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Please see the Glossary for any terms that look unfamiliar in the remarks below.

 

Remarks on Terminology

Wings and x,y,z

Sudoku theory is an effervescent phenomenon. People write such good stuff on web sites, and they give their ideas away for free. I love it.

It's no wonder that all this activity has given rise to a large vocabulary with many synonyms and a few conflicts (see Sudopedia's large glossary at www.sudopedia.org/wiki/Terminology). This diversity is perfectly natural, given the great variety of individual input.

Nevertheless, there are two terminological components that are rather fuzzy:

• wing
• x,y,z

The terms x-wing, y-wing, xy-wing, and xyz-wing have no conceptual unity. "Wing" doesn't mean anything specific. If to that you add x-chain, xy-chain, and xy-loop, you get a great hodgepodge of end-of-alphabet names that fail to convey any clear concept. I choose to avoid these terms.

By contrast, the term Fish is much more satisfactory.  2-Fish, 3-Fish, 4-Fish, swordfish, jellyfish, frankenfish, and mutant fish are all conceptually related Tactics dealing with Lookalikes and constraint sets.

Candies

Well, so I get worn out writing "candidates" fifty times on a page. The only good word is a one-syllable word, but sometimes I have to settle for two. A candie "can die", of course, given the fact that we spend most of our time trying to kill the little things, so the spelling seems appropriate.

Twins versus Pairs

Twins, Triplets, Quadruplets, and Quintuplets are identical candies in different Cells of one House.

A Pair, a Trio, a Quartet, and a Quintet are different candies in one Cell.

Nobody expects the members of a trio or a quartet to look alike.

But people do expect triplets or quadruplets to look alike.

Twits & Bandits

Twits and Bandits are the intersections of a Block with Columns or Rows, respectively.

I need words for these things. And block-column ranks right up there with great iron bird. I wanted something short. (A Twit is, so to speak, a little Tower. A Bandit is a little Band.)

 

Remarks on Tactics

Frequency of Occurrence

Every Tactic requires that a certain pattern must occur on the Grid. But for a given Tactic, how often will you run across the required pattern in real-world Sudokus?

On the web, you can find descriptions of dozens of different Tactics. For example, www.sudopedia.org/wiki/Terminology lists more than 70 of them.

Is the frequency of occurrence of these Tactics relevant or not? You could argue that one does Sudokus for pleasure, and that learning lots of different tricks, even if some are rarely useful, is the major source of that pleasure. Ultimately you could argue that one could cease solving Sudokus altogether and just concentrate on the absorption of new Tactics bearing exotic names.

It sometimes happens that the proponent of a new Tactic honestly points out that the situation for applying it arises "rarely". But how rarely? After doing hundreds of hard Sudokus, I have found productive xyz-wings and swordfish to be rather rare; I have never seen a productive jellyfish; and I can't imagine spending my time looking for a sue-de-coq. A few of the Tactics that apply to less-frequently-occurring patterns are covered on this site, but it is explicitly noted that they're a bit rare, and the overall emphasis is on Tactics for attacking commonly occurring situations. (Many other sites follow the same philosophy.)

Directed Chains versus Aimless Chains

Sufficiently easy Sudokus can be solved via a succession of Tactics that just require the application of certain visually recognizable attack mechanisms on the Grid. But for harder Sudokus there comes a point where you have to find or construct a less obvious Attack Chain in order to make your next Kill.

Attack Chains come in two very different varieties:

• Directed Chain:  through analysis, you determine a Worthy Target (whose demise may crack the entire Sudoku or at least substantially clean up the Grid), and then you try to construct a Directed Chain to attack that specific Target.

• Aimless Chain:  having memorized a slew of different kinds of Attack Chains which are recognizable by their visual form, you see if you can find one on the Grid. If you've located one, it will kill some specific candie-value in its Target Area. But are there in fact any victims? Maybe, maybe not. If there is a victim, will its demise give rise to a big cascade of moves? Maybe, maybe not.

You will find Sudoku solvers on the web that will lead you through a succession of Aimless Chains that kill one insignificant candie after another, gradually whittling down the Grid to the point where it finally collapses.  Well, that's one approach.

On this site, however, the emphasis is on a two-step procedure:  determine a Worthy Target on the Grid, and then try to construct a Directed Attack Chain to crack that Target, which will give you a big bang for the buck. What's the advantage? The satisfaction of carrying out a well-planned, productive maneuver. What's the disadvantage? Well, if I really can't find a Worthy Target, I'll just stick the Sudoku in the shredder.

It's a psychological choice, of course; but at least you know what my bias is.

Human Doability

There exist a few off-Grid techniques, such as subset exclusion and braid analysis, which involve the construction of extra lists or tables that produce results telling you that certain candies on the Grid can be deleted. These were definitely interesting ideas, but with the passage of time they have been pretty much rejected as practical Tactics because they aren't very user-friendly.

However, there is one cumbersome off-Grid formalism that has not yet died, but it's hard to imagine why:  this is the weak/strong terminology, along with the descriptive runes known as eureka notation (or its cousin, nice-loop notation).  Concerning these ideas, www.sudopedia.org/wiki/Inference says this:

• "Two premises can be linked by a weak inference if they cannot both be true."
• "Two premises can be linked by a strong inference if they cannot both be false."

This takes us back eight hundred years to the realm of classical logic. "Weak inferences" are contraries, and there is nothing "weak" about them. "Strong inferences" are subcontraries, and there is nothing "strong" about them. And many of the relations in a Sudoku are contradictories, meaning they are both "weak" and "strong" at the same time, which is lethally counterintuitive. And the notation! Look at this:

[R3C8]-1-[R5C8]=1=[R5C3]-1-[R2C3]=1=[R2C8]-1-[R3C8] ==> R3C8<>1

Can you read that? Of course you can. But do you want to read that? People work Sudokus on a two-dimensional visual basis. They understand that when you promote a 9-candie to a Big Number in some Cell, you then have to kill all the 9-candies in that Cell's Buddies. They know that when you kill one candie of a Pair, the other candie gets crowned. They grasp the fact that when you kill a candie, all of its Twins (if it has any) get crowned. They understand the idea of a Chain visually, on the Grid itself. All the off-Grid terminological and notational rigamarole sometimes associated with Chains and nice loops drags us a long way off from the simple visual process of solving Sudokus.

"By logic alone"

You will sometimes see this curious phrase used to mean "without using Uniqueness Tactics or a Fork" (although a Fork may be called a "bifurcation", presumably to enhance the feeling of its presumed perversion).

A puzzle that has no solution or two or more solutions is not a Sudoku. A Sudoku has one single solution. The Unique Rectangle and BUG+n Tactics are logical techniques based on that fact.

A Fork has an honourable history in mathematics in the form of proof by contradiction. Just as a mathematician could, in principle, make a brainless attempt at a proof by contradiction, it is likewise possible to do a dumb fork in a Sudoku. But it is also possible to do some analysis of the Grid to illuminate the choice of a Fault Point from which to launch a Smart Fork, and there's nothing even vaguely illogical about that.

Related to this whole issue is the occasional insistent denunciation of "trial and error". But when I have nine empty Cells in a Row and suspect the existence of a Locked Quintet, you can bet I commit lots of trial and error in trying to find it. When I'm sweeping the Grid looking for 2-Fish, Sashimis, and Splatterfoots (finned fish), you can be sure I am engaged in massive trial and error. If you're going to solve a Sudoku, you're going to use trial and error; you just want to use it as comfortably and efficiently as possible.

 

Mumbling

I have recommended mumbling to yourself as an integral part of certain Tactics.  If I mumble the theoretical justification as I proceed, I'm less likely to go astray.

And talking to yourself is such a wonderful way of having a conversation where nobody is going to disagree with you.

Try it, you'll like it.

 

Theoretical Questions

I have not done a search for mathematical literature on Sudokus. Much of it is not available online, and I will freely admit that I have found it tough going wading through the few articles that I have read.

So it's quite possible that someone has already published results that answer the questions below.

• Since Twins and Pairs both exhibit bilateral mutual exclusivity, one might suspect that there exists some sort of duality relationship between them. Has such a relationship been worked out?

• When I construct a Wrong Chain or a Gotcha Chain that cracks a Sudoku, sometimes (for practice) I go back and try to construct a different Chain, focussing on a new Worthy Target that is not closely linked to the original Target. When I do that, I often find that I have to run the new Attack Chain through several of the Cells used in the original Chain if I want to find a Chain that works. It's as though there existed a region in the Grid that naturally supports the construction of an Attack Chain. Is there any methodical way to look for such a region?

• When I mount a successful Attack on a high-quality Fault Point or Point Line (see the Chain page), that very often cracks the entire Sudoku or at least substantially cleans up the Grid. But, rarely, the Attack produces a whimper, not a bang. That means that my criteria for assessing the weakness of a Fault Point or Fault Line aren't good enough. Do there exist better criteria?

• For a difficult Sudoku, if there exists a high-quality Fault Point or Fault Line on the Grid, then mounting an Attack on that Fault may crack the Sudoku. But both those notions of a Fault are Twin-based constructs. For a Diabolical Sudoku, there aren't enough Twins on the Grid to create a Twin-based Fault. Is there a more general notion of a Fault which would be findable and attackable by an unaided human?