An Unusual Review

This begins a review of the advanced techniques of Hodoku, Bernhard Hobiger’s puzzle generator and solver, as described on the Hodoku site’s Techniques page. The post describes the Hodoku program and the special nature of this review.

Hodoku is primarily a learning tool for advanced solving techniques. It’s a drill facility, something like a batting cage. You can dial up curves, fast balls, or sinkers. The program will compose a puzzle to challenge a user on any technique selected from its extensive advanced repertoire, and solve it up to the point where the selected technique is needed. After finding the technique and identifying eliminations, you get a color coded grid displaying the principle candidates of the technique and its removals, with a brief note of explanation. A remarkable feature of Hodoku puzzles is that they usually come out of basic solving with an example of the ordered technique ready to be found on the grid.

From the example displays on the Techniques page, you can link to the pre-solved grid on which the technique is to be discovered, and try to find it yourself. You have to spot the technique on Hodoku’s keypad pencil marked grid. You can even choose to do the pre-solving yourself. The given clues are in black on the example grids. Generally speaking, line marking is a bit tough. In this review, I transcribe all Hodoku grid displays to Sysudoku notation.

Hodoku UR t2 n2 LMAs an example of all of that, here is the line marked grid for a Hodoku unique rectangle, Type 2. It’s the same as Andrew Stuart’s Type 2, in which two candidates of the same number on one side stand guard against the deadly rectangle of a multiple solution.



You could do a personal grid searching experiment by loading the example puzzle into Hodoku. Click on the grid for directions.

The conception and execution of Hodoku is admirable. But as you see here, the batting cage solving experience is not the same as you have on a newspaper or collection puzzle. One reason is the “composed to order” characteristic of the puzzle itself, but it is more the fact that you know what to expect. A batting cage is no replacement for a talented pitcher at game time.

The descriptions and examples the Hodoku Techniques page are there to identify and explain the repertoire of the system, which I believe, reflects the general consensus of human solving experts at the time of the Hodoku launch. Hobiger’s major influence seems to be Paul Stephens, whose recent books are reviewed here. Bernhard’s aim was to create accurate examples, and solve them by technique rules and to trace solutions, for the current techniques of the day. He did an excellent job of that.

Hodoku is not a puzzle collection, and there will be no rating table of ten preselected puzzles. However, a primary reason for the review is Hobiger’s excellent selection of examples to illustrate his training system. The review is an opportunity for me to recall, and link back to additions to human solving repertoire made since the Hodoku launch. It is an opportunity also to correct misconceptions often spread by computer solver programmers, Hobiger included, about solving Sudoku puzzles with neurons rather than CPU’s.

The Hodoku active training system provides teachable moments, but I don’t recommend any solver for your general use in your Sudoku adventures, as opposed to training for them. To my mind, it’s pointless to have your computer solve a Sudoku with mysteries to be discovered. If the solution were all that important, we’d just look in the back. We get to do it for fun. With the emergence of Sysudoku basic solving, especially the bypass, I wouldn’t even consider letting my computer clutter the grid with too many starting candidates.

The Hodoku batting cage doesn’t help much with basic solving. Examples show pencil marked grids in keypad style, with the advanced technique marked. In Techniques pages, Naked and hidden subsets are defined and illustrated. No basic procedure, with stages, is suggested. Not even number scanning. The candidates appear, courtesy of Hodoku.

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Please Give WXYZ Some Space

This post takes issue with recent redefinitions of the WXYZ-wing It suggests a name for Andrew Stuart’s update, apparently proposed by StrmCkr, and recalls the Bob Hanson’s Bent Naked N-set method in Student Assistant. The derivation of the toxic set in these two methods are contrasted with the WXYZ.

Getting a timely alert from my friend Gordon Fick, I revisited Andrew Stuart’s page on the WXYZ-wing. The landscape has shifted underneath the WXYZ. My early post on the regular XYZ-wing just passed “W” off as the rare cooperative enterprise of four bv, but during my lapse of attention, it became something else. Andrew’s second example fit my earlier conception, but his current first one, which he calls a “classic” WXYZ, shook my hammock:

Stuart Bent WXYZ 1Three of the wing bv are replaced by a single ALS! I didn’t panic, though, because the ALS supplies the not(25) & not(29) & not(59) logic of the WXYZ.

But wait! In the old WXYZ, a victim has to actually see the Z in all four wing bv. These four Z’s are a toxic set because at least one of them is true. Otherwise the hinge cell, r4c3 here, gets stripped of all candidates.

The victim 9 does see all 9’s in the new WXYZ wing, but why are these 9’s toxic?

Andrew doesn’t say, only attributing the elimination to a rule by a forum correspondent StrmCkr, which says, if effect, that in four cells containing only the candidates of four numbers(the naked set), if candidates of three numbers are restricted commons (i.e. all see each other), and the candidates of the fourth number are not, the fourth number candidates are a toxic set.

Great rule, but why is that? StrmCkr doesn’t say either, but sometimes you only need the right question. The answer is a fundamental argument in set link theory, and is arithmetical. It is that each of three restricted common numbers can supply only one true candidate of the four required, leaving the “unrestricted” number Z obligated to supply one, which the toxic set victim sees.

WXYZ figureAs I mentioned, Andrew does verify that his revised WXYZ is still about the old classic version with hinge and three bv wings, by including one as his second example. I’m going with the WXYZ schematic to make clear that all such WXYZ obviously meet the StmCkr restricted common requirements. Four numbers, all but Z in restricted common.

But does that make the above example a WXYZ-wing? I don’t agree that it does.

Stuart Bent WXYZ 3Andrew’s follow up “WXYZ” examples drift even further from the old WXYZ. His third example seems to have an ALS hinge and a three candidate wing.


At first I feared maybe the pressures of the sudowiki site had gotten the best of Andrew, but then I read through the EnjoySudoku forum thread initiated by StrmCkr with his four cell, three restricted common idea, under the title WXYZ-wings, way back in May of 2010. Andrew provides a link.

It is StrmCkr who claims that his technique is a WXYZ-wing. He uses w, x, y, and z in a display of all possible cases. In this thread the only detailed challenge was that his example could be covered by overlapped ALS, another non-wing alternative. There was some mild grumbling about calling it a WXYZ, as the thread trailed off.

The toxic set mini tutorial above suggests that anyone that familiar with base and cover sets should not have bought into StmCkr’s labeling his idea as a WXYZ-wing. And I hope Andrew backs away from it as well. Look at the derivation of the bottom line of the two methods, the toxic set. The candidate stripping logic of the WXYZ is reviewed above. It has little in common with StmCkr’s 4-to-3 toxic set logic. And besides, the StrmCkr proposal applies to the XYZ-wing as 3 numbers, 2 restricted , and can expand to 5 numbers, 4 restricted, and beyond.

I’m saying that here is a new method, not too hard to spot and easy to verify. It needs an apt name, and an identity of its own in advanced solving lit. Let’s get off WXYZ’s back and let it be. I propose giving Stuart/StrmCkr’s new WXYZ a new name: Bent Almost Restricted N-set, or BARN for short.

But as another reason for getting off WXYZ’s back, let’s recall another worthy but initially misnamed technique that also overlaps the Stuart examples, and many of StmCkr’s wxyz cases. It is just as entitled to a room in WXYZ’s condo as the BARN, which isn’t much.

In Bob Hanson’s explainer site for Student Assistant, Bob called it the bent naked subset and I got after Bob for his misuse of the good word “subset” for something that is not. There I suggested it be known as Hanson’s bent naked N-set, or BNS. I also got on Bob’s case about using the BNS to explain the XYZ-wing, which it is not. Bob had his reason for doing that. It was to make as much advanced solving as possible fit under one grand principle. But I saw this theoretically commendable objective to be at cross purposes with clear exposition of human solving methods for most solvers.

The characterizing term “bent” is already in use on Stuart’s “Strategies” page. The N-Set in the suggested name refers to the fact that the method applies to n cells containing n numbers which are not a subset as normally defined. Bob normal, but a little wild.

In the BNS, Bob calls the N-set cells of the box and line intersection the hinge, and the cells of the box and line remainders, the wings. The wings then define toxic sets by how they divide the candidates. This differs distinctively from BARN, but the effect can be the same.

BNS1 and BNS0Bob defines the BNS in two flavors, which I call BNS1 and BNS0.

If the N-set cells in the remainders have more than one number k in common, then if an outside k1 sees all of them, the N-set could still be filled, because a k2 in each remainder could be true,. No toxic set.

In a BNS1 with a single common number (k), this is not possible. If all k’s are removed from the N-sets, no other number can fill two of the naked cells, so one N-set cell goes begging in a solution.

In a BNS0, with no common number, the N-set, including cells in the hinge, is locked. Candidates of any of its numbers are toxic. I would still not call it a subset, because its set is not a unit (house), but the N-set.

In application, BNS is easier to spot and apply than the BARN. But understanding and explaining the BNS1 rationale is heavy lifting, I admit.

Clearly, Andrew’s examples above, are BNS1, as well as BARN. But every WXYZ is neither. Grouped AIC winks allow restricted commons outside of the box/line. When I get an example, I’ll revise this post to include it.

Hansons BNS0I’ll close with Bob Hanson’s BNS0 example from Student Assistant explainer, done up in Stuart style but with Sysudoku pencil marking. It’s a five number case.

Notice that cell r4c8 might have been substituted for r4c6 to remove 7r4c6 and promote 1r3c1, but this fails because 8 and 9 would then be common in the wings.

The question is, why is the BARN any more worthy of moving in with WXYZ than Hanson’s BNS? Maybe ALS chains should move out as well, but that’s an issue for another day. If you start a thread about it, could you please send a comment to let me know? I don’t track the fori (high school Latin).

Next, we begin a very instructive (for me) review of the extensive Hodoku Techniques page, which takes us back to the days when a WXYZ-wing was a WXYZ-wing. I hope to have you come along.

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Arnold Abandons Reason

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.

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.

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. 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”. Arnold doesn’t believe in learning “the difficult stuff”, so he reaches many dead ends. That’s why he goes for the impossible force only when he has many bv cells to try.

That excuse for arbitrary guessing is embarrassing enough, but Arnold also argues that the impossible force is actually a logic based method. He gives us several examples where the choice of the wrong bv partner leads to a conveniently quick contradiction. One of them is the preview puzzle #19, with the Sysudoku basic trace:

Form3 19 basic tr

Form3 19 swordfishArnold’s order of battle yields a swordfish which is quite evident on the Sysudoku 8-panel. 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 X-Y-Z-wing?” (Arnold’s names.)

Arnolds Impossible FarceArnold 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.  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?

Form3 19 railroadWell, the XY-chain railroad, 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 alternative chain logic. He would know all about “hard stuff” XY-wings as a bonus.

Form3 19 three ANLAnd 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 bv, 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.


Form3 19 coloringHe 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.

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”

Form3 19 iXY-wingOn a brighter note, I can report that Arnold’s #19 also produced the only example I have so far of an irregular XY-wing, and this with an interesting kicker. The wing, in red, is an XY-chain of length 3, with toxic end 6-candidates. The 6r6c6 victim sees one of them directly, and the other, by ER, illustrated by a grouped forcing chain. This is the kind of fun in store for Arnold when he invests more deeply in Sudoku.

Then look at what the iXY-wing removal leaves behind, a BUG +1 resolved by Stuart’s three candidates per unit rule.

Next, we defend the honor and living space of a seldom seen but still remembered friend, the WXYZ-wing. We hope to give two rivals their own apartments, so they can move out and leave our friend some breathing room.  After that it’s on to a Hodoku review. Now get out there. Summer’s waning.

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Advanced Solving with Arnold

This post reviews the 54 pages of Arnold Snyder’s Sudoku Formula 3 devoted to advanced solving techniques. These are sandwiched between 35 pages that repeat his Sudoku Formula 2, and 19 pages on how to get the solution by arbitrary guessing.

In Formula 3, Arnold deals with unique rectangles, the X-wing and swordfish, and XY-chain nice loops. He leaves out Sue de Coq, APE, XYZ-wings, XY wings and XY-chain ANL, X-chains, AIC, ALS, coloring, and pattern analysis. You have to conclude that Formula 3 is woefully thin for what Snyder claims to be “the first in-depth look that shows you how to unlock the killer puzzles and get results!”. In fact, Arnold seems unaware of such fundamentals as link logic, almost nice loops, alternating inference chains, and Medusa coloring. Arnold’s endorsement of trial and error gives me no reason to expect more from the Sudoku Formula 4 Arnold promised in his introduction to Formula 3.

The best chapter in Formula 3 is on unique rectangles. Arnold credits this material to Sheldon’s Sudoku Master Class and Gordon’s Mensa Sudoku Guide. This ignores much better sources, such as Andrew Stuart’s 2006 Logic of Sudoku, and Ruud’s Sudocue website. These use the name “unique rectangle”, or UR.  Stuart’s treatment, in particular, is very thorough, going far beyond Snyder’s. Yet Formula 3 renames the UR a “slipknot”. Arnold doesn’t even go along with his model writer Peter Gordon’s Gordian Knot!

Snyder3 1 ur1 gridArnold’s first and simplest “slipknot” is illustrated in his Formula 3 puzzle #1. There the UR is shown with the full complement of number scanned candidates, but I’m using it here to demonstrate that URs can show up in the Sysudoku bypass.

The bypass route to the UR is

Snyder3 1 ur1 tr



Arnold labels candidate 2r8c6 a rogue number even though it cannot be removed.

Snyder3 1 ur2 gridArnold continues with the puzzle, to demonstrate a second type of unique rectangle. Going on through Sysudoku box marking, the third row of line marking takes us to it.

Arnold has an effective explanation. Since Er5 must contain 1, it can’t contain a 9 as well. This has a remarkable number of effects:

(Wur9, Enp59, E2, Enp18, SEnp18)   and finishes the puzzle in line marking.

But recall, in the Snyder Order of Battle, all candidates are in place before you look for the first “slipknot”.

This type of UR is demonstrated twice more in the slipknot chapter, along with an extended unique rectangle, which Arnold renames the three-way slipknot. Google that.

Turning now to fish, Formula 3 covers X-wings and swordfish in two chapters.   Arnold has us search for them among the candidate lists, as most other Sudoku writers do. In Sysudoku, most X-wings are found in line marking, and without searching. As line slinks are identified and marked, we have only to check if an earlier marked parallel line has a line slink in the same position. The corner pencil marks help with that. If the X-wing eliminations don’t finish the puzzle, and line marking is not complete, the fish marks are left to prevent the inclusion of prohibited candidates on elimination positions in unmarked lines. Oh, but Snyder doesn’t do line slinks, only box slinks.

Snyder’s candidate lists are even less adequate for the task of finding swordfish. He describes a filtering approach for X-wings, but has no tips on swordfish.

Here is the grid for Arnold’s reader challenge to find a swordfish in Formula 3 #3.

Snyder3 3 sword gridWhile your patience holds out, search for one in this relatively simple candidate field.

I have left off the key numbers, since only the candidate lists are involved. And Cardoza , I also left out the typo in r3c4.

Sysudokies search for X-wings, swordfish, jellies, and X-chains, but not on the grid. Instead we use X-panels, where matching sets along parallel lines can be easily seen.

Snyder3 3 x-panelsIt’s so much easier when clues and other candidates are left out. The X-panels have other purposes, and the time for making them is easily justified.

I don’t show the dead 5-wing. The three lines in the 2-panel give a dead swordfish. No need to look.

The 4-panel has two 4-wings eliminating the same candidate.

The 7-panel yields no fish. Three wanna-be kraken fish, one a sashimi, fail as victim confirms fin. I looked for X-chains, but gave it up quickly.

That leaves the 8-panel. An 8-wing removes 8r5c6, but beyond that, I find three rows covering three positions, and have marked them with dashes in c3. In this blank line marking, I put pluses in unoccupied r7 to mark those positions . Then I scan the plus columns, x’ing the candidates in non fish rows.

There’s Arnold’s swordfish. It’s easy when you do it right. You might have selected the plus columns for your swordfish, removing the same candidates. That’s because the three 8 clues leave six lines for fish.

Snyder doesn’t have any of this regular fishing lore for his readers. It doesn’t help much to be shown the swordfish patterns you can’t find in a reasonable amount of time. And Virginia doesn’t understand why he leaves out finned fish and kraken analysis. Of course, I didn’t expect to see anything about the Suset algorithm fishing net.  No “first in-depth look” here.

I can’t find much good to say about Snyder’s cycle analysis chapter, either. The XY-chain is the easiest to construct among the AIC, but Arnold renames it and describes it as an isolated revelation. He describes the XY-chain as dependent on a row and column alternation, which it isn’t. Worst of all, Arnold can justify eliminations only when a nice loop is completed. Can he really be ignorant of the simple alternation arguments for almost nice loop eliminations and confirmations? If he brings out bi-value slinks, winks and AIC in Formula 4, how does he explain the omissions of Formula 3? I’m not buying another one. I’ll just browse a copy in the bookstore to find out.

Sorry, but I’m obligated to continue this for one more post. I have to call out gamesman Snyder on his ”Impossible Force”, and for his stupefying endorsement of Peter Gordon’s logically lame theory of guessing. The only useful information in it is what you can say to friends who are proud of guessing their way to a solution and are just not interested in the solving logic.

Form3 19If you plan to read that post, you might like to solve the puzzle Arnold puts forward as justifying your guessing friend’s position. Its puzzle 19 from Formula 3.

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Basic Sudoku With Arnold

This post compares Arnold Snyder’s basic solving methods, the techniques leading to a full set of candidates, with Sysudoku basic.  Snyder’s basic methods are described in Sudoku Formula 1 and Formula 2. A second post will evaluate the Snyder advanced methods covered in Sudoku Formula 3.

I haven’t had the pleasure, but I think it would be a delight to talk with “master gamesmith“ Arnold Snyder. I had the impression that he is a poker and gaming professional recruited into Sudoku by his gaming publisher, Cardoza Publishing of Las Vegas, NV. Now, having examining his approach to Sudoku solving, I wonder if Arnold has sufficiently mastered the science of probability to be a successful gaming professional. On the other hand, I see a streetwise intelligence in Arnold’s Sudoku work that could make up for such a weakness.

To his credit, Arnold is the first author I have reviewed who explicitly recognizes the value of strong links in basic solving by developing a marking technique for them. His “key numbers” are pencil marks at the bottom of a cells to mark candidates as box slink partners.

Snyder2 5 key numbersHere is the key numbered grid of his instructive Formula 2 #5 puzzle.

Unfortunately, Arnold’s key numbers marking does not identify the row, column or box of the slink, or multiple partnerships. I wonder if Arnold ever thought about line slinks, and using the corners of the cell to represent them, as in Sysudoku basic.


At this point, Arnold says it is time to find all candidates, and describes his version of line marking. But first, you might want to compare this with a Sysudoku box marked grid:

Snyder2 5 BM stoppedWell, “key number” marking includes aligned slinks only, and also leaves out triples. Recalled in the bypass or written down in box marking, these omitted slinks can make the recognition of naked pairs and filled chutes automatic. Such is the case with the naked pair Wnp49

Arnold gives his readers extra credit for recognizing these “forced twins”, once they have completed the assigned line marking. They also earn an A+ for recognizing the hidden single 3r5c7.

Snyder2 5 collapse trThe value of the unaligned slinks is demonstrated by the fact that we had to stop the 9: marking at the first level in order to make any comparison. The truth is a precipitous collapse of Formula 2 #5, as evidenced by the trace.

Aside from leaving a good part of slink power unused, Snyder never, even in Sudoku Formula 3, explains the concepts of the strong and weak link. This leaves him very limited in his ability to exploit candidates in advanced methods. But that’s for the next post.




In basic solving, Arnold is superior to most Sudoku writers in another way. He does not leave readers entirely at the mercy of number scanning for enumerating candidates.   In Formula 2, he describes a line by line process similar to line marking, but without sysudokie refinements. There is no ordering of rows and columns by increasing free cells. No reusable fill string is suggested. There is no marking of bv to aid in the free cell count. And to bring up another serious omission, there is no marking of line slinks.  That’s why there is no corner marking. All candidates are placed in a list across the top of the cell, including key numbers.

These omissions make Arnold’s line marking much less effective, and much more of a boring interlude, longer and less often interrupted by x-wings, naked pairs, and marking runs.

Arnold duly defines naked and hidden subsets beyond the “forced twins”, but offers no guidelines or algorithms to find them in the sea of candidate lists. In his last instruction puzzle of Formula 2, #6, his challenge is to find the naked quad and the naked quint in line marking, but the puzzle is felled by a naked pair induced by an X-wing earlier in the sysudokie line marking. The boxline exclusion is another elimination often seen in line marking, but ignored in Snyder basic .

I didn’t purchase Sudoku Formula 1, designated for EASY puzzles, to go along with Formula 2 for HARD ones, and Formula 3 for KILLER ones, but the way 3 repeats 2, I think I have the gist of how Arnold starts off beginners. Arnold lumps double line exclusion and cross-hatching together, explaining them informally by shading cells excluded by the outside clues in examples. He refers to the process as scanning and shading, as if shading is a solving technique, rather than a simple vision aid. Arnold dresses up the idea of outside key numbers excluding cells in neighboring boxes from containing that number as inferred shading.

Arnold includes one special configuration of clues in box marking, the wall, or filled chute. He calls it a trio. He doesn’t call attention to the 2×2 square or four corners or pocket. Maybe they’re in Formula 1.

Bottom line, Arnold Snyder is not the best advisor for Sudoku basic solving. If you gifted a beginner with a Sudoku Formula missive, don’t feel bad. Make it up to them right away by putting them on to

I thought about contacting the Sudoku Formula n cover endorsers, the Sudoku Puzzle Society and the Puzzle & Sudoku Enthusiasts Club about better information for starting Sudoku solving. But I found no evidence these authoritative bodies actually exist. I’d like to hear from their members.





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Sadistic Sudoku Review

No, it isn’t a sadistic review. I’m reviewing Arnold Snyder’s Sadistic Sudoku collection of puzzles. Arnold’s solving advice, as presented in his “Sudoku Formula” books 2 and 3, will be analyzed in the following post. Two typical review puzzles are traced and illustrated.

Arnold and his publisher, Cardoza Publishing, describe his collection in Sadistic Sudoku, ©2012 as “200 extremely difficult, hair-tearing puzzles”. On the cover they are “brutally awesome . . . each one designed to sizzle your brain, frazzle your nerves”. Snyder also publishes three instructional books with puzzles, claiming to show ”never-before-published solutions … that unlock the most difficult expert-level puzzles quickly and easily”.

Naturally, such claims demand sysudokie attention. Snyder’s Sudoku Formula 3 came out in 2010, and Sadistic Sudoku, his capstone collection, in 2012. Where has it been?

Well, let’s find out what I’ve been missing. First of all, how does the Sysudokie Order of Battle protect your composure and your hair as you embark on Sadistic Sudoku? This will tell us if any of the never-before-published methods of Arnold’s are actually required. I started with number 5 and took every 20 to 185, for the following review table.

sadistic review tableNo threat to your composure or hair, but rough on Arnold’s pants, which are ablaze. One Sadistic is a victim of box marking, five more don’t get past basic. Survivors carry generous fields of bv, and are easily cut down by XY-chain almost nice loops and color wraps. Four x-wings, two unique rectangles, three naked triples, two hidden singles, and a naked single show up in line marking. The bypass stingy puzzles starting with six free cells per line may threaten a few follicles, but melt into naked triples before any hair comes out.

For illustration, let’s go through two, your homework puzzle 25, and the stingy 185.

Sad 25 basic trIn 25, the bypass closes 17 cells by clues or naked pairs, and the 9 pattern. Box marking and line marking are not difficult.

Sad 25 LM gridThe line marked grid shows the 1-wing and a typically generous bv field. As a matter of form I scanned for Sue de Coq and XYZ-wings, but was really pushing to draw the XY railroad and start coloring.



Sad 25 railroadHere’s my railroad. It easily produces some prodigious XY-chains and toxic pairs.


Below, I found the long black ANL looking for results on an even longer nice loop. The eliminations seemed indecisive, so I continued looking along the tracks.

Sad 25 two ANL


The red chain elimination leaves a clue and a boxline.

Updating the grid, I get SWnp15 and another ANL shown below. That’s two more bv.

In Sudoku Formula 3, Arnold recognizes the XY nice loop, calling it a cycle. I found no cycles on the railroad.

But apparently, Arnold missed the much more frequently occurring Almost Nice Loop, in XY-chain or other AIC form.

Or maybe he’s saving all that for the Arnold Snyder Sudoku Formula 4.

Sad 25 third ANLIn any case, why bother updating the railroad. With such a bv field, the crayons leap out of the box themselves.

Bad news. Arnold doesn’t do Medusa coloring either.

Sad 25 coloring

In our blue/green cluster, there is a double trap in the South Box, but it’s a game over home run when two greens compute into the same cell in the North box. The blue army of new clues is overwhelming.

I’ll report on how close Arnold comes to coloring, and where he goes instead, in the next post.



For now, we’ll see what happens to stingy Sadistic 185.

Sad 185 tr to nt 1After two cells are closed in the bypass, and no clues arrive in box marking, a hair tearing line marking is taking shape.

Sad 185 nt 1 grid

A naked triple jumps out, and it seems a little less grim. But now we have to deal with seven free cells per line. Ugly.

There are five five unmarked rows and five unmarked columns, all 7f.



But if you think this is bad, try it without line marking.

I go with rows, for no good reason.  I should have seen it when 2r7c9 was removed, but 2r7c2 is now a naked single.

Sad 185 SWns2


However, not much time is lost when it turns up in the marking of row 7.

Look at those fill strings. Its sadistic. Like a monster.



Sad 185 two nt bxl


Incredibly, the marking grinds to a halt after naked triples in W and r4, and a decisive boxline Cr5.





Sad 185 LMm1



During this marking, the unmarked row 9 let 185 off the hook again and again, leaving this line alone to finish.

But just as I formed the fill string, I noticed that S1 had confined S3 to r9c6.



This was enough to make Sadistic 185 give up.The trace to garbage time follows,

Sad 185 final tr

Sorry for the overly long post. Next, in more reasonable doses, is a review of Arnold Snyder’s instruction books, the Formula series. First I consider his basic solving method, as described and illustrated in Sudoku Formula 2. If you have that book, I’ll be working through puzzles 5 and 6, for comparison.

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Revisiting Wex 435

This post updates the Weekly Extreme Review with a revisit of Wex 435, a puzzle that proved resistant to the single alternate Sue de Coq and was deemed the toughest in the review series, inspiring an extreme ALS trial, the jump ball, in the review. But afterward, two setbacks at the hand of sysudokie friend Gordon Fick enables a kraken 1-wing and two advanced level finishes, one by Gordon and another by YT.

Wex 435 UR with SdcFrom the basic grid opening the post of June 2, Gordon noted a UR removal to go with the indecisive single alternate Sue de Coq. In Fick’s UR, the floor naked pair c3np19 force one of 9r2c2 or 9f8c3 to be true. Then if either 1r2c2 or 1r8c2 is true, a deadly rectangle, and at least two solutions, occurs.




Wex 435 W-wingNext in the bv scan, Gordon calls upon his Hodoku training, and finds a W-wing on the 1-panel. The W-wing is described on the Hokoku wings page as a pair of identical bv with one of the two numbers linked. Hodoku over specifies the link, saying that a strong link is required. Actually, the inference logic requires a weak link is made by forcing chain or ER. A unit wink would make it a naked pair. Hodoku’s two examples illustrate the W-wing with AIC winks. I now include a W-wing schematic and one of the Hodoku examples, translated for Sysudoku readers, in the XY-chain post.  The matching bv of a potential W-wing makes it a tempting target for building AIC winks.

Wex 435 ANL krakenAlong with the W-wing, Gordon came up with a 1-chain(green) removal that pairs with the W-wing to enable a kraken 1-wing on the 1-panel.

Kraken analysis is another worthwhile occupation for forcing   chains.  Here, double chains force through a grouped chain that gives the finned 1-wing victim its fatal vision of the fin.


Wex 435 nt boxlineThat is followed by a NW naked triple and a NEr2 boxline. At this point, Gordon and I part company.



Wex 435 color wrapI’m attracted to the meadow of bv for the coloring. The trial trace helps me find a quick wrap in which two blue candidates are found guilty of removing all 5’s from r5, proving green.

The collapse starts with 4r4c3.





Wex 435 Ficks wicked AICGordon, on the other hand, found Wex 435 to be a playground for AIC ANL, using XY-chains extended by slinks. His first blow was this wicked ANL with XY ends and a Medusa X-chain in the middle.

I added the relevant AIC hinge that would have helped me construct such a chain. My hinge marker encloses candidates bringing slinks into a multiple candidate cell. The slink pencil marking by corners makes hinges easy find, and they help you spot AIC strong links.

Wex 435  wicked trWex 435 second W-wingAfter the follow up, another W-wing:

As Wex 435 falls back on the ropes,


Wex 435  AIC hidden pair


Gordon wades in with another AIC ANL, this XY and X-chain mix bringing a naked pair to remove 7r7c9, and leaving a hidden pair 57 in the East box.

Wex 435  hidden pair trGordon’s knockout blow was a round house AIC ANL that covered the grid, and brought an immediate collapse.


Wex 435  AIC knockoutBefore the collapse, I got a snapshop of an irregular remote pair, in red. The victim sees one of the forbidden ending 5’s by forcing chain.

I hope these posts have alerted some dedicated WECers to new possibilities in the competition puzzles.




Please comment, particularly if my review of another Competition puzzle would be of special interest to you. I’ll try to enlist Gordon to attack from the other direction.

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