The Fiendish 100 BUG-gy Maze

Bad news for Su Doku Master Wayne Gould. Fiendish 100, the only “Fiendish” of our Train Your Brain Su Doku Fiendish review to survive line marking, is found to have at least 15 solutions. The solutions are revealed in multiple BUGS, an irony in itself, since a BUG can be expressed only with the pencil marks that Wayne urges us to do without.

fiendish 100 DBTo begin the Fiendish 100 report, here are the dublex bypass and box marking traces, followed by the resulting grid.


fiendish 100 BM tr




fiendish 100 BM grid

fiendish 100 LM trLine marking depends on two naked singles, and deals with many triples.

fiendish 100 LM gridWould you want to do it without pencil marks?

I didn’t think so.








fiendish 100 SdCThere is not much for advanced techniques to do, but to start, I find a potential  Sue de Coq in


+627 +672.

fiendish 100 veriBUG trTo eliminate 5r2c2, I have to verify it, by proving that 15 is not missing, leaving Nr2 = 627 +672.

The trial trace runs right into a Bi-value Universal Grave. I color the two solutions immediately.

fiendish 100 blue greenDespite the name, a BUG is not a death sentence for the composer, but it is a big black mark. It’s not supposed to happen. Some “experts” recommend solver eliminations that prevent an iminent BUG on the theory that the composer could not possibly have foisted a multiple solution puzzle on the public.

I have found that, instead, composers sometimes do that. We should expose composers who do not fully employ readily available computer technology to prevent multiple solutions. Note that I am not talking about unique rectangle eliminations, but complex BUGs such as this one.

When a multiple solution is encountered, it signals a slippery region of puzzle logic that accommodates multiple inputs and gives the solver no logical toe holds to climb on. In this case, the faulty Sue de Coq verification BUG left me with the suspicion that more solutions may be lurking in beyond the failed Sue de Coq’s 5r2c2 elimination. It turned out to be at least 15 solutions.

I decided not to post the seven additional grids necessary to show how the solutions are derived, but those interested can ask at  for the PowerPoint solution file for Fiendish 100.  The search follows a pattern I’ve used before. Use the Find It page above to find them.

The NPM challenge was fun, but it won’t be continued in the Order of Battle. I believe this collection was tailored for NPM, in tune with Wayne’s beliefs. We have adjusted Sysudoku basic solving toward NPM, by including a dublex bypass in box marking. I believe the combination is a little more efficient than box marking alone, on average, and adds a challenge .

The Fiendish puzzles are a treat for basic solvers, NPM or not. On a road trip, my grandson Daniel and I had a solving contest with them. He won.

Before I begin the review of Peter Gordan’s advanced solving instruction, ending the basic clinic, I’ll be posting my annual report on the Akron Sudoku Tournament, put on by the Akron-Summit Public Library, with the support of Will Shortz. The puzzleakron 14 champs are basic level. You will learn how my dublex bypass for timed contests would have fared in the hands of the rapidly thinking winners, as opposed to me, in whose hands it failed decisively.

Participants who have the puzzles will be able to get sysudokie traces for all the puzzles by request on In the post, I’ll have a review table including all tournament puzzles, and a full trace of the championship puzzle shown here.

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Fiendish NPM Wins and Loses

The NPM collapse of Fiendish 130 is checkpointed, with regular slink marking backup on the solving logic. Then slink marking reveals the more demanding mental task necessary for NPM solving of Fiendish 145.

Your “No Pencil Marks” trace to a collapse of Fiendish 130 could look like this:

fiendish 130 npm tr

If you got stuck, this corresponding regular slink marking will reveal what you missed:

fiendish 130 reg trIt is longer, isn’t it?  And the point is, we won’t be needing those slink marks.

fiendish 145 npm gridTurning now to Fiendish 145, the win streak ends. on my first attempt, I managed only four NPM clues. The anemic trace was

NPM: W3, N6, C7, C4

Going back to slink marking, it became evident that I had missed something. Do you see it in the “2:” marking?

When the naked pair 12 closing chute Cc6 is noticed, the NPM trace becomes

fiendish 145 npm trA little better, but note, that in this trace, the causes of (C4, C7), SE6, and E8 are hidden. This is not a good feature of a solving trace,  another reason to reject the notion that “no pencil marks” is a superior mode of solving. Pencil marking and tracing the naked pairs, as in dublex bypass tracing, would at least track the state of locked sets and uncommitted cells.

fiendish 145 BM gridHere is the full result of box marking Fiendish 145. It comes with another strike against NPM. This puzzle requires extensive line marking. Would you want to do it without these pencil marks? Why?

I believe what the NPM solvers would actually do is begin trial and error probing, starting with the naked pair. I hope not. If the answer is more important than the logic leading to the solution, I say just look in the back.

And by the way, in line marking, do line strings count as pencil marks? Do we have to give them up as well, in order to be enlightened?

Outside of the preselected review puzzles, I ran across Fiendish 41, a puzzle that raises an army of naked pairs. For this one, NPM is like blindfold chess.

fiendish 41 gridBut the dublex bypass, just by pencil marking the cells locked by the naked pairs, keeps the parade going. From this bypass grid, you can recover the given clues and walk through the trace below to experience this remarkable solving effect. Following the bypass, box marking, line marking and solution are easy.



fiendish 41 trBottom line, there is no reason to forgo pencil marks in solving advanced and extreme puzzles. Wayne Gould, and others that claim otherwise can provide no evidence on puzzles not fashioned for NPM. Puzzle publishing is up to its hips in hypoku, and favored labels like “fiendish”, and “no pencil marks” admonitions are equally naive.

fiendish 100Next time, I’ll derive and display the multiple solutions of Fiendish 100. Want to try your own technique first? You have one week.

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Gould’s Not So Fiendish

This post begins a review of Wayne Gould’s Fiendish collection in his Train Your Brain Su Doku. The review is an opportunity to take up an important issue in basic solving, the use of pencil marks. For the ten preselected puzzles of the Fiendish review, instead of the dublex bypass of recent posts, I begin box marking without pencil marks. I checkpoint it in a separate trace, labeled “No pencil marks”, or NPM. This Wayne Gould review concludes my series of posts on basic level collections, and my clinic on basic solving.

fiendish 85 npm trHalloween is over, but was Fiendish 85 a trick or a treat? I was surprised that NPM carried me to a solution. Here is my 2-D trace.

Read it by filling the grid as you go and supplying your own reason for each effect, given the cause and the state of the puzzle at that point. Except that with NPM you have to remember candidates of that state.

Actually, my trace follows the same path as the usual slink marking. Without pencil marks, however, you have to remain aware of the slinks and naked subsets you are not allowed to write in. A good example is 6: S3. This necessary “marks memory” is also responsible for N7.

Any remaining mysteries should be resolved by following the regular slink marking 2-D trace. I carry it below to a point where it is no longer necessary to keep marks and closed sets in mind,

fiendish 85 bm tr

fiendish 85 slinks gridwith the grid now looking like this:

I know several Sudoku solvers who operate without pencil marks, but I haven’t investigated how they do it. The Fiendish 85 example illustrates how a sysudokie, accustomed to slink marking and closed subset marking, could do it on a basic level puzzle.



For the review, I preselected every 15th puzzle starting with Fiendish 10. The table includes the number clues I was able to add in the NPM phase, with a reasonable effort. Let’s consider it an average sysudokie performance, with many of my readers doing better.

fiendish review table

For two puzzles, the collapse (—-) was reached in the NPM phase of box marking. With one exception, there were very few clues added in box marking after NPM, suggesting that the NPM clues are those normally found in box marking anyway. The exception is definitely an outlier, being a multiple solution puzzle.

Another very noticeable feature of the collection is that it is definitely basic level. Line marking is generally easy, when it is necessary. Fiendish? Hardly.

Wayne Gould is a very successful purveyor of a computer solver, and a long standing supplier of daily puzzles to newspapers. Peter Gordon, again in his Mensa Guide to Solving Sudoku, relates how Wayne became well known in Sudoku history. It wasn’t his knowledge of human solving, or his opinions of pencil marking, as quoted in Train Your Brain Su Doku Fiendish. Wayne says:

“If you are writing too many pencil marks, it means you are not understanding how the puzzle works. You may be relying too much on mechanical procedures, without appreciating the underlying logic.“

At first I was aghast at this proclamation, until I realized that nowhere did Wayne specify what “too many” means. Maybe he wasn’t dissing slink and subset marking at all. He could have been making a valid criticism of the inefficient, mechanical process of finding candidates by number scanning that we have called out expert after expert for advocating. Number scanning is indeed a mindless operation that does indeed generate too many candidates, and too many pencil marks.

But then Wayne goes further, with “If, in time, you can shake yourself free of written pencil marks, you will see the Su Doku puzzle for what it is – a thing of beauty!” By the special spelling we are to understand that Wayne means his puzzles

Now that’s going way too far! Here Wayne must mean all pencil marks. And only if you go without them, can you behold the beauty of his puzzles. That’s ridiculous. I agree that keeping the marks in mind, and not on paper, is good mental exercise, but it’s going to expose less logical truth, not more. Take my own results in this post, for example. In the review, my NPM home runs were 2 of 10, but my basic collapses by slink marking were 10 of 10.

And what about advanced methods? Are you prepared to remember all viable candidates so you can do the advanced repertoire blindfolded?

Wayne’s opinions on underlying Sudoku logic will get more respect when he demonstrates  genuinely advanced and extreme level Sudoku solving.

What’s your opinion on solving without pencil marks? And do you think those who do it, on very tough puzzles, are really solving? Or could they be doing simple trial and error? That can be a mental challenge, but it is not solving by the Sysudoku definition.

As to the Fiendish collection, I have several tasks in mind to do, all bearing on the pencil marking issue.

fiendish 130First, I’ll checkpoint Fiendish 130  the same as I did 85, giving readers another shot at one for which a legitimate NPM collapse is within reach.  That will be next post .

Then I’ll examine one with which I had the least NPM success, Fiendish 145. I intend to use the successful regular slink marking to show what mental gymnastics would be necessary for NPM there.

Next, I’ll demonstrate a puzzle for which the difference between NPM and the dublex bypass is stark, Fiendish 41. It’s not a review puzzle, but one I encountered in the car service department waiting room.

Finally, you may be interested in the analysis discovering the 12 solutions of Fiendish 100.

If you have Train Your Brain Fiendish, you can go on ahead and start uncovering all of these fiendish secrets. Meet me at the passes.


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Just One More Dave Green

Here we savor one more basic solving 5-star gem by Dave Green, appearing in my paper, the Akron Beacon Journal, October 19, 2014.

My plan was to conclude the Dave Green portion of the Sysudoku basic solving clinic with the white knuckle 6/19/14 offering of the previous post. But a writing pen encounter with the newsprint version of Green 10/19 had me believing this one was going to escape Dave’s controlled delivery and get into advanced territory. A careful PowerPoint solving session brought relief, and was very entertaining, so I squeezed it into the blog schedule.

The checkpoints of the Green puzzles represents very well my current thinking on basic solving, including the latest innovation in the basic clinic. If you are a new reader, I recommend you work through the early posts on basic solving, then follow traces of the clinic, catching up on advanced solving later.

green 1019 DBMy dublex bypass took respectable advantage of Dave’s four “wells” in 10/19.

green 1019 DB tr



The DB clues speeded up the very productive box marking.


green 1019 BM tr





green 1019 BM

Line marking is tough, starting at 4 free cells and going to 7 at r8, when . . .

green 1019 LM grid

a naked quad triggers the collapse.

fiendish 85 So here’s to Dave. Maybe someday I’ll review his 6-stars or higher. Right now, I want to continue the Sysudoku basic solving clinic by reviewing a collection of New York Post puzzles by Wayne Gould, the well known composer. Wayne made some intemperate remarks about pencil marking in his Train Your Brain Su Doku Fiendish Edition. I’m going to do what I can to refute them, as I review this collection.

You can get a jump on the review by tackling one of his “fiendish” puzzles, # 85. Before box marking, I’m going to do as Wayne suggests, and find as many clues as I reasonably can with no pencil marks( NPM).

NPM is very close to the recently introduced sublex bypass, excepting only that the naked pairs are remembered, not written. Try it NPM, and I’ll back you up with checkpoint tracing. As in the bypass, I won’t be listing NPM effects in separate lists by number.

There’s always a chance we might learn something new, don’t you think?

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Basic by Our Fingernails

Here we continue a salute to Dave Green in the current Sysudoku Basic clinic by illustratively checkpointing a highly challenging Sunday 5-star published in the Akron Beacon Journal on June 29, 2014. Box marking goes nowhere in this gem, leaving it to alert line marking to keep the puzzle within basic boundaries.

Green 629 BMAssuming you took up the challenge, how did you feel when you finished box marking  with no clues and 6 slinks?  I felt puzzled. What was this mini monster doing here? Dave does the 6-stars in the LA Times. Did this one escape to Akron?

It put me on the alert to be especially careful in line marking. We start in ”4f:” and with six blank lines. Take the paper up and transfer the givens to the ©PowerPoint grid. Its going to be a long Sunday afternoon.

But then, a naked single on the first line and a productive naked pair on the second put me in a more positive mood. The good vibes continue as I trace out:

Green 629 4f tr

Green 629 4f gridAmong the resources explicitly held in reserve as I go to the next line, are two naked pairs, two X-wings and a naked triple.

I leave the X-wing icons in place on the grid as an aid to further line marking.



Continuing the trace as it runs off the page, I copy the last effects list to the left, adding the X-wings as possible causes.

Green 629 final trWadiyaknow? I never had to face the line marking of those blank lines.

green 1019If you haven’t done so already, please try out your dublex bypass, box marking, and line marking skills on this typical Dave Green 5-star from the paper just this past Sunday, October 19, 2014. There is moderate progress on the first two stages, but it requires a very careful line marking. Look for some Green whimsy, and a rarely seen trigger for the collapse. It’s my final Dave Green checkpoint in the current Sysudoku basic clinic.


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Green Pastures

Continuing the basic clinic, this post recognizes Dave Green for his Sunday 5-star syndicated series, and checkpoints an illustrative example of Sysudoku marking on one of them. The checkpoint compares the traces, with and without the dublex bypass.

Finally I turn to my daily puzzle supplier, Dave Green. His syndicated King Features puzzles from 1-star (Monday) to 5-star(Sunday) fit right in with our current basic clinic.

For me, Dave does what Nikoli claims to do. You often see a whimsical pattern in his given clues. A cross-hatch pattern may be repeated several times across the puzzle. And as to feelings, sometimes I get the eerie feeling that Dave is chuckling as he throws me a curve that I’m going to miss the first time through .

green 62214 db gridDave varies the collapse point a great deal, but his level of difficulty is unusually consistent. Except that his 3 and 4 stars sometime rival the 5 stars, at least for me. Starting with the dublex bypass up to 3 stars, I usually wind up with pencil marks for anything above 2 stars with Dave.

Our homework on Dave’s 6/22/14 5-star was surprising on two counts. First, on how far the dublex bypass took me, and secondly, the number of operations compared to straight out slink marking.  The bypass leaves a lot to do, but you get there faster, and there is a lot less slink marking before the collapse begins.

Here is the complete trace. The DB part in number order, but without number sections. We just add in box marking and line marking as before.

green 62214 db tr

There are no decisive highlights in the slink marking trace but you probably struggle if you miss anything. This trace certainly represents the best of grind-it-out Sysudoku basic.

Green 622 1-7 tr

Comparing the traces, the without DB box marking goes further through the numbers. There’s not much difference in the number of operations, but the dublex marking is faster when you are not tracing. Anyway, what you really need to assess is not the writing time, but the searching time – how long it takes to identify the next move.

It does appear so far that the bypass does not impose extra work, and could be used on all puzzles. I’ll continue evaluating.

Green 6-29-14For contrast with the early generosity of 6/22/14, the next Sunday paper brought what might be the absolutely stingiest 5-star Dave has ever produced. It’s still basic, but you have to scrounge and be extra sharp, picking up every crumb to avoid starvation – and advanced solving. See what you can do with Dave’s Sunday 5-star of 6/29/14. I’ll make the same with and without DB comparison in the checkpoint.

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The Dublex Bypass

This post describes an adaptation of Sysudoku box marking which is suitable for timed solving contests. I call it the dublex bypass. You apply Sysudoku box marking to clues and unit filling naked pairs only, bypassing other pencil marks. Here it is illustrated and compared with the Gordon Guide “scanning”, and with full box marking, on the example that introduced “scanning” in the Guide.

I’ve gladly admitted, after each of the Akron-Summit County Library Sudoku tournaments, to be nowhere close to being a contender. Of course my tender age of 76 may have something to do with that, but another telling factor is that only clues count, not pencil marks. In the Akron tournament in fact, all pencil marks that are put in must be erased. Going through my Sysudoku traces of the championship puzzles, you can see what a great excuse that is.

You know that this blog is directed at human solving of the toughest Sudoku, but even the championship puzzle of many timed contests are basic level. They never make it to the advanced techniques that require pencil marks. If we expect a basic level puzzle, why not just bypass candidate pencil marks? Or at least, put them off until they are easier to do, and there are fewer of them. In a timed contest we won’t be tracing or reading a trace, but I’m including traces so you can “read” in a 2-D trace how fast the dublex bypass would be.

mensa ex3 1 - 5Gordon’s “scanning” and Paul Stevens’ similar “cross-hatching” incorporate the dublex, or double line exclusion. Let’s examine Example 3 of the Gordon’s Guide.   Peter scans numbers in order, 1 – 9. In the Guide he does a few scans out of order to begin his discussion, but we’ll go by his regular order, for a better comparison.

Here is the grid after scanning 1 – 5 :

The trace is in the 2-D format. For this introduction only I’m adding cause coding:

h – horizontal dublex, v – vertical, oc – one-choice. Cause and effect is shown by indenting, and ordering is 2-D style, i.e. depth first.

mensa ex3 15 trNote that NE5 triggers NE4, but SW8 does not generate SE8. That is because 4 was an earlier scan, but 8 is a later one. The dublex scan will be depth first regardless, following each cause as far as it goes.

Also note that the r7 fill is not made when SW5 reduces free cells to two. That is a speedup feature of the dublex bypass.

Gordon says “Whenever I put in a number using horizontal scanning, I like to immediately look vertically to see if that gave me enough information to make a conclusion based on vertical scanning.” Actually his marking is stronger than that. His SEhv2 requires a simultaneous cross hatch. Neither the horizontal or vertical scan would do it alone. What Gordon actually does is box marking for clues only, less slink marking, and two-cell fills.

mensa ex3 69Continuing through the numbers, Gordon reaches this grid with the trace below.

mensa ex3 69 tr



The Cnr1 is a surprise for Guide readers. I’m coding it as “nr”, for “new rule”. The clue could have been traced as Cv1, but Gordon uses it to call attention to his number closeout procedure.

He mentioned before that he notes when 9 clues of a number has been found, so that he will not have to consider the number again. But he also notes when the eighth clue shows up, because then the location of the final number is known. Its row and column are the only ones missing the number.

Let’s think about that.

Do we agree that it isn’t really necessary to track numbers of clues to do this? The two clues in the horizontal and vertical to cross hatch the last clue have to be there when the count reaches 8.

With scans left pending, Peter starts through the numbers again. His count tells him that two 2’s are missing and he accounts for them, but the counts and this accounting take time. At the 3’s, he finds Nh3, and uses the new rule for Cnr3, but we can see he simply missed Nv3 when he entered Soc3. He should look both ways after a one choice, as he does after a scan effect. And the Nh3 generates Cv3 without the new rule.

Next found is the Wv4. We look at the trace to see which came first, the W9 or the SW4. It’s the SW4, so Peter did not note that the W9 closes the Wc2 chute, and look for a vertical scan that might result. The correction brings in a pile of clues. In Gordon’s accounting we learn that he doesn’t regard the last number in a line as a one-choice.

The second traverse of the numbers goes to 7 before the puzzle is finished, but we have learned enough. The dublex scan will carry every effect clue as far as possible, using the same form of 2-D trace. Two free cells in a unit will be filled with clues or a naked pair, as soon as created. There is no second traverse of numbers.

Here is the grid after the dublex bypass on 1 – 3:

mensa bypass 3To the trace below, NE3 closes a second chute in the NE box, and NE5 takes away the alternative for NE4, leaving a “one-choice” for NE9. NW5 brings the first two cell fill, and NE9, the first naked pair fill.

Pencil marking the naked pair fills is optional, but I think it pays. Foregoing other pencil marks does concentrate the marking effort on clues.

It may delay the collapse, but in a timed contest, who cares?

mensa byp 3 tr

mensa bypass 4The 4: trace shows how you can go wrong with the dublex bypass. When you are not tracing, it may require some training to be able to recall listed effects for further marking.

mensa byp 4 grid



The grid at Cnp46, along with the tail of the trace, shows how the pencil marks can speed up recognition of the last few clues of the collapse.


Adding to the success of the dublex bypass, I can report that a full slink marking of this puzzle does no better, and undoubtedly takes more time.

mensa byp BM

The Guide example was certain to show off dublex bypass well. You might want to evluate it against the Akron Tournament championship puzzles reported here in November 2012 and 2013. I will suggest and do more such evaluation ahead.

green 6-22-14Continuing the basic clinic, I’d like you to solve a 5-star by my daily breakfast composer Dave Green, that appeared in my paper, the Akron Beacon Journal last June 22.

Compare traces with me. I think you’ll find full slink marking necessary, but start with the dublex bypass, and see how far you can go. Then do box marking. Enjoy.


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