Berthier’s Hidden Logic of Sudoku

This post begins a sysudokie reading of Denis Berthier’s weighty and challenging treatise on Sudoku, The Hidden Logic of Sudoku, or in this report, THLS. I start with an overview of the agenda of the book, which  differs greatly from the human solving theme of the Sysudoku blog.

royle-17-3Most of the examples in THLS come from an exhaustive collection of puzzles with 17 given clues, compiled by Gordon Royle.  Here is Royle 17-3, the only one for which Denis explains his basic level solution in detail.

I’ll checkpoint the sysudokie basic solving next time. The bypass is all that is necessary, but for reasons of my own, please include SE1 as the second new clue, even if you don’t normally examine 6f: lines in the bypass. The THLS solution, compared to your bypass, suggests why this book offers little direct help for human solvers, although it offers considerable innovative material on Sudoku.

THLS begins with an amusing account of what happens when a scientist trained in formal logic and the rule based technology of artificial intelligence encounters the addicting fascination of Sudoku.  The book dates back to 2007, and was preceded by an era of rule based AI machines called expert systems.  In this early AI era, practitioners would deeply interview experts with commercially valuable, but hard to transfer knowledge and mental skills. They would codify these abilities in a set of rules.  Then, when the experts were no longer available, these software systems could function in the same way, asking the right questions, classifying the situation, and providing answers.

Quite naturally, a teaching scientist in the AI field would be attracted to Sudoku solving as a practice problem for students of rule based Artificial Intelligence. But on a deeper level, such an individual is also compelled to understand how to encapsulate human expert Sudoku solving in a set of rules of an AI engine.   In that endeavor, Denis writes of his frustration at the lack of well defined expertise on which to build a set of rules.  Instead, he turned to a very different goal: to develop Sudoku solving rules for a rule based solver he calls SudoRules. Berthier’s solver takes advantage of implied, but normally hidden structure of Sudoku, and ranks puzzles in order of increasing “logical complexity”.

I wonder what Berthier would have done with an opportunity to evaluate Sysudoku solving, with its Order of Battle and constructive techniques, as a model of expert human solving. Would he have been drawn into the project of emulating the ideal sysudokie with a rule based expert system?

From the Sysudoku perspective on human solving, the following quote is significant:


It is easy for readers of THLS to be misled by such statements.  And I doubt its writer believes “simulating human solvers” is what his solver is doing. This review will demonstrate the contrary.   In the quote above “relative efficiency” means the level of complexity of the rule set that solves the puzzle. But this is a very subjective definition, burdened with a detailed theory of logical complexity of interest to very few. The generally understood meaning of relative efficiency is more related to computational complexity, the number of operations needed to solve a problem of a given size. Computational complexity is the critical factor separating human and computer solving methods.

In the case of Sudoku puzzles there is a problem with computational complexity. There is no agreed upon measure of size. In Sysudoku, as a substitute measure of computational complexity, puzzles are ranked by how long they hold out against the Sysudoku Order of Battle. The SOB methods are subjectively ordered by increasing computational complexity, including searching and construction in its operations.

Berthier’s SudoRules is yet another computer solver, exploiting the inhuman ability to execute millions of instructions in less than a second. The next posts will uncover the elephant in the room, that the execution of rules in increasing order of logical complexity does not simulate what any human solver should attempt to do, much less how an expert solver would do it.

A major factor is that solvers like SudoRules are not coded to exploit the visual pattern recognition ability of the human solver. Enlisting these abilities to construct solving patterns is a primary way humans overcome computational complexity.  In his introduction, Berthier mentions the “great gap between abstract logical complexity and psychological complexity for the human solver.” Working with an expert solver, in the manner of an expert system practitioner, might have led him to some surprising psychological discoveries.

The next two posts will reverse engineer Denis’ basic solving technique of your homework puzzle, Royle 17-3. It is solved by L1-0, the logically most simple rule set of SudoRules. L1-0 is revealed to be too narrowly focused to serve as a basis for human solving.  Beyond that, the Berthier order of battle continues into advanced techniques following the abstract notion of increasing logical complexity, and departing seriously from a reasonable track of increasing difficulty for the human solver.  The logical complexity agenda also leaves serious gaps in the SudoRules advanced techniques repertoire. These gaps damage the credibility of Berthier’s claim for his “hidden logic” rules such as:


While such claims, and Berthier’s carefully crafted presentation of Sudoku as a logical structure, has led readers to believe that THLS is an instruction manual for human solvers, it was clearly not intended to be that. As I said above, human solving is not its goal, though I wish it had been.

On the other hand, there is fiercely independent thinking, with novel approaches worth examining for practical solving ideas. In the review I will avoid the issues of logical representation of rules, and explain what I understand in THLS in a manner familiar to Sysudoku readers.

Do your homework.

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Green’s Labor Day Sunday 5-Star

The last Sudoku of Dave Green’s Labor Day Weekend, the Sunday 5-Star, has a challenging sysudokie bypass and line marking, barely staying within the lines of basic solving. Even on a ©PowerPoint grid, this puzzle would be a nightmare with number scanned candidates.

green-9-04-nw-fillAction begins in the NW box, as a crosshatched NW1 allows aligned givens 3,5 => NWnp35,  leaving NW2 and 2 cells for NWnp89, and a 2f: r2np46.

green-9-04-by-gridThen unwritten

SE7m => E7, and the

Ec8 wall => E5m => SE5.

In 3f: c8, 4,6 will force NEnp46 and hidden single 8.

The dublex SW9 leaves a resolvable 3f: in r8.

To finish up the bypass, 2 and 4 are box/lined, into a naked pair in r8, then

7&9 in SW, r6 and c3 =>Wnp79. 

Then E5m&W5m => E5.


Next comes an easy box marking and very tough line marking, showing how the Bradley Hand ITC font holds up with long fill strings. 


We’re rescued from advanced solving by a 6f: naked triple.


Here is the entire trace. You don’t want to miss a naked triple like this one.


The Dave Green puzzles of this Labor Day Weekend series bring out the stimulating challenge of the slink marking bypass, and basic level Sudoku puzzles. Next week we start a review of Denis Berthier’s 2007 book, The Hidden Logic of Sudoku, looking for human solving insights.  You might want to dig out or pull up your copy.

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Green’s Labor Day Weekend Workout

Here I checkpoint the bypass on the correct version of Dave Green’s 9/03/16 Saturday 4-Star, and assign my readers to try the bypass on his Sunday puzzle. The series of three demonstrates how the challenge and efficiency of the bypass adds to the enjoyment of these excellent Sudoku puzzles.

What a difference one clue makes! Many of the same moves, but this time a steady collapse starts on the 5’s and drags the other numbers in.  


If your trace doesn’t look like this, it might be worthwhile to read this one and see what you saw that I didn’t and vice versa. I frequently lose concentration to what I see ahead, and have to go back to follow checkpoint tracing rules.  But if you’re not solving to post, don’t look back.

green-9-03-gridHere’s the grid at almost solved, with the last naked pair unresolved and a couple of “in reserve” causes still in centered pencil marks.

At the risk of over doing it, let’s apply the bypass to one more Dave Green, the 5-Star Sunday Sudoku of the 2016 Labor Day Weekend.



green-9-04For the checkpoint, I’ll have the trace and a report on the role of unwritten pencil marks, including restricted boxes, 2f: and 3f: lines, slinks and aligned triples.

Just don’t expect to wrap this up in the bypass. I had to try out the Bradley Hand ITC on the bypass, box marking and a tricky line marking.


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Recovering From a Small Disaster

This post corrects an incorrect homework assignment, and announces some changes in Sysudoku due to a switch to a new computer and updated operating system.

Last week was not a good one. On Tuesday I posted an incorrect puzzle for homework on the bypass. When I dragged the given clues of Dave Green’s Saturday 9/3/16 4-Star into my puzzle template, I got one wrong. I must have forgotten my box-by-box check.  In preparing the post,  I somehow “solved” my version with the bypass. Like you, I never check  solutions unless I encounter a problem.  My apology to all readers who spent time on it.  Without access to a solution, you couldn’t know that clue 1r2c3 was out of place. It belongs in r1c3.

But I was amply rewarded for this transgression when my hard drive died the next day. Now I had to switch over to the new laptop I purchased some time ago in anticipation of this sad day.  And I have to rework everything not backed up, including checkpoint for this post.  That’s how I discovered the error.

That leaves us with a plausible looking puzzle that doesn’t have a solution. On which some readers have experience. Let’s consider it an opportunity to see if your bypass would have revealed the contradiction.

green-mis-trHere’s mine, with some accounting below on how unwritten aligned triples and a smidgen of line marking carried the contradiction along: 


N2t =>NW2, N5t => NW5 & E5m => NE5,   N4t => NW4,   NE8 is a 3f: trap.  

green-9-03-con-gridIt turns out that S8 conflicts with the now forced 4r9c5!

Hold on! This doesn’t look like a Sysudoku grid? You bet it doesn’t.

Unfortunately, the new ©MS Windows is without the Brush Script Font which represented solver input. So I learned that updated solvers can’t duplicate my blog pictures with their newer computers. The solution is to choose a new solver font from among currently available ones. This grid shows Bradley Hand ITC.

green-9-03-bradley-sampleI’m not so sure I want this one. The problem is, the 7 and 9 get a bit flamboyant. Here’s a little sample, along the way to the solution, that includes 7 and 9, and pencil marks.

lucinda-handwriting-sampleAn alternative is Lucinda Handwriting, less extreme and more bold. It’s not as distinguishable from the Calibri font of the given clues. Maybe a little too perfect for handwriting?

segoe-script-sampleA better option, perhaps, is Segoe Script. It is bolder, with more conventional hand lettering. I don’t care for the jumpiness in  vertical registration. Again, it’s harder to distinguish from given clues.

For now I’m sticking with Bradley Hand ICT.

All you have to do, to incorporate any of these fonts into your Sysudoku puzzle template, is to select the font for your script letters on the right. Some font size adjustment in the pencil mark text box may help with placement in the cell.

So consider all this an extension of the earlier basic clinic posts. My amended assignment is to move that 1 clue as shown above, and tackle the bypass on the real Green’s Saturday **** of 9/03/16. To follow up, I’ll interpret  the trace with the role of the unwritten aligned triples and slinks. In the bypass, I’m now including another useful tactic, the mental line marking of the lines as they are reduced to three free cells, the 3f:’s.  

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Dave’s Friday Labor Day Four-Star

This post traces basic solving on the first of three Labor Day puzzles by Dave Green.  The series is an opportunity to highlight the efficient and uncluttered basic level Sudoku solving with the slink marking bypass.

In keeping with the adoption of the slink marking bypass into Sysudoku basic solving, I’m the tour guide on three in a row that showed up in my Beacon Journal just before Labor Day, from King Features, Inc.  Dave Green gets my Oscar for basic level puzzles every year.  If you haven’t asked for free ©PowerPoint and ©Word  templates, and don’t have your own,  look over the Tools page and send an email request for them.  Try out the puzzle template by dragging in last posts grid of Green’s 4-star of 9/2/16, and filling it as you read the bypass trace.  The grid with the completed bypass is below.  Account for every event in the bypass.  The grid may help.


green-9-02-bypass-gridThe bypass leaves the grid with only 28 blank cells, and hopefully, it leaves you with an added source of adventure for basic solving.

From here, slink marking is easy and line marking easier than it would have been.

By its nature, the bypass can be done on the original copy of the puzzle.

In this case, there was no line marking, just a deep collapse.

I said deep, not steep.  My traces are sometimes described as too complicated.  Landscapes are complicated because nature is complicated. Sudoku solving, taken all together, is complicated.  I take it as a compliment.

green-9-03For an immediate follow up, try to duplicate next week’s bypass trace on Green’s Saturday 9/03/16 4-star.  I call the NSEW box patterns wells.  Before starting, I saw clues in two of them, while looking for naked pairs in all four of them.  That’s the way to react to a well, isn’t it?

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Adopting the Slink Marking Bypass

This post reviews the origins of the slink marking bypass, now adopted for Sysudoku basic solving. A bypass trace on Very Hard 28 illustrates the challenge available in a basic level puzzle.   It also kicks off a series of  Dave Green’s Labor Day  weekend puzzles  that show off the bypass.

Working through the Beginner’s page and the early posts on basic solving, I came to realize how inevitable it was for me to adopt the slink marking bypass as the way to begin puzzles of any level.  After all, Sysudoku was started when I realized, with due diligence, that human solvers needed a better way to begin.

For a time before that I had wondered what I might be missing, because Sudoku advisers devoted so little attention to finding candidates.  So much effort went into generating candidates, then removing so many of them. There wasn’t even a name for what seemed to be the default starting method.  Tom Sheldon in his Master Class, called it the “completion strategy”.  Not wanting to dignify it that much, I called it “number scanning”.

I was ready to begin when I was convinced that box marking with dublex and crosshatch, then finding the remaining candidates with line marking,  finds all remaining candidates more efficiently with much less distraction.  I’ll tell you how it started, but first, you may have more work to do on Very Hard 28.

vh-28-basic-trYour assignment is to inspect this Very Hard #28 basic checkpoint trace.  Unless your trace looks better, you must read through it, marking your copy of the grid, and decide exactly what each move is and exactly why it is possible.

For navigation hints, see the traces page.



The slink marking bypass originated in my review of Peter Gordon’s basic solving advice in his Guide to Sudoku Solving.  His basic solving I recognized as box marking with clues only, with limited line marking.  While definitely not enough, it was a positive step toward minimizing clutter.  Although he did not explain them explicitly, Peter was evidently tracking some slink effects and naked pair claims on cells.  I decided to try this as a first stage of box marking, with pencil marking for naked pairs only.  I called it the dublex bypass, and recommended it for speed tournaments, in October 2014.

Shortly afterwards, I was doing  basic clinic posts, reviewing some Dave Green puzzles.  So  I compared traces with the bypass and without.  The bypass did surprisingly well, finding almost all clues and replacing some slinks with clues.  Slink marking was easier to  accomplish, after the enjoyably challenging bypass.

Next I reviewed Wayne Gould’s Train Your Brain with Su Doku, Fiendish collection . Wayne advocated no pencil marks at all. He doesn’t give any instruction on basic solving, but his basic level puzzles do require taking slinks into account.  See the review table and quotes in the 11/04/14 post.   I worked through them while trying to track naked pairs mentally, but gave up on this memory exercise.   Not because they don’t work.  I just don’t like to do them, even though I should.

Since November 2014 I’ve been using the bypass effectively.  It definitely takes slink marking further in minimizing distracting clutter of less than relevant candidates.  I think it also adds the challenge Wayne had in mind.

Clues are discovered in line marking, often the triggering a collapse.  But clues found in my box marking all too frequently reveal something overlooked in the bypass.  Keep that in mind when you read a bypass trace in Sysudoku.  After a tough-to-diagnose effect, you might wonder how you could have seen it.  Well, I probably didn’t.  However, discovering what I’m missing in the bypass, I believe I’m seeing more.  Thanks, Peter and Wayne!

Next we’ll  checkpoint our way through the Dave Green Labor Day weekend trio.  I think it will show you another side of Dave’s excellent basic series.  I was sure one of these would cross the Wizard Green’s line into advanced solving.  But it didn’t happen.   Dave knows what he’s doing.

green-9-02Here is Green’s **** of Friday, September 03, 2016. The bypass starts in the NE.  I’m counting on you to figure out why and where, and to go on from there.  Enjoy!

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Mike Peterson’s Very Hard Sudoku, v.1

Resuming posts after a quiet summer, this review identifies the title book as a basic level collection, offering an interesting challenge for the slink marking bypass, and little else.  The brief  introduction offers interesting historical facts, but is another example of ineffective solving advice.

Through my occasional Amazon scan, I picked up one of Mike Peterson’s Sudoku puzzle books, Volume 1 of an apparent series of “Very Hard” books.  The back cover pictures six levels of these books, ranging from “Very Easy” through “Very Hard”, then “Extreme”.  My magnifying glass reveals that volume 9 of each of these levels is pictured on this volume 1 book.  Except for Very Easy Sudoku, only on volume 6.  The review book has a publication date of April 1, 2016, written on the last page, and carries no copyright notice.  Seriously?

veryhard ex 1Among the historical and mathematical data in his three page introduction, Mike drops in a one paragraph prescription for Sudoku solving, which I just have to quote, along with his example grid:

“The process known as ‘scanning’ involves analyzing cell for possible values, and filling cells where one number is possible. Scanning alone will solve most simple Sudoku puzzles.  In the grid above, x = 1.  Harder grids require the ‘forcing chains’ technique.  Above, any value of a forces f=2, since

a=1 => b=8 => c=7 => f=2

a=2 => d=9 => e=7 => f=2.”

Did you follow that?  If you did, it was only by finding the candidates first.  Mike’s “scanning” assumes the number scanning  I’ve constantly called out as inefficient and a generator of distracting clutter. After that, Mike’s  advice is to pick a cell and follow the chains for each of its possible values, looking for a cell registering the same value for all chains.  If you’re a computer, this trial & error is practical.  If you’re not, it’s not. Not in real puzzles, that is.

veryhard ex 2After line marking,  the final step in Sysudoku basic solving, the above chains are much easier to follow. Sysudokies don’t follow forcing chains. They take  them  as winks in advanced solving techniques.

veryhard ex xyzAs an example right here, a forcing chain enables 1r9c4 to “see” the third Z=1 candidate in the 891-wing.



veryhard ex xyOr if you prefer, how about completing an XY chain to eliminate 8r3c4, setting up a Snp19 to eliminate 1r9c4? In either case, the immediate collapse confirms 2r6c8, no T&E and no problem.

very hard review tableGetting to actual puzzles, my review selection provides no evidence that forcing chains, or XYZ-wings, or XY chains, are necessary for  Peterson’s Very Hard collection.   I took Very Hard 1 and every ninth puzzle, the series being  1, 10, 19, 28, . . ., 82. Starting with the slink marking bypass on each puzzle, I had 7 of 10 collapse in the bypass, and three in box marking.  The consistent 17 givens suggests that the puzzles may come from the Royle collection that Mike cites in his introduction.

Green 9-02


Very Hard Sudoku provides a good workout on the slink marking bypass. I enjoyed the book recently on some long airline flights, and shared it with  grandson  Daniel , who was equally successful.  For your bypass homework, here is Very Hard 28, one of Daniel’s impressive victories, with a 2-D trace checkpoint to follow in the next post.


The  mini-clinic on the slink marking bypass will continue with puzzles from Dave Green, the composer I follow daily in the Akron Beacon Journal.  I’ve been modifying earlier Sysudoku posts to adopt the bypass fully. Last week, in the Friday, Saturday and Sunday paper, Dave posted two 4-stars and a 5-star that demonstrate especially well the techniques and value of the bypass.  I’ll be checkpointing them with detailed traces to show the enhanced clutter fighting power of slink marking.  And to clue you in on the pleasurable challenge of beginning efficient solving of basic level puzzles with the bypass.  Don’t miss it, and do your homework!


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