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Sudoku/Reviews/Systematic Sudoku/Arnold Snyder

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Arnold Abandons Reason August 18, 2015 (John Welch)

review by Abd (talk) 16:34, 25 July 2019 (UTC)(Comment and correction invited.)

This post calls out Arnold Snyder’s embrace of arbitrary guessing as a way around the inadequate advanced Snyder methods in Sudoku Formula 3. I also deplore his misguided endorsement of Peter Gordon’s Sudoku Guide.

Formula 3 was published in 2010. Did Snyder recommend "arbitrary guessing"? Not really, but he did not explain well. As to Gordon, I have already reviewed the first edition of the Mensa Guide (2006). What is missed by all authors is that with ordinary sudoku, there are many possible bivalue choices, and even the world's hardest have some. In ordinary sudoku, what is most common is that any paired choice can crack the puzzle, if coloring is used instead of just single value Nishio. Single-value Nishio is closer to "trial and error," but it is not clearly guessing, if the choice is an informed one, rather than random. Authors generally do not define "guessing," nor do most define "logic." They make up their own definitions without specifying them and then opinionate. And that is a brief summary of much of the world of Sudoku literature, which I found amazing when I started to investigate it this year.

I'm opinionated, too, but this is a public wiki, anyone may edit these pages (on attributed pages, I will decide what to keep, but sincere comments on the Discussion page will be kept, and edits made here, other than vandalism, will be copied there if not kept. Accounts will be provided on request, see the home page here.)

If the puzzle survives the “Snyder Method”, Arnold advises that you “forget all the difficult stuff”. If it has plenty of bi-value cells, use Arnold’s ultimate weapon, the grandly titled Impossible Force. Arnold doesn’t say what to do, if it has only a few bv. Maybe that’s what Sudoku Formula 4 is all about. I’m not going to find out.

There never was a Formula 4, so far (2019). There is Sadistic Sudoku (2012). However, there is How to Solve Sudoku Puzzles: A Player's Guide to Solving Easy and Difficult Puzzles (2016) -- after the review we are looking at here. I'm ordering it.

Immediately it looks to me like bifurcation, which is indeed an extremely powerful method if used from a bivalue source (2 candidate pair in a single cell or paired single candidate value in two cells in a region.) In fact, before I started discussing sudoku publicly on Reddit, I was using "Snyder notation" -- which I call "double dotting" and "Snyder" in that is Thomas Snyder, not Arnold -- with some additional clarifications, to solve nearly all ordinary puzzles, but the class of puzzles represented by, say, Longo's Black Belt Sudoku ("If you have to ask, it's too hard for you!") did not fall before that technique, nor did the more advanced puzzles in Sudoku Addicts Workbook, Paul Stephens (2008). So, having been given those books by a friend, I invented (independently), bivalue coloring, off of a "promising pair," and found that it solved all ordinarily published sudoku, without caring about advanced "fish," etc. It went "underneath" the fish and other specific patterns and solved them from basic principles.

What happens if a puzzle has only a few "bv," i.e, bivalue cells? [This is an artificial limitation, regional candidate pairs also work] This is the observed fact: Most sudoku have *many*, Very difficult sudoku have few, and the most difficult, very few, and they are not "productive." But those are not ordinary published Sudoku, they are examples constructed to be difficult, such as Sudoku/Inkala's Maze, and others. If Arnold is showing how to use what I call Simultaneous Bivalue Nishio (SBN), he is indeed giving readers some advice that will serve them with nearly all published sudoku. Elsewhere I will provide examples of where it fails.

[Snyder does bivalue Nishio, all right, but does not do "simultaneous," where the study is done with coloring, so that one can see interactions. So he will much more commonly come to an impasse.]

To make this clear, SBN can be easily done on paper in ink, without the complicated structures that Welch and others recommend. The human engineering in this field has been atrociously neglected, and Welch, in spite of the ambitious title of his blog, does not do better.

So what is the Impossible Force? Following in the footsteps of his mentor, Peter Gordon, Arnold doesn’t tell you what anything is, but only in great detail what you do to verify that his example works. Or in this case, he does describe the process, because it’s not difficult.

Ad hominem argument and guilt by association. All too common. In fact, what Welch asserts as Synder's practice, he then immediately contradicts, Snyder does explain, so why did he write that in the first place?

Here is his overall description of how you “find” the impossible force: “You find it by taking any cell that has only two possible candidates. Assume that one of the candidates is the actual number, then follow the trail to see where it leads.” That means solving until you reach a contradiction or a solution? No, as you read on in Sudoku Formula 3, you discover that “where it leads” means solving until you reach a contradiction, a solution, or a “dead end”.

What is called a "dead end" is not actually dead, the situation being merely that a more complex strategy is needed. I call this an "impasse." This is about the solver, not the puzzle itself. A common cause is that the solver has missed something. Bifurcation always simplifies the puzzle, splitting it into two puzzles: one with no solution (if it is a single-solution puzzle), and one with a solution which will be a little bit easier to find, or a lot easier. Simultaneous Bivalue Nishio is simultaneous bifurcation, looking at both legs at once. This can find extraordinarily complex "strategies," without ever naming them. It just cracks the puzzle.

"Where it leads" includes resolutions and eliminations through weak links, not merely strong ones. Welch may have missed this. The method described if restricted to strong links is Sudoku/3D Medusa, which is far more limited.

If Snyder actually wrote "until you reach those three conditions," and that the method has then failed, he did not fully develop the method, because the more general strategy is Alternating Interference Chains, with this approach being limited to a bivalue starting pair, which then almost always creates progress with ordinary sudoku. It fails with sudoku specifically designed to be extraordinarily difficult, specifically because they have no productive pairs. I'll be showing examples. While this is a problem in ink, SBN may also easily be repeated with a new choice of pair, when using Hodoku. Each time it may eliminate or resolve. Most Soduko can be solved with a single run from a single pair.

Arnold doesn’t believe in learning “the difficult stuff”, so he reaches many dead ends.

Suppose it is possible to solve Sudoku much more simply than many authors have believed? Suppose this makes it unnecessary to learn to find an entire zoo of "patterns" which is what many solvers use. This is the reality: I learned to do this, deriving it from basic principles, not having read all that literature. This does not stop me from learning patterns. Nishio is a logical technique that has been around for a long time, named after Tetsuya Nishio. There is a particular application of it with box cycles that I found on my own, very early in my solving history, not requiring coloring or any marking (other than eliminating the candidate tested). (And this is what is described as Nishio by various authors. Simultaneous Bivalue Nishio is quite different and far more powerful.)

This is, again, a straw man argument, based on what Arnold allegedly "doesn't believe." I have been using, and lately, promoting SBN as a generic tool to use whenever "stuck," that is, whenever one goes through one's personal armamentarium of strategies, and comes to an impasse where nothing seems to work. All solvers benefit by knowing and, my opinion, systematically practicing the basic strategies, updating them when there are eliminations and resolutions. Beyond that, the more patterns one can identify and use directly, the simpler the puzzle remaining and the faster SBN will be if needed. The best book on solving techniques that I've found is Stephen's Sudoku Addict's Workbook (2008), and, at the same time, it's lousy in certain ways. I recommend it for serious students, and I've just bought Mastering Sudoku, week by week, by him, which I intend to review.

I suggest that staring at puzzles is not a productive solving strategy, and that when an impasse arises (a much gentler term than "stuck," it merely means that a way forward has not yet been found), there are a number of choices that are always open, starting with putting the thing down and doing something else for a while, preferably something that one enjoys. But as long as one is enjoying the process, it's fine to stay with it! (And this is heuristics, human engineering, something that has largely been neglected in many suggested methods).

Then there is SBN. It has been poorly and incompletely explained about everywhere, if mentioned at all. It is done best with "coloring." I have seen very complicated methods, and hints that some may have figured out that you can mark candidates distinctively, which is what I discovered. I was using black ink on paper. "Coloring," when I saw it, did not communicate to me! But I was coloring.

Using some computer solvers, like Hodoku, candidate coloring is available, and all that is really needed for the basic technique is to be able to color with two colors. In black ink on paper, and using dots (positional notation, which Welch imagines carries no information, when he is blatantly and obviously incorrect on that from a human engineering perspective. It is redundant to the tiny numbers that occupy the same positions, and it is compatible with them (i.e, I'm habituated by dots, but the numbers in position are like little blobs, and when my focus is lousy, I can still read them that way. The eye is faster at reading and comparing geometric patterns than text, i.e, numbers. There is even more to this than I'm saying here.)

So in ink, I circle dots or write a triangle around them. Easy to see, easy to read, easy to compare and it is in the comparison that SBN becomes a truly powerful method. Does Snyder realize that? Welch sure doesn't!

In the method, one looks at both sides of a pair. It is not just a guess of one side. It is like any other method where we look at details and pick something to analyze. It is far from an "abandonment of reason."

And Welch apparently subscribes to the "If A then not-B" school of logic, Aristotelian, dualistic. There is no conflict between learning and using SBN, by choice, and learning the advanced pattern recognition strategies, and those will, indeed, make SBN more productive, though it is already extraordinarily productive with ordinary sudoku and simple strategy, requiring for power only a sane choice of pair, and there are principles that may be followed. What SBN does is to present the user with data from two mutually exclusive, alternate sudoku, one with choice A, the other with choice B. Using SBN, I do not care which choice is "correct." I will learn from the process either way.

And it is much simpler than the explanations I see in many places. How to explain effectively is is a technical writing or human engineering problem. Many explanations I see are so complicated that people who would be perfectly capable of understanding a method are turned off, give up. Stuart on SudokuWiki has done a lot of good, I use the site routinely for many purposes, but some of his explanations lead me to "Maybe I'll look at this later." Welch is worse. With Welch, it is "When I have time, I may look at his overall presentation of his methods." I.e., same result: "Later," because obviously there are many prerequisite understandings -- and he doesn't backlink to them.

This is a wiki. Anything here can be improved, and we can add links such that the first time a term is used on a page, there is a link to an explanation.

(Welch's blog could be edited to create far higher readability. Is anyone helping him? This is a wiki here, anyone can help. Anyone may add local links (even if redlinks, i.e., no page has been created yet) or links to, say, a SudokuWiki page. SudokuWiki, by the way, does not appear to be a wiki. I have no idea why Stuart called it one. This is a wiki, and it did have open account registration until very recently. It still allows anonymous editing, and an account will be provided to anyone who shows interest in the topics covered here and requests it. Requests may be privately made on the blog, anonymous comments are allowed. Provide a real email address (which will not be published) and the comment will also not be published if that is requested in it. With an email address, I will have a way to provide the account information, be sure to give a preferred name.)

That excuse for arbitrary guessing is embarrassing enough, but Arnold also argues that the impossible force is actually a logic based method.

Well, when this was first written, I was not sure what the "impossible force" is yet, but I'm sure it's logical. That is, if there is a set of two mutually exclusive possibilities, one may look at the logical and necessary implications of each of them. That is, Either A is true or B is true, not both and not "neither" (i.e., assuming a valid unique solution, and this approach generally can prove uniqueness by showing that only one is true). This is standard logic, Proof by contradiction.

But contradiction is only part of the method. There is also mutual elimination and resolution. It is the mutuality obtainable by looking at both options at the same time that has largely been overlooked. [And Snyder did overlook it, as did Mepham, proposing the same method in 2005, before him].

He gives us several examples where the choice of the wrong bv partner leads to a conveniently quick contradiction.

This result is extremely common. But Welch misses how the method is actually used, so focused is he on the choice of "wrong bv partner" as "guessing." It is not a "guess" to observe that there are only two possible solutions for some pair, and it is not a "guess" to look at one or both.

Because I do not use uniqueness in solving of sodoku -- other than as a possible hint, sometimes -- and am seeking a full proof of a single solution -- I must always look at both legs of the pair. In fact, the most difficult situations I encounter are that one pair completes the sudoku, and the other is languishing with very few proposed resolutions. But, so far, if I keep looking and especially looking at intersections between the two result sets, I find them and the sudoku moves toward a BUG, one cell at a time, until I am led back to a contradiction, which often is found by the "false" choice leading back to and confirming the "true" one, i.e,. the one that completed the puzzle.

This is fully logical, but it also happens to be very simple to practice, fast, and it bypasses all those lovely complicated structures, they aren't actually necessary. But they are still interesting, and knowledge of them can speed up SBN.

Form3-19-basic-tr.jpg
One of them is the preview puzzle #19, with the Sysudoku basic trace:

Huh? What does this notation mean? I have to read a book before I read a page? He could link to a page where this is explained, but he doesn't do that. Have you ever read a technical description where there were terms that the author seems to assume you know, and know how to apply them to the problem you are attempting to fix, but you don't? That is unskillful technical writing, it's very common, especially when software engineers also write the help pages.

So with the help of my friend Google, Sysudoku traces. OMG. No wonder it's not intelligible. To read it, I need to learn a new language, quite different from what everyone else uses. He is writing for himself and a small circle of those who have been following him. To represent a cell takes only two digits, Gordon uses this. Rows are numbered from top to bottom, and columns from left to right. So top left is 11 and bottom right is 99. The cell is thus fully specified, but it apparently has been realized that it is helpful to be a bit redundant, so those two cells are often represented as r1c1 and r9c9. In fact, on some of his guide pages, Welch uses that.

He does not actually explain the trace notation, referring readers to the "beginners page." Where is that? I looked for it. It probably exists, somewhere. Why is this so hard? When referring to a page, link to it! Very basic: his trace notation. A clear guide would not present this as a problem in itself to be solved, it would simply explain clearly or refer to it with a link (or, in a book, a page number). I looked for his notation and was unable to find it. So I'm giving up, I'll come back if it's needed. [I finally found it at Sysudoku Speak."Glossary" would have been a more useful title there.

Snyder1.png
That’s why he goes for the impossible force only when he has many bv cells to try.

He has not explained "why." Let's take a look at the Sudoku itself. SW Solver link to givens, to get to the study point, turn off all advanced strategies and run basic strategies repeatedly with Take Step until it gives up. I also loaded it into Hodoku. Welch's display of the board is difficult to read, because of his punk notation.

Arnold’s order of battle yields a swordfish which is quite evident on the Sysudoku 8-panel.

In other words, you Snyder and Snyder-follower dummies, you can find swordfish by using the Sysudoku 8-panel. It's easy, you can't miss it. All you need to do is press 8. Lucky guess, 7 or 9 won't do it. Of course, you can do what most of us do, look for the swordfish pattern. It is not difficult to see, if one looks. But this is not Snyder's point. His point is that if you think you are stuck, because you don't see that fish, you are not actually stuck, there is something you can do. (For an exercise, find the swordfish without Snyder's "hint"!)

His next comment, though, is telling. He notes the lack of more fish, slipknots, or classic cycles, then asks,
“Do we have to start looking for the really weird patterns like jellyfish or squirmbags? Do we need to find some erratic cycle pattern that’s not rectangular? Do we need to look for a different type of pattern that we’ve seen discussed online but that’s not even mentioned in this book, like an X-Y-wing, or and [sic] X-Y-Z-wing?” (Arnold’s names.)

There is an answer: No, we don't, not with ordinary "hard" soduku. Welch thinks this is "telling." It is simply true, and the "telling" is probably a fantasy in Welch's mind. On reddit, someone will ask about a puzzle, and a helpful redditor will say "Swordfish in rows X,Y,Z." I will respond there with a description of a swordfish, and how to spot them, and then the rows or columns in which it lives under a spoiler link, and likewise the columns, so the OP still gets the fun of looking for it.

And, no, the answer to spotting them is not "load the puzzle into a solver and press the button." but maybe, "Press the number highlight buttons in sequence, so that you look at all of them, looking for the 3x3 pattern."

But is this necessary? Part of human engineering is understanding the distinction between "necessary" and what is chosen. I know that I can whack just about any published sudoku with SBN, but I never start with it. (This could also be difficult. SBN works better with a board that has been simplified, though it is also true that in the SBN process, I often notice simple resolutions that have been missed, because SBN in-process is always looking for row and column pairs, and will thus find singles, for example.)

Snyder2.png
Arnold avoids this “difficult stuff” by “assuming” that 4r4c1 is the “actual number”, and following the trail to see where it leads. Conveniently it leads to a contradiction in a nice rectangular pattern, ending as 3r9c4 implies 6r9c1 in an “impossible force”. Arnold then announces that the collapse from 4r4c1 can be verified by the answer in the back of Sudoku Formula 3.

Did a simple note that the answer may be seen in the back become something to be ridiculed?

I'm not trusting Welch's account as being fair. What I'm going to do is look at the puzzle with SBN, which is trivial in Hodoku. I will often choose the first available pair, if the legs look good enough. I'll first see what happens with the alleged choice of Arnold. It has not been well described. The pair is actually two pairs in one: 4 in r46c1 and {48}in r6c1. So I will color the upper 4 green and the lower red. I have colored the pair in light orange. Green leads simply to completion, as shown on the right.

Snyder3.png

However, I don't use uniqueness, so I'm not done. I need to show that red leads to a contradiction. This is normally easy, because I can use intersections, now, for eliminations. It was easy. I colored eliminations in light purple, purely to show here, I would normally just remove them. Common resolutions are shown with a cell colored light pink. Again, normally, I would simply resolve the cell, which then would, in Hoduku, automatically remove any conflicting candidates. There is an apparent NUR colored in light blue in the red chain. That's an illusion, caused by not resolving 8 as confirmed in r2c5.

Form3-19-railroad.jpg
Case closed, and Arnold moves on to more examples, even using Peter Gordon’s Repetitive Bilocation Cycle as a “too difficult” strawman he can bypass by arbitrary guessing. I won’t waste your time with that, but of what “difficult stuff” in Formula 3 #19 does Arnold’s “impossible force” leave him unaware?
Well, the XY-chain railroad [shown on the right], would have shown him a large number of his “classic cycles”, and he might have realized why they are often ineffective in a cloud of related bv.
Since Arnold does “classic cycles”, he could do certainly find XY-chain ANL, if only he undersood [sic] alternative chain logic. He would know all about “hard stuff” XY-wings as a bonus.

Arnold's point was very simple: all that complicated stuff may not be necessary. Welch points out the XY-chain railroad.

And by being no more systematic than he is in pursuit of his “impossible force”, Arnold could find many ANL (almost nice loop) eliminations. Here are three of them, with toxic ends marked. Extensions to the inner chain remove 6 and 8-candidates, but the removal of the inner chain, the 4-candidate, removes the other two and finishes the puzzle.
And of course, with such an extensive network of bivalue choices, why not color? Here, Snyder has allowed himself to be mislead by the one writer he endorses, Peter Gordon.
Gordon has him believing that Medusa coloring is a way of seeing where his guess leads.

This is not Medusa coloring. Medusa coloring is limited to certain puzzles having strong-linked cycles. This is AIC, with a bivalue starting limitation, at least I think so.

However, Welch shows something very similar to what I do. Yet his explanations were so impenetrable that I didn't see this on first reading, I'm seeing it today (October 30, 2019)

He has totally missed an easy way to exploit the puzzle’s network of strong links, made extensive by the bv. That network is a fact on the ground, like the bv themselves. It is there regardless of which candidate Arnold guesses is true.

Bingo! Simultaneous Bivalue Coloring, off of a pair, and it equals SBN. He misses something. In practice one colors like this from a pair, and the seed pair is important, unless a 3D Medusa is found, which requires restricting the chain extensions to strong links. He does describe mutual results, which is what SBN finds.

In this case, the easily applied cluster covers the bv field. It traps two candidates for one clue, and forces two green candidates in r8c1, and two green 6-candidates in r8, a color wrap that declares all blue candidates true. So easy. So decisive. Such a testament to the willful Sudoku ignorance of “experts” Arnold Snyder and Peter Gordon. Instead, Arnold finishes his puzzle with another “Impossible Force”

Yes. Snyder is incomplete. However, Impossible Force will often work, and it is a logical technique. As stated, it includes a step that looks like guessing, but that's a naive description. It is not guessing to look at two exclusive values, and it does not become guessing merely because one must (with the limitations of ignorance) look at one of them first. To the method, it does not matter which leg one examines first. The method does not depend on finding the "correct guess." What Snyder missed -- and Welch sees -- is mutual results, that work off of both legs being examined at once, which is coloring. But how to do coloring, Welch has poorly explained. At least on this page. So poorly that I missed it on first examination, confused by his complicated notation.

SBN is not a panacea, a universal solver. It does require the presence of at least one useful bivalue choice. Those do appear to be ubiquitous in published sudoku. However, it was not useful for Inkala's Maze, we needed to use Ariadne's Thread. It is not useful at the major impasse in what was at one time considered the hardest sudoku, harder than Inakala's Maze, this beast, which is now solvable with Exocet, which I'm still wrapping my head around. When that puzzle is hit with Excocet, though, what is left is solved with Simple Coloring, which SBN will do practically automatically. Characteristic of those very difficult puzzles is serious scarcity of useful pairs. When I look at an allegedly very difficult Sudoku and see lots of pairs, I infer that it's actually easy!

Today (October 30. 2019), I found something very explicit in Sudoku Formula 3 that I had missed:

Finding an impossible force is a variation of cell coloring, a technique that is primarily of use to those who solve puzzles online or with a computer program that allows them to literally "color" cells. Ideally, you'd like to be able to color, uncolor, and change colors of individual cells. It's a logical process that essentially reveals impossible forces that allow the removal of candidates from cells. Working on paper in black-and-white, we cannot take full advantage of the cell coloring strategies, but studying these strategies does reveal a lot about the types of patterns that lead to impossible forces, and opens your eyes to using similar approaches without the colors.

True, but he is also revealing that he hasn't used coloring. Cell coloring is almost useless compared to candidate coloring, which is what really works. Using SBN, I only cell-color the seed cell, so I can identify it if I find a contraction. A contradiction only eliminates a leg from the seed, essentially nothing else. But mutual eliminations and resolutions are unconditional, which is a demonstration of how a "wrong guess" can be positively useful, if paired with a "right" one, even when we don't know which is "right" or "wrong."

"Coloring" was done with literal colors because it was easy for those willing to use colored pencils. I didn't like that because I wanted to work in ink, I love the clarity of ink, the feel of it, and the discipline of avoiding mistakes. A mistake makes a mess, which I don't like, but I also want to know my error rate, and I can look back over what I've worked on and see it.

Reality is always better than what I want or imagine.

And because I was faced with puzzles where ordinary candidate listing, and the strategies I knew, were not enough, I realized that I could mark candidates distinctively and chain them. In black ink on paper. And it cracked the most puzzles I could find!

What I see over and over is authors declaring something impossible ("cannot") that is quite possible. This is very, common, we convert our own ignorance into an impossibility. And that language disempowers us, makes it more difficult to find. A declaration of ignorance doesn't have that effect. Declaring possibility, even if we don't know how is empowering, the vast association engine of the cerebral cortex starts searching, and it will do this outside of consciousness, until we have one of those moments: "OMG! This is so simple! Why didn't I see this before?" The answer to that question is, itself, simple. We start out not seeing anything! Then we see things. Order of the universe. Realizing that what we don't know is not therefore impossible is a basic step toward awakening to reality.

Snyder is only thinking of elimination, and his method will eliminate only one candidate at a time. Coloring can eliminate all the non-solution candidates from an entire puzzle, fairly quickly. It is an incredibly powerful method, and those who have described Nishio have failed, in general, to notice the "mutual results." I just received Mastering Sudoku (Stephens, 2007) and he does cover Nishio, but with a very limited view of it, demonstrating that he had not really explored it.

And in ink on paper, the candidate notation systems that all these authors use make coloring quite difficult. They are all designed to need to be erased, and they don't give room for the clear marking that makes SBN fly. That's why I often harp on lousy candidate marking systems. Remarkably, most computer programs except those dominated by "experts" (such as the web solver linked from the Youtube channel, Cracking the Cryptic), use positional notation, the only difference being that I use dots instead of tiny numbers, because it is much easier and faster to place a dot in position than to write a tiny number in the same position. And visually, dots and small numbers in matching positions are the same, it becomes unnecessary to *actually read* the numbers. This human engineering perspective has been almost entirely missed.

I prefer coloring with Hodoku to my marking in ink, because I can erase the Hodoku marking, making the penalty for choosing a punk seed (an unproductive one) far less, and "productive" can be quickly judged. In ink, therefore, I take more care in choosing a seed, looking for a significant chain of proposed resolutions with each leg. It's a judgment call how much is "enough." It is almost always, now, enough. There are then techniques for recovering from a poor choice, but, in ink, the puzzle becomes quite a mess and error rates go up.

(And recent studies have shown that with many supposedly difficult sudoku, almost any paired choice will crack the puzzle. It is not critical, that choice, usually).

To move beyond this would be simple, but I rarely do it: create the candidate list in ink, and place resolutions in ink, but then color in pencil which can be erased. This technique is what I'd use with world-class "unsolvables," but since I can also work on them in Hodoku, it's not necessary.

Enjoy Sudoku is a phone solver that allows coloring.

See also[edit]