 7-chain. Chains are suggested by the line slinks. For analysis it is convenient to mark the alternating inference path on the panel with a single curve or freeform, which is translated into alternating curves on the grid. It is a prime example of constructing a chain, then seeing its results – the opposite of starting a chain with a premise.



Hodoku 1-chainIt’s not always that easy. Here is a 1-panel on which I thought I had found a grouped 1-ANL in the same puzzle. But when I add the 1-candidate that I missed in r4c1, it doesn’t work. Fill in the slink/wink alternation along the path and picture how it would have looked on the grid.

The Hodoku Techniques page has a link to Single Digit Patterns, where the popularly named X-chain types are illustrated, including Skyscraper, 2-String Kite, and Turbot Fish. I regard these categories more of a hindrance than a help, since the human solver should be constructing short X-chains instead. You can even have Hodoku generate 2-String Kites for you to find, one after another, but that’s fantasy baseball, my friend.

Bernhard’s inclusion of ER, the Empty Rectangle of this section, is a mistake. Although his description does clarify it somewhat, this  inclusion itself suggests that ER is a grouped X-chain. It is actually a form of weak link, a way of ”seeing”. I prefer to identify it as an aid to spotting helpful grouped forcing chains.

When I realized that Hodoku was leaving X-chains at that, going to XY-chains and then into AIC, I was disappointed. They were a lot more fun when I was exploring them in early 2012. Yes, that’s when my abc diagram for eliminating (a) confirming (b) ANL and nice loops (c) was drawn up. I liked it so much I used it for awhile as the blog header.

X-loopsHodoku explains (b), the confirming almost nice loop, as an X-chain with the slink terminals ending on the same candidate. I use the ANL designation of my preferred author, Andrew Stuart, adding confirming to distinguish its result. Starting around in either direction, you assert the candidate is false, and arrive back proving, in that case, it’s true. So it can’t be false, can it?  Another example of proper use of chain tracing from a premise; namely, in the proof of a chain property. Only unschooled beginners are allowed to use it to solve puzzles

So what fun stuff did Hobiger leave out? There’s the business of good and bad patterns of winks, and even but not odd length slink loops, McCollum’s rule, and Andrew’s guardians, and my own prolonged dance with grouping, going on into Stephen’s Nishio turned Grouped ANL Test. In fairness, everyone ought to get to play with these as X-chain entities on the X-panel before encountering them on AIC grids.

But it is what it is, and we must move on with Hodoku to AIC loops. And unfortunately, here we must deal with the confusing legacy of Paul Stephens once more.


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Hodoku XY-Chains

As we explore Hodoku Chains and Loops (tech-chains.php) examples, here I contrast the premise starting, chain following school of Stephens/Hodoku, with the Sysudoku find-em-all technique of XY-rails. Also, a mathematically sensible order is suggested for remote pairs, as a Hodoku web of them is unraveled.

The XY-chain has a built-in alternate inference pattern. The slinks are within the bv cells, and any link between bv numbers will serve as an alternating wink. Not only is it easy to build, but the XY-chain advertizes its low hanging fruit with an ample field of bv.

If you don’t know what a bv is, it’s a bi-value cell, and you’re in the wrong place in the right blog. Scroll back and read the posts from the beginning. You’ll actually directed to reading them out of chronological order because, along the way, I discovered a very easy way to exploit the XY-chain and gather in every piece of that fruit. It’s the XY-railroad.

Hodoku XY chain 1You may not have to follow the link, though, because I’m going to show the railroad for Hodoku’s second XY-chain example. The first was the XY-wing.

In the batting cage, you would score a double for the 3 removals, but not credit for a triple for the 9 removal as well. You get the extra base by simply extending the chain for two more bv, and detecting the subchains ending in 9.

In this case it doesn’t matter, because the 3 removals also take care of that illogical 9, but extending the XY-chain calls attention to the railroad.

Hodoku XY railAlso in this case, the railroad is not very impressive, because of the confines of the batting cage. But in the real puzzle collections reviewed here, there are some doozies. In this example, you can see on the rails how the 136 bv group is isolated from the XY-chain group, but is solved along with it. That may be an artifact of the batting cage technology.


I’ll not bother to contrast the XY-rail as a human solving aid to the Hodoku chain notations that are now officially beneath our notice.

Despite his demonstrated skill as a computer programmer, Bernhard shows little mathematical insight in the organization of Hodoku Techniques. He overdefines the XY-chain to require the ending slinks, with matching terminals, what I call their toxic sets. He also says that an XY-chain is not an XY-chain if it has no victim. This even as he freely uses them without this requirement within AIC. I know you don’t believe he said that, but check it out. I’ll be right here when you get back.

The remote pair is one of the easiest advanced structures to recognize, because of the constellations of identical bv.

Bernhard also betrays his math instruction when he defines the XY-chain as a less restricted remote pair. It’s the other way around. A remote pair is a very restricted special case of XY-chain, with conjugate pairs at the ends of chains of length 4, 6, 8, . . . Hodoko places them as “any chain at least four cells long”.

Hodoku remote pairsHere is the Hodoku second remote pair example, translated to Sysudoku graphics. It shows a green one of length 4, a black one of length 4, a blue one of length 6, and a red one of length 8.

The chain ends are conjugate pairs. Conjugate pairs and connecting paths are easy to follow. The pairs work as if they, and the victim, are in the same unit.


Next week, we examine Hodoku X-chain views and examples, with the welcome aid of the Sysudoku X-panel.

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