I had the distinct pleasure of having a conversation with Charles Lantz, also known as Sudoku Sam, about his latest book, *MODUS FORTIS Sudoku Logic, Sudoku Reason*. Charles has once again captivated the Sudoku community with his innovative approach and deep understanding of the game, offering enthusiasts a fresh and challenging perspective.

In our conversation, Charles delves into the intricate details of his new book, sharing insights into its unique methodologies and the inspiration behind his creative process. Whether you're a Sudoku fan solver or a curious newcomer, Charles’s expertise promises to enhance your puzzle-solving skills and expand your appreciation for the art of Sudoku.

Stay tuned as we explore the fascinating world of Sudoku with one of its foremost masters.

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CORE CONCEPT

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The book claims a "Khunian shift" in Sudoku strategy. Can you explain this concept and how "Modus Fortis" represents a major change in solving puzzles?

In 1962 Thomas Khun published a book titled The Structure of Scientific Revolutions in which he proposed a radical interpretation of the progression of scientific theories. Prior to his work, philosophers of science felt that scientific knowledge progressed incrementally by solving progressively more challenging problems, thereby building and reinforcing scientific theories experiment by experiment; like building a house brick-by-brick. Khun proposed that this interpretation applied in science to specific paradigms – such Newtonian mechanics – thereby popularizing the concept of the paradigm shift. A new paradigm, as, for instance Einstein’s Relativity, supersedes the prior one in what we would call a quantum leap – and what Khun called a Scientific Revolution. Modus Fortis represents such a revolutionary replacement of the Standard Strategy because it provides more rigorous explanations of the problems and solutions proposed by the Standard Strategy.

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You mention four basic tactics leading to two "molecular structures." Can you elaborate on these structures and how they form the foundation of your problem-solving method?

The four basic tactics represent the four basic components of Sudoku. First, the digits or numerals and next are the three basic components of the matrix: rows and columns, of course, and boxes. Here we use atomic and molecular as analogies for understanding the Solution Process. The most basic and fundamental elements (atoms) of Sudoku are Cells and Digits (numerals), which can be thought of as location (or place) and object (or digit). For each Cell (location) we must discover the appropriate object (digit) to place there and still satisfy the prime objective – “Each Digit (1-9) must be placed in each Box, Row and Column.” When beginning to work up a Solution, the most basic and pervasive structures that appear are Exclusive Pairs and Double Digits. Each represents an exclusive set. An Exclusive Pair, or simply a Pair, is when there are only two Digits possible for a particular Cell. We see these arise frequently during the Solution Process. If we can eliminate one of those two Digits, we will have a Solution for that particular Cell – we will have discovered the atomic structure of that Cell. The other most fundamental structure is a Double Digit – which we call simply a Double – in which there are only two Cells in which this Digit can serve as a Solution in a particular Box, Row or Column. Again, if we can eliminate one of those locations as an option, we will have discovered a location for that digit. That is the basic understanding we need in order to answer your question. It is possible that there can be two Cells in a Box, Row or Column that Host an exclusive Pair of Digits – say, for example 4 and 6. We call this a set of Twin Pairs – or simply, a set of Twins. Twins are a familiar and common structure seen during the Solution Process. They are composed of two sets of Exclusive Pairs – what we would call molecular structures. Similar structures can arise with Doubles.

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You mention a concise notation system. Can you give some examples and explain how it aids in recording the solution process for future reference?

Throughout the book there are Logs of Solutions from beginning to end and sometimes just of brief sequences in the process, just as in recording a game of chess. As in chess, we use a shorthand notation system so everything is recorded in a concise table of events.

For instance, in chess, a move made by a Queen would be noted as Qe5 which tells the reader that the Queen has moved to the 5th row (from the bottom) in column e. But in Sudoku there is no movement; there is only location and in Modus Fortis, rows and columns are numbered from 1-9 from the upper left corner – Cell 1 in Box A. So A1=ab would mean that there is an exclusive Pair in A1. If a, b, c and d are possible solutions for A1, there is no need to record or remember this. Only the exclusive Pair is noted. Similarly, with Doubles, A37=e would mean that the Digit e must be the solution for A3 or A7, exclusively in Box A. If we are considering Rows, however, the expression A3C2=e would indicate that with regard to Row 1, e is constrained to just A3 and C2. This would be an exclusive set of the first order for the e-digit and we would call it an e-double in both cases. The exact same reasoning would apply to columns. If we later discover that C2=e, then we also understand that A7=e. A similar reasoning applies to the ab pair in A1, if we were to discover that a cell in column 1 – say G5 – is b, then A1 must be a. Thus, the notation system can record every action taken in the solution process from beginning to end. Recording (or remembering) only exclusive sets makes the process simpler, and with practice every Sudoku can be solved with no notes at all.

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Exclusive Sets and Forbidden Sets are presented as the "Families" of complex structures. How do these sets help solve even the most challenging Sudoku puzzles?

Sudoku is all about structure – by which we mean sets or potential sets. Sets of Cells are called Forms, so when we talk about sets, we mean sets of Digits. Denis Bertier notably stated that in Sudoku, a fact is what is clearly identifiable in the Matrix. A Double or a Pair, therefore, represent facts – the two most basic facts possible, short of a Solution. From Doubles we construct Exclusive Sets and from Pairs we construct Forbidden Sets. A set of Doubles is the first order of organization for an Exclusive Set (ES1). For example, this can be represented in a row as:

The second order (ES2) is when the same digit aligns in a second row as

The third order (ES3) is when the same digit in a third row aligns with two other rows such that the two digits each aligns with one of the other two doubles as:

This works only when each double is fixed in their respective rows. The 4th, 5th and 6th orders are a bit more complicated and require coupling – a concept that does not exist in the Standard Strategy. These patterns reflect the underlying structure of Sudoku. In each of the above examples, D is fixed, or locked-into the rows but not, necessarily, the columns.

With exclusive Pairs, similar arguments prevail for Forbidden Sets (FS). The first order (FS1) is simply an exclusive Pair – D1D2. The second order, when seen in a row is

This is perfectly allowable, but the second order is not allowable (i.e. is Forbidden) because it would permit two solutions to the overall Sudoku. Therefore, higher order FS assume forms like

During the solution process, the two forms (ES and FS) arise naturally and spontaneously.

All these structures project constraint throughout the Matrix in predictable ways to create expected effects. It is the projection of constraint that leads to the resolution of certain Cells. This is all explained in my book in immaculate detail.

## LEARNING EXPERIENCE

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While comparing "Modus Fortis" to E=mc², you acknowledge its complexity. How does your book bridge the gap between the core concept and its practical application for Sudoku enthusiasts?

First of all, my name for “Sudoku enthusiast” is Sudoku-ka – a Japanese expression. The Modus Fortis is the disjunctive syllogism (also known as the Exclusive Disjunction) and it is a rule of inference (i.e. logic). Its expression is simple

a ѳ b; -b: a

This is to be read as: it must be a or b; it can’t be b; therefore, it must be a. Here ‘it’ refers to location (Cell) or object (Digit) depending on the context.

Even if this is all there is to it, this rule must be repeated as many as seventy times to complete the Solution. With easy Sudoku, this is just how it works out. But for more challenging Sudoku the situation is more complex; and sometimes the Logic runs out and one is required to resort to other tactics; primarily hypothesis testing. Bridging the gap between the core concepts and their practical application is by building, or trying to build, structures as explained above.

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Your book is aimed at all interest levels. Can someone new to Sudoku benefit from "Modus Fortis," or is it geared more towards experienced players?

The truth is the right path for everyone. Given the state of the current Sudoku literature with its pervasive ‘Logic only’ mind set, Modus Fortis offers the only viable option for beginning to understand Sudoku. The basic elements are laid out in the first few sections of the book and provide a good foundation for further exploration. To the extent that Sudoku is a ‘Logics Puzzle’, one must understand the logic that is inherently involved in order to progress to more complicated tactics. If you do not understand the most basic logic involved in Sudoku, it is difficult to see how you can progress to the more complicated levels of reasoning. I lay out the Modus Fortis in the very first chapter of the book. The remainder of the basics is explained in clear and concise language that is understandable by all. Logic does not have to be complicated. New Yorkers root for the Yankees; I’m from New York, therefore I root for the Yankees. How difficult was that? When you winnow it down to an exclusive pair or a double it is becomes a trivial issue. It must be A or B; it can’t be B; therefore, it must be A.

## SUDOKU SAM

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What inspired you to write a book about a seemingly simple game like Sudoku?

My father was a master watchmaker in North Carolina and for as long as I can remember I have sought to understand how things work. When I picked up my first Sudoku magazine in St. Barts, I figured that I would knock this out relatively quickly – like in a matter of hours. Four days later, I was still tearing my hair out trying to solve my first ‘simple’ Sudoku. That was the first step in my long journey to Modus Fortis. When I realized that the Sudoku world was dominated by positivists, my motivation shifted from wanting to know how things work to how to restore honesty to the process.

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How long did it take you to develop the "Modus Fortis" approach, and what challenges did you face in refining it?

I had worked out the basic strategy within four years and had published my first three books. My third book: Sudoku Sam’s Meta-Strategy for Solving Sudoku was published in 2009. A concise, Sudoku-centric numbering system for the Cells was in place and I had also resolved the problem of recording every step in the process, as seen in chess. This was the single most important development on my path to understanding Sudoku. It allowed me to retrace my logical steps whenever I encountered an error and the Solution Process crashed. It took another decade to realize that it was the exclusive disjunction that powered the logic of Sudoku. Once that was in place, the rest followed smoothly.

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What are your hopes for readers who engage with your book and the "Modus Fortis" method?

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BONUS

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Can you share a particularly challenging Sudoku puzzle you solved using "Modus Fortis," and explain how your method tackled it?

That would be the "Easter" Monster – hands down the most difficult Sudoku in print. This Sudoku appeared in the e-literature of Sudoku from an anonymous author and no one has provided a convincing approach to its solution outside of Modus Fortis. This is because the rest of the SudokUverse is stuck in a Positivist, Logic-only mind-set. Sudoku is not a logics-only problem, it is an Empirical problem – from the very moment that one begins to find a Solution. Where do you start? How do you begin? Does this sound familiar? This is the way every problem known to man or woman begins. “How can I get that jar off the top shelf?” “How can we get batteries to store more electricity?” “How do we colonize the moon?” Where do you start? How do you begin? Certainly, the answers must be grounded in Logic, but when there is no clear path to a logical answer, you must try the most likely options. This is the path of Empiricism. This is Reason. With the Easter Monster, this was precisely the path taken and the problem was solved with Modus Fortis as shown and discussed in my book.

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Beyond solving puzzles, do you see any broader applications for the logical principles behind "Modus Fortis" in other areas of life?

As I mentioned above, Modus Fortis is a method for discovering the truth. It is a glimmer of hope in a world awash with misinformation. My objective was never just solving a puzzle, it was discovering the essence of the problem and developing the tools necessary to uncover the ‘truth’ of the matter. Thinking logically is an essential tool but when we reach the limits of Logic, it is time to begin to Reason through the logjam. Logic, after all, is a tool of Reason; and every act of Reason holds a kernel of uncertainty. Teaching our children and grandchildren to use Modus Fortis helps to develop their ability to think with Logic and Reason.

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What are your future plans for promoting "Modus Fortis" and potentially expanding on this approach to Sudoku solving?

My current project is writing a technical paper on Sudoku as an Elementary Formal System (EFS). This will put Sudoku in a position of being its own self-contained system. It is also possible to use Sudoku, or some simpler form of Sudoku to evaluate neurological disorders like dyskinesia. Sudoku can also serve as a model for certain aspects of mathematics, and in particular, an understanding of Gödel’s incompleteness theories. I have been told that in France, there is a move toward using Sudoku to understand verbal declension. The structure and composition of Sets is readily seen and utilized in the evolution of a Sudoku Solution. I believe that Sudoku will be the passion for the remainder of my life.

### How did you come up with the name ‘Sudoku Sam’’?

Sudoku Sam is my name. I own it as a Trademark (R). I began writing my books within a year of Solving my first Sudoku; in anticipation of the First National Sudoku Competition in Philadelphia; sponsored by the Philadelphia Inquirer and hosted by Will Shortz. It was the largest puzzle competition in history. I felt like Charles Lantz; the Author, was not so scintillating - aside from the obvious Germanic overtones. Then, knowing the history of Sudoku, I wanted to pay tribute to its Japanese roots. Sudoku Sam popped into my mind immediately; but that sounded too American - Great grand-nephew of Uncle Sam, no doubt. But to a Japanese speaker, the ending mmm is difficult to enunciate and the inclination would be to harden the syllable to sound more like a nn (Sudoku-san) - meaning "honored Sudoku", or "esteemed Sudoku". I participate in Aikido and it is not uncommon for someone - some Aikido-ka - aikido practitioner - with overtones of deep commitment - to refer to him- or her-self as an Aikido-ka. Referring to someone as an Aikido-san is an entirely different matter, and one would reserve it to the person of highest esteem on the mat. So, to Americans, Sudoku Sam refers to a dyed-in-the-wool American, while Japanese see more grandiose implications due to the linguistic slide. It felt right, so I bought it!

As we wrap up this enlightening conversation, I want to extend a heartfelt thank you to Charles Lantz for taking us on this amazing journey through the world of Sudoku.

For those eager to dive into Sudoku Sam's innovative Sudoku challenges, MODUS FORTIS is available for purchase through various platforms. You can find it on Amazon and Barnes & Noble, or visit Charles’s official publisher website for more details and direct purchasing options.

Thank you again, Charles, for sharing your insights and for contributing so much to the Sudoku community. I'm looking forward to seeing how your work continues to inspire and challenge puzzle enthusiasts around the globe.