A Gordonian Fantasy

Here we spring a booby trap laid by the “definition by example” policy of the Peter Gordon Guide. It is the Gordonian Polygon, as demonstrated by his Guide to Solving Sudoku. The correct interpretation and application of the strategy are revealed. Also, a counter example shows again why this “Guide” cannot be trusted.

The first sentence of the Guide section on Gordonian Polygons reads: “What works for rectangles also works for figures of more than four sides.” That is true, regarding particular arrangements of identical candidate pairs described in Beyond the Rectangle. But these uniqueness figures are not mentioned, much less identified, in the Gordon Guide, and the reader is left to interpret Gordon’s sentence to mean any polygon (of identical pairs) of more than four sides. Gordon’s first example suggests he means exactly that, with one of the cell vertices of the polygon containing extra candidates along with the bv candidates, and no apparent connection between all vertices. From the “extra candidates” cell, he says, the bv candidates can be removed, because one of the extra candidates must remain to prevent a double solution.

gordonian polygonGordon’s first example, Example 23-1, is a loop of 57 bv and extra cell 357r3c2. Peter’s rationale for removing bv numbers 5 and 7 from r3c2 is that, if 3r3c2 is removed, the polygon loop of six 57 bv can never be resolved.

By this argument, if you can draw in a six- sided polygon, you’re Gordonian, and the removal can be made.

But this just isn’t the case.

gordonian fantasiesFortunately, one of the Guide’s Gordonian examples demonstrates that this simple rationale is a fantasy.   Going back to Example 20-1, the grid for the one sided Gordonian rectangle, a dead remote pair forms four sides, allowing us to complete a Gordonian Polygon in three different ways. Problem is, they place the 1-clue in the three different wrong places.


Do we need a magic rule? Is it because we can add only one side to form the polygon? The Guide doesn’t say.

gordonian unpolyThe Guide does mention another reason why the particular Example 21 case works, but it has nothing to do with polygons. Going back to the diagram, the 3r3c2 removal allows the set of bv to remove all outside 5 and 7 candidates that could resolve whether each cell contained 5 or 7. The puzzle would have at least two solutions. No polygon is necessary.


Gordon’s polygon argument is that, going around the polygon, there would be no way to decide which alternating cells are 5, and which are 7. Again, two solutions.

But what kind of loop is Gordon’s polygon? Why should the bv in r5c3 and r3c5 be considered adjacent cells in a loop? Unfortunately, the Guide has not developed the Sudoku fundamentals to explain it, but the loop is of conjugate pairs. Lines and boxes define most of the pairs, but r5c3 and r3c5 form a conjugate pair as a remote pair, bookending a series three conjugate pairs.

Bottom line, Gordon has stumbled over something that works, but he is unaware of what he is dealing with. There is certainly no reason to name it a Gordonian anything.

BTW, Example 21 is easily solved without assuming a single solution and using any uniqueness argument.

gordonian coloring 1Two clusters are well supported, and the bridging logic is

not(blue and red) =>

green or orange,

removing 5r5c2, which proves green, and  catches two orange 5’s in c2, proving red. No resistance is left.

To guard against a more complex multiple solution, I tested orange, but it quickly asserted that both blue and green are false.

The Guide follows with the Gordonian Polygon Plus, the One-sided Gordonian Polygon, and Gordonian Extended Rectangles. I have to see if there is actually anything new under these titles, and I invite you to come along on the next post. Have a groovy Christmas.

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Gordonian UR-ology

This post continues a review of advanced instruction in Peter Gordon’s Guide to Solving Sudoku, by explaining why unique rectangle methods need not be renamed “Gordonian rectangles”, and why you can’t count on Gordon’s Guide to make you an expert Sudoku solver.

Before we get to the Guide’s Chapter 8 on Gordonian Logic, which Gordonian jellyfish of the previous post did you find first?

gordon jelliesMy post Casting for Regular Fish details the blank line tally, a marking method using blank lines on the panel to mark fish lines by dashes or vertical bars and victim lines by plus marks. Like most authors, Gordon provides no visual tools for actually finding fish.

If you had difficulty finding one of these jellies(and who didn’t), try going high tech with my suset fish detector. On the rows, the row/position susets are 2/12389, 3/137, 4/2346, 5/79, 6/26, 7/67, 8/1234789, 9/29. Taken in increasing order by number of position digits, the list makes 5679 rows/2679 column positions obvious. To the point, the “expert” Guide has nothing like this. If you haven’t tried it, do it on the column jelly above.

There is a choice of jellies, because a complement fish of (9 – #clues – n)  lines is defined along with an n-line fish, having the same victims. This makes most searches for jellies and all searches for squirmbags unnecessary, the complement fish being simpler.   Experts know this, but it’s not in the Guide. When you go fishing, leave Gordon’s Guide at home. It’s not waterproof and it doesn’t make a good boat seat either.

gordon ur 1Turning to UR’s, here is the first unique rectangle example in the Gordon Guide:

In this simplest of all UR, 8r3c5 must be present in the solution, to prevent a multiple solution.

Only Peter has chosen to call it a Gordonian Rectangle. That didn’t catch on.

Along with this example Peter Gordon relates how he independently discovered the unique rectangle. My UR post of January 8, 2013 recounts the descriptions of unique rectangle variations by other experts. These include the Guide’s “Gordonian”(single guard) and “Gordonian plus”(multiple guard) variations, and more. Gordon reports that he dubbed this strategy “Gordonian rectangles”, and that his partner Frank Longo came up with “Gordonian plus rectangles”. As always, the Guide only demonstrates in several specific examples what to do when that UR situation is found. It defines no general procedures or logical ground work for UR. As far as I know, Gordon has never offered evidence that he is the innovator deserving to name the UR strategy for himself. No such evidence appears in the Gordon Guide. In my opinion, “Gordonian Rectangle” is nothing more than misleading and shameless self promotion.

gordonian plusAfter another example of the very same UR type, Peter presents this “Gordonian Plus” unique rectangle at left. Gordon’s argument is as follows:

“Once we eliminate 1 and 3 from cell 58’s candidates, we have a pair. Both cell 57 and 58 have 6 and 8 as their only candidates, so one must be 6 and the other must be 8, That means that cell 55 can’t have a 6 or 8 in it, so it must be a 4.”

This explanation, though accurate, shows why Gordon’s Guide is not going to help anyone become a Sudoku solving expert. It starts with the wrong problem. The right problem with multiple extra candidates is to prevent all of them from being eliminated. We have to find a different culprit in each case. In this case, we must find means to prevent the simultaneous removal of 6 and 8 from r5c8. Having decided that, we see that removal of 4r5c5 creates a naked pair that does just that, so 4r5c5 must be true.

Now look at Gordon’s argument again. He has the reader wondering why both 1 and 3 must go, and then goes through the argument candidate by candidate, avoiding any Sudoku algebra that shortens the logical trail. He walks it out, avoiding the term “naked pair” or any equivalent term, along the way. If the reader synthesizes any generally working procedure from this, it is in spite of this introduction and long winded account of events specific to this very case. There is no insight into why the solver is doing what he is doing.

gordonian one sidedA similar “from scratch” explanation obscures Gordon’s example of a “one-sided Gordonian rectangle”, also well known among unique rectangles. The UR requires a 1-slink removing 1r8c3. The slink also creates a box/link restriction eliminating 1r1c2, which Gordon neglects to point out.

Next post, we’ll debunk another Gordonian “innovation”, the Gordonian Polygon. There is some innovation, but it isn’t Gordonian.

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The Peter Gordon Fishing Guide

We leave November and basic solving with a review of Peter Gordon’s advanced solving recommendations in his Mensa® Guide to Solving Sudoku, now re-issued as the PuzzleWright Guide etc. After a brief summary of the Guide’s general failings, this post examines Gordon’s Chapter 7, The X-wing Family. It’s about regular fish.

Basic solving is by nature a fixed sequence of techniques. The Sysudoku basic review found Gordon’s Guide inadequate for harder puzzles, because it depends on cell by cell number scanning, an inefficient, repetitive, and boring process for enumerating candidates.   But at least, Gordan puts his “scanning” process first to some degree. I was inspired by this and by Wayne Gould’s NPM, to adopt a sysudokie version, the dublex bypass, as a new preliminary phase of box marking.

The Gordon Guide review continues now into advanced solving. Unfortunately, it demonstrated that his treatment of advanced solving is far too incomplete to regarded as a true “Guide”. While claiming “ingenious tricks that many experts don’t know”, it actually omits too many tricks that experts know well, and offers very little that is new. Worse still, the Gordon Guide treatment of the techniques it does cover is woefully superficial. Peter simply demonstrates the technique at work, avoiding the embodied advanced concepts entirely.  In Gordon’s Guide, techniques are not built upon common logical concepts. You can only hope to recognize another instance of any technique from the Guide in the sea of candidates before you. Peter finds them, but he never tells you how. If you don’t see one, never mind. Just use a Gordonian trial and error method and you’ll soon have the answer.

Sorry to be yet again be delivering bad news on published Sudoku instruction, but the remainder of this review will still be of interest. I promise to stay off my soapbox and to demonstrate the particulars in a matter of fact manner. There’s a gem among the rubble. And you will find many helpful links to advanced Sysudoku posts relevant to Gordon Guide examples.

So let’s do it. Chapter 7 on regular fish is where advanced solving starts in in the Gordan Guide to Sudoku Solving.

The X-wing is Gordon’s first advanced method, to be undertaken with all candidates in place. To sysudokies the X-wing is the only basic fish, most often and almost automatically revealed in line marking. The Guide’s Example 13 is a good example.

gordon xwingIn the last post, readers were invited to hop into Gordon’s boat and follow his Guide to an X-wing. To get to Peter’s stuck point, shown here, “candidate free” neighbor numbers were considered necessary. Now did you complete the candidates by number scanning, a.k.a. “one-choice”? Sysudokies have slink marking, but the Guide reader is stuck with the candidate lists shown here.

No problem, though. Gordon finds the X-wing, and everything else, for the reader. The “Guide” doesn’t deal with the finding process at all, for anything.

I’m sure you found the 7-wing regardless of your choice of marking, but with line slinks marked, it leaps out at you, once you have the candidates. But wait. In the SSOB, this puzzle never gets that far.

gordon xwing slinkedIn line marking, on row 4 of 3f: r9, c5. 4f: r2, r4, the 7-wing shows up as we find cells for the number 7. Clearly, there are no more locations for 7 in r3 either. We mark the 7-wing to await the marking of line 3. In the placement of numbers 167 in r3, 6r3c3 is a naked single, and the collapse begins. We never went door to door, looking for the 7-wing. It came knocking.

gordon jelly panelWhen we move to X-panel territory, Guide omissions become more serious. Peter illustrates swordfish, jellyfish, and squirmbag, cheerfully finding each one among candidate soups of growing complexity. To see how misguided this is, just try your hand at finding a jellyfish on the 1-panel of Example 15-2. Guide readers are supposed to be doing what you’re doing, but with the 1’s embedded in those cell lists of varying lengths. The answer will be waiting in the next post.

As you search, note that the Guide never suggests anything like the X-panel. Peter just tells you where the jelly is, and reminds you that it’s four places for 1’s. You get a conversational acquaintance with Sudoku terminology, but that’s it.

If you have the Guide, don’t even try for Peter’s squirmbag. I thought I would show how the squirmbag with two 6-clues showing would be replaced by a 6-wing, but Gordon’s grid of completion candidates has a hidden single in row 2 that destroys all candidates that would be removed by the squirmbag, and starts the collapse. Isn’t somebody supposed to check these things?

The Guide tackle box has nothing else. No finned, kraken, or sashimi fish. Does any expert not know how to use these? I’m all right with the omission of mutant fish or sudopods. I’ve never needed them myself, but experts know about them.

Next post, we see what Peter Gordon does with a well known advanced technique he names for himself.

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The 2014 Akron Sudoku Tournament

Here is a report on my experience in the Akron-Summit County Library Sudoku Tournament of 2014 last Saturday. In particular, it’s a brief review on how well my recently introduced dublex bypass worked on the tournament puzzles. Each puzzle came from a different Will Shortz book, with his kind permission. This post also contains a very detailed walkthrough on solving the championship puzzle. By following it, and as necessary, looking up strange words on the Sudoku Speak page, tracing questions on the tracing page, and related posts on the Find It page, new readers get immediate entry into Systematic Sudoku, an Akron product, in one post.

Happy Turkey Day! I’m thankful for every interested reader.

Personally, I had a great time at the third annual Akron Sudoku Tournament. I earnestly tried to pick a lucky contestant number, and apply my latest secret weapon, the dublex bypass, to get on the championship stage. But I flubbed it royally. Nevertheless, more careful analysis at home reveals that DB worked very well. Let me explain that for those not following the recent posts of the basic solving clinic.

Dublex stands for double line exclusion, whereby if a number is known to be in two lines running through three boxes, then it goes in the third line in the last box. What is bypassed, temporarily, is the marking of box slinks. The DB, or dublex bypass, is box marking of clues only, plus naked subsets that reserve cells of a unit, but cannot be exactly placed within the subset. The naked pair is the only type of subset that is frequently occurring. One advantage of doing the bypass first and strong links later, is that there may not be any later.

In the tournament, there were three 20-minute rounds, then a championship round on the auditorium stage. Three puzzles in round 1, two in round 2 and one in round 3. Scoring was on the number of errors and blanks, with a bonus for finishing early.

akron 14 tableFor the tournament participants, who took home a fresh copy of all the puzzles, here is the tournament review table.

Under “Clues” is the number of given clues. The DB column shows the number of cells assigned to clues and pairs. The box marking column gives new clues/slinks marked. Line marking reports the number of lines marked with 3 free cells, 4 free cells, etc. The dashes mark collapse points.

The number of puzzles and the level of difficulty reflect the practical limits of an afternoon human solving tournament. Within these limits, the dublex bypass did very well. Of the five puzzles solved in box marking, only one required further slink marking. In my limited experience with DB, slink marking often reveals overlooked DB clues.

So, Akron participants, how did you do on the championship puzzle? Let’s walk through a sysudokie solution. To follow up on this, tournament participants can get free detailed traces of the other six puzzles by requesting “2014 Akron Traces” on sysudoku@gmail.com. The traces show exactly what is involved in collapsing these puzzles quickly. But first you need to learn about box marking traces.

Get out a fresh copy of the onstage puzzle, and let’s go. You can figure out most of it by following along, and nail it down with references pages later.  We start by scanning 1 to 9, looking for clues and for slink marks and triples that produce clues. This goes much more quickly when we write in only the new clues, and naked pairs. What’s the first one? ‘1’s dublex and two crosshatches give three box slinks, but no clues.

‘2’s first crosshatch has four free cells. The second generates a slink, which makes a dublex, which makes another box slink. Nothing to write down.

Look at NE3. Its NW crosshatch makes a NW slink, making only a N slink. but its dublex with SE3 implies E3. Writing in the 3 clue alerts us to the filling of the E chute, forcing a 1 in Ec7, giving us NE1. We look at the new clue’s crosshatch with SW3, then going down the grid, we look at dublex and crosshatches from C3, and SW3.

The ‘4’s don’t cooperate. The ‘5’s make a SE slink, but the dublex is clueless. We could look for a second dublex to partner with the ‘6’ dublex to get a naked pair in the N box, but it’s not there.

akron 14 DB trI think you must be ready to explain for yourself the next trace effect NE7, so I’ll leave you with the entire DB trace, and a quick update on reading 2-D traces.  First of all, the trace is a list of causes and their effects. A single effect, or a list of immediate effects in parentheses, is indented under its cause.

As you read, the effects on a list are posted on the grid together, then each effect becomes a cause, going from left to right. In the depth first 2-D trace we have here, all effects of a cause become causes before the next effect on the list becomes a cause. So the posted order of the DB trace above is

E3, NE1, NE7, N7, SE8, Snp38, S1, SW1, NE6, N6, SE2, SEnp57, SEnp69, SW2.

It’s up to you to determine why each cause has its effects. Never leave an effect without understanding why it happens. Besides learning particular tricks, being able to watch flow of action is the value of the trace. Solving can go in many directions, but a strict order is followed in writing Sysudoku traces so that solvers working independently can compare results. But trace readers do not have to know the ordering rules.

In the DB trace the current state of your updated copy should show why the naked pairs appear. The solver is to notice when a clue leaves only two free cells in a box or line. The effect is a pair of clues or a naked pair.

akron 14 DB gridSo now what does your puzzle copy look like? Did you put your naked pair candidates in a simple list, or in their keypad positions?

Well, you could go sysudokie and put them at the top of their cells and in corresponding corners, to mark them as strong link (slink) partners.



The significance? The slink reserves the two cells for the two partners. Why hide it? Especially since we are now going to add more slinks to the grid, in box marking.

The task of box marking is to add those box slinks to the grid that we saw, but did not record, in the dublex bypass. Without looking at the grid below, read the trace in the same way, updating each effect, to get the swing of box marking. We did without number list labels in the DB trace, but life was less complicated there.

akron 14 BM trN9 means a 9 in the North box. You decide where. Cm mean slink marks (two candidates ) in the C box. We start with an aligned triple of 1’s across a chute in the E box. We know its 1’a because the effect is on the 1’s list.

akron 14 BM gridWhen you add S4, you should have this grid, except for the green squares dragged over the borders of bi-value cells. I call them bv, singular and plural, because of the central role they play in advanced (beyond basic) techniques.

Currently the only bv we know we have are the naked pair cells. There are others among the candidates we still need to identify to solve this puzzle. The most efficient way to get all candidates is to fill lines, starting with those with the least free cells, in continuing in increasing order of free cells. I call it line marking.

The most efficient way to get all the candidates for advanced solving, namely the box marking followed by line marking described in this post, was published first in Akron Ohio, and so far, nowhere else.   It’s true. Not in London, New York, LA, Australia or New Zealand, but in this very blog, back in October 2011, and in the posts since. If you run across any evidence to the contrary, send it to my email sysudoku@gmail.com and I will acknowledge any prior discovery and your correction, anonymously or not, by your choice. Also, you can comment on the relevant early posts, and have the comment appear with them, with my reply.

I haven’t exactly hidden this under a barrel, and no experts have disputed it in the three years since.

Having read the principle of line marking in one sentence above, you could now re-invent it for yourself. But do that now, because next, I’m going to walk you through the few lines that finish the 2012 Akron Tournament Championship puzzle.

Where is the line with fewest free cells? Actually, its easier to count cells locked by clues, naked pairs or other subsets. Row r9 has none, while rows r7 and r8 have three. There are ties, at 3 free cells. My resolution of ties is rows top to bottom first, the columns left to right. We only want the first one, because when we mark that line, the order may change. Actually, I look no further than r1, because 3 is the least number possible after DB. Right?

So we start with r1. Watch closely. My first task is to make up a list of numbers that may be new candidates in the three free cells. I see 5, 6, and 8 in r1 already, so I won’t need more. But 6 and 8 cannot add new candidates in r1. You figure it that out. So my list is: 5. On my PowerPoint template, which was not available on the tournament stage, I place my fill list on the side of the line, or below the column and then copy it into each free cell. Then I eliminate candidates from the copy that can see the same number in its box or column. Marking a column, it’s the box or row.

Moving on, r2 has four closed cells, and five free. But r3 has only six closed, and three free. Make up the fill list and mark it. Now review the number of times each fill list number appears on the line. If a cell contains only on number, it is a naked single and a clue. If two cells, then you have a line slink. In r3, 2 is such a number, so we pull the 2’s down to the left corners of their cells to mark a row slink. That’s a little harder with pencil and paper.

Why bother with line slinks? The most immediate payoff is, if you get a matching one on another row, you’ll have an X-wing. That means we look at other rows as we pull the numbers down. We do the same with columns.

akron 14 LM trNow that you’re aboard the line marking train, just follow the trace and check your grid with the one below. Check your selection of rows and columns, and your slink marking. Note my priority to box marks over line marks on the same candidates.

akron 14 LM gridIn line marking r5, we got a new naked pair, taking 4 and 6 from the other r5 cells, and leaving a 1-clue. This triggers a collapse of the puzzle, but be very careful as you handle cells not covered by line marking. In this case, the four free cells of r6. The ‘5’s appear only in the W box, but cell r6c2 is not necessarily a bv, and 1, 2, 8, and 9 are ‘maybe’s. Stay aware of that. It’s safer to line mark r6 first, then follow the collapse.

We haven’t touched on all of basic solving here. You may want to consult early posts on naked and hidden subsets, X-wings, the hidden dublex, and boxline restrictions. And if you live in the Akron area, you could even ask the library to have me do a live session demonstrating how all this is done with PowerPoint and Sysudoku templates. Deb in the Popular Culture Department, our own Akron Tournament Director, has my email.

mensa xwing 13The blog’s basic solving clinic is over. Turning our attention back to advanced solving techniques, next post continues the review of Peter Gordon’s Mensa® Sudoku Guide, with its advanced instruction. Peter begins with fishing. I suggest you re-enter Gordon’s world by taking on his first advanced example, demonstrating the X-wing.

No, I’m not asking you to do it as Peter would, by “scanning” as far as you can, then doing a number scan on the remaining cells to get the remaining candidates, and to pick up the remaining “one- choice” and “elimination” opportunities. Just use the dublex bypass, and search among the candidates for the X-wing. It has to show up in line marking if you don’t see it earlier.

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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 sysudoku@gmail.com  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 sysudoku@gmail.com. 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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