A Guide for Experts

This post describes the explanations, innovations, and reviews of the Systematic Sudoku blog for experienced solvers. Influences and introducing posts for innovations are identified. In blog revisions, the original dates and order of introductions is preserved.  This update occurred March 2017.

Welcome to Systematic Sudoku. Please refer to the previous post for the purpose and goals of this blog. It is important to understand what is meant by human engineered Sudoku solving, and the scope of this enterprise. With difficult Sudoku, it is assumed that the human solver will be aided by © MS PowerPoint or an equivalent. Templates for entry and analysis are available free by request from the Tools page.

A  significant feature of Systematic Sudoku is the detailed sequence of solving actions outlined by the “Sysudoku Order of Battle”, or SOB.  It is a sequence suitable for every puzzle, from easiest to most diabolical.  For monsters, the sequence can only be performed as permitted by the lifting fog of candidates.

The current state of the SOB is spelled out in flowchart form on the Order of Battle page.  There are three main stages.  Basic solving ends with all remaining candidates defined.  Advanced solving techniques depend on this candidate field, but a normal order is defined.  Extreme techniques are the ones deferred for “last resort” effort.

Having a defined order of solving is an innovation in itself. A significant advantage is the added focus that comes with certainty on what comes next.  Of course, many deviations are justified by the observed state of the puzzle.  Even then, the SOB remains a basis for rating puzzle difficulty.  

Basic Sysudoku

The primary basic Sysudoku innovation is slink marking, a uniquely Sysudoku  form of pencil marking.  It replaces the keypad style of candidate marking with one in which strong link (slink) partner candidates are placed in the corners of cells, and other candidates are relegated to a string in the middle.  Slinks, aligned triples, and naked subsets defined by boxes are added, along with derived clues,  in a first stage basic operation called box marking. In a basic update initiated in October 2014, box marking is preceded by the slink marking bypass, utilizing slinks and aligned triples, but limiting marks to clues and subsets. The bypass is followed by the former box marking. The idea is to decrease candidate clutter and thereby enhance the proven ability of human solvers to “see” the sequence of pencil marks leading to clues. This successful change was inspired by admonitions of Wayne Gould 11/04/14 for “no pencil marks”.

Box marking was recently been expanded to include three-cell fills by an easily implemented process, the bypass 3-fill, 11/01/11. The expansion was introduced in a review of  Michael Rios’ Mensa Sudoku in March 2017.

Remaining candidates, line slinks and hidden subsets are found in a third stage, line marking.   In line marking, lines are filled by increasing order of uncommitted cells, until all lines in one direction are filled. Line slinks are added as each line is filled, and consequent X-wings are found. Hidden subsets are identified as lines and boxes are completed.

Basic solving and follow up marking in more advanced stages is generally enhanced  by another innovation, the marking trace. Traces are a practical way to archive the marking process.  They are “read“ by making the sequence of moves on a copy of the puzzle, and thereby seeing the puzzle state of each move.

Traces fill in the marking gaps for a full record of solving. This allows for improvements  in solving to be documented. The move is recorded but the reason for the move and even the exact location, is left to the reader.    More important for new solvers  is that the marking trace is an opportunity for those not yet familiar with Sysudoku marking to experience how it works, and to practice actively its many tricks.

Two types of marking trace are defined on the traces page (menu bar). A single dimension depth first marking order trace, using nested parentheses has been largely replaced by a two dimensional one that is easier to read accurately.  The third type, the trial trace,  follows a breadth first marking order, allowing an expected result, a contradiction,  to be demonstrated in the shortest and simplest manner.

Accompanying the explanation of subsets in basic Sysudoku is a tool for confirming difficult cases on box and line units. It is Suset enumeration, a scratchpad version of the maximal cliques algorithm applied to many subset forms in Sudoku.

Advanced Sysudoku

Advanced techniques, based on links between remaining candidates, were introduced and explained in posts of 2011 and 2012. These posts remain in their original order. In SOB order, advanced techniques begin with a scan of the candidate grid for unique rectangles, remote pairs, Sue de Coq ,  APE’s and BARN’s. BARN stands for Bent Almost Restricted N-set, a configuration rescued from a mislabeling as an WXYZ-wing. BARN properties were proven in 2016.

The bv (bi-value) scan continues with the aid of a bv map, and a related XYZ map. The latter aids in enumerating regular XYZ-wings, and irregular iXYZ-wings via forcing chain or ER weak links. The iXYZ wings  are a Sysudoku innovation, and a surprising one, since the fundamental elements were well understood in 2011. The XYZ map is also the place to spot genuine WXYZ wings and Death Blossoms, regular and Lite .

The use of forcing X-chains as weak links in iXYZ wings, led naturally to the use of them for “seeing” in all contexts, including “toxic sets” of candidates of one number which must contain a true candidate, such as the Z values of the XYZ-wing. 

The bv map is also the palette for the XY railway network of connected curves.  XY chains can be exhaustively enumerated along the railway.

Sysudokie action then moves to the X-panels, displaying the remaining candidates of each number, where a scan cycle is performed on each panel as it is derived from the grid. X-chains and loops are usually found first. Popular names for X-chains are discounted in these posts, in favor of linking principles applying to all X-chains and loops, including nice loops and grouped nodes 2/28/12.

Fishing is the second application of each X-panel. The subset properties that limit regular fish are explained, and spotting techniques include a novel panel marking technique, the blank line tally , with back up for difficult cases and finned fish provided by Suset enumeration. The Suset scratchpad also devastates Sashimi fish in the review of Bob Hanson’s Sudoku Assistant. Finned and Kraken fish  are covered in depth. Franken fish and multiple fin kraken fish are thoroughly explored in the December 2015 posts of the Hodoku review, by means of Suset enumeration.

Sysudoku strongly advocates Medusa coloring, a very easy operation in PowerPoint, to be considered whenever the bi-value field becomes noticeably strong. Color traps, wraps, bridging and merging are covered in May 2012 posts and illustrated throughout. This representation of the strong link network was extended by nice loop coloring and extensions in the KrazyDad Insane review of 2013.

To continue with AIC of mixed type, the sysydokie grid is marked with “AIC hinges”. These are cells containing partners of two or more strong links. These combine with bv cells,  ALS nodes 7and X-chain segments for AIC and loops.

The ALS-XZ 7/17/12 was explained in July 2012, but Sysudoku has not uncovered a systematic, humanly practical  process for exploiting it, and it is therefore yet to be securely placed on the SOB.

Late in the “advanced methods” year of 2012, Andrew Stuart’s pattern analysis in The Logic of Sudoku was reviewed and tried on one of his Unsolvable #40. This experience developed some understanding to two methods of enumerating patterns, a.k.a. templates, for a process labeled Limited Pattern Overlay, or LPO. The first method, ”lettering” assigned letters to patterns and attempted to identify X-panels with sufficiently small numbers of patterns to write manageable overlay logic equations.

Lettering was eventually supplanted by a Stuart’s more graphic pattern representation, the freeform line connecting pattern candidates in successive lines across the grid. By the first year anniversary of the blog, the relationship of coloring to patterns was understood, several forms of  overlay tables had been tried, and systematic discovery of orphans, candidates with no pattern, was developed. A significant result of this effort was labeled the pattern slink 10/23/12.  The pattern slink divides patterns into opposing sets, in yet another extension of  coloring clusters.  

Unsolvable #40 was defeated, but I had to wait for a sufficiently  tough puzzle collection, The Krazy Dad Insane 7/16/13 collection, to explore it further. Coloring extension by pattern divisions came to be called pink/olive analysis, for the pattern cluster colors selected to stand out from the blue/green and red/orange cluster colors of slink network coloring.

Sysudoku Trials

The Sysudoku answer to arbitrary trial-and-error is trials, defined as the assembly of a set or opposing sets of candidate values which are true or false together, followed by their assertion in a trial marking. One example of a trial set is one color of an unresolved coloring cluster. In the Insane review a new, trial form is defined for the Sue de Coq,  the Single Alternative Sue de Coq, or SASdC, in September 2013.  On the post, a new, breadth first form of trace is introduced, for trials. The trial trace is described on the Sysudoku Traces page. Its purpose is to find the shortest possible path to a contradiction in a trial.

Having gone that far, trials are rarely carried out in Sysudoku. Trials of all kinds are considered extreme, and although trial opportunities may be noted, they are deferred until normal means are exhausted. One notable exception was in the April 2015 review of the Weekly Extreme  collection, where to win an award, participants have to turn in the correct solution on time, every week. The SASdC proved remarkably successful in solving these puzzles quickly. Another situation calling for an immediate trial is when there is reason to believe that there are multiple solutions. Coloring trials have revealed multiple solutions in several reviews.

Collection Reviews

The review of Sudoku puzzle collections is perhaps the most useful feature of Sysudoku for most readers.  One can compare difficulty of available collections, as measured by a consistent point of view, repertoire of techniques, traces and graphic presentations.  In the reviews, solution paths are reported for a preselected sample of 10 puzzles.  Summary tables and comments, traces and highlight grids are posted to help readers assess the character and level of difficulty of the collections.

Collections reviewed include Sheldon’s Master Class, Longo’s Absolute Nasty Level 4, KrazyDad SuperTough, Paul Stevens Week by Week and Sudoku Addict collections, Will Shortz, Ferocious, Trickiest,  KrazyDad Insane, Xaq Pitkow Expert, GM and SuperGM, Longo’s Gordon Guide, Alary More Extreme, Weekly Extreme 429 – 438, Longo’s Nastiest Ever, Vorderman’s Super Difficults, Djape’s Only Hard,  Castillo’s Hard Sudoku, and Rios’ Mensa Sudoku.

Down Sysudoku Guru Avenue, there are encounters with a number of Sudoku writers.  Although his website Strategies page was my early mentor, Andrew Stuart his forcing chain strategies were reviewed critically, starting with “Digital Forcing Chains?”.  Andrew has been lobbied to change an  encroachment on the defenseless WXYZ-wing .  Likewise, in the first book review, an early guide, Tom Sheldon’s Master Class was hammered for his mischaracterization of the limited advanced techniques he covered.   In Paul Stephens‘ Week by  Week and Addicted books, 4/13 basic is adequate, but criticized for getting to slinks and subsets late, thin advanced techniques coverage, and for linking cells instead of candidates.

Rank Theory , as proposed by Allen Barker as the mysterious SudokuOne, was briefly explored in November 2013. Base and cover sets simple enough for human spotting were found to have accessible AIC counterparts, while those represented by pages of plumbing can only be considered another means of computer based solving.

Bob Hanson’s Sudoku Assistant report  taken as instruction, was not generally endorsed, but certain insights and an effort to unify advanced solving methods are appreciated. Peter Gordon’s inept, formerly Mensa Guide to Solving Sudoku is repeatedly called to order. Very little time is spent on Arnold Snyder’s Sadistic Sudoku series, or Carol Vorderman’s misnamed Mastering Sudoku.

The second longest review took on Bernhard Hobiger’s advice attendant to his Hodoku solver. The convenient access to a store of excellent examples, via your own basic solving, is a remarkable achievement. The advice to human solvers, and the explanatory graphic,  however follow programmer ideology unsuited to human solving. Time after time, I had to make explanations and diagrams to show how a human solver could arrive at the decisive moves in his examples, and how they appeared on a Sudoku grid. Inaccurate and impenetrable on the Hodoku site, but welcomed additions to the Sysudoku repertoire, were the sections on Multiple Fin Krakens and Franken Fish

The longest review was the refutation of The Hidden Logic of Sudoku as guide for human solving. For the most part, author Denis Berthier does not claim it as such, but he does claim that solver’s are “naïvely” missing valuable solving techniques unless they maintain grids of otherwise hidden dimensions of Sudoku logic. In the review, it is carefully shown that his extensive evidence of this is wanting, and that this well known computer technique is impractical for human solvers. All of the hidden logic discoveries Berthier offers in this book are shown to be available by ordinary advanced means. Many of the most effective are left out of Hidden Logic, compromising Berthier’s case further. Denis is also called out for his untenable position on the unique rectangle.


Several well known monster Sudoku have been solved on Sysudoku, with completely documented diagrams and traces of humanly accessible solution paths. In the Golden Nugget (December 2013) and Fata Morgana(June 2015) , interchangeable AIC loop paths facilitated Exocet trials.  Kurzhals’ SK loop was followed up with pattern based coloring trials in the Easter Monster  in February 2014. A less plausible sequence of color trials was successful in the Hanson and Marans Easter Monster  in April 2014. This led to a review of Bob Hanson’s report on Sudoku Assistant, linked above.

At Andrew Stuart’s invitation, several filtering techniques were developed for Exocet trials, using the Unsolvable series of his web site, in spring 2016. My understanding remains that the Exocet, even the JE versions detailed by Robert Bird, define trial sets. The filters were intended to shorten these trials. The project was abandoned for now with Bird insisting that his JE requirements  guarantee a solution, obviating trials for these Exocets. Perhaps. He could lay it out like the Sysudoku proof for the BARN.  I’m interested in returning to this when I complete tasks of more direct interest to sysudokie readers.

That’s my current view of what could be of interest to experts, so far.  Now everybody,  let’s explore Sysudoku from the ground up.  These early posts apply directly to the puzzles you encounter everywhere.


About Sudent

My real name is John Welch. I'm a happily married, retired professor (computer engineering), timeshare traveling, marathon running father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. The blog is about Sudoku solving. It covers how to start, basic solving to find candidates efficiently, and advanced solving methods in an efficient order of battle. It is about human solving methods, not computer solving.
This entry was posted in Advanced Solving, Basic Solving Procedures, Expert Reviews, Extreme Solving, Monster, Puzzle Reviews, Sudoku Toolware and tagged , , , , . Bookmark the permalink.

One Response to A Guide for Experts

  1. http://pressemitteilung.ws/node/689444 says:

    Danke schön für euren tollen Beitrag.

    Ich habe euren Weblog schon seit einiger Zeit im Feed abonniert.
    Und gerade musste mich mal einen schnellen Kommentar schreiben bzw.

    mich mal bedanken.

    Macht genauso weiter, freue mich bereits jetzt schon auf die nächsten Beiträge

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s