Chapter 1: Rational Sudoku
Sudoku is, fundamentally, a puzzle...not unlike the crossword puzzles that it often accompanies in papers and magazines. Like crosswords, Sudoku have (in most instances) a predetermined solution. It is possible to create Sudoku with non-Unique Solutions, but these are never knowingly used as puzzles. So suffice it to say, that Sudoku that find their way into puzzle books and onto web sites have what mathematicians call a 'Unique Solution'. Also, like crosswords, each cell in the matrix requires a unique solution. Ever notice how crossword puzzles are often broken up into little domains linked by bridges? Well, there is nothing irregular about Sudoku... at least, not classical Sudoku. The little 'domains' are regular and uniform, although there are variants in which the domains are irregular.
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There is only one rule in Sudoku and I call it the 'Prime Directive': Place the Digits 1-9 into each of the Cells of the Matrix in such a way that each Digit appears only once in each Row, Column and Box of the puzzle. Only one of the required 9 digits can satisfy a particular cell for a given Sudoku (assuming, of course, that there is a Unique Solution). In many ways, the solution of Sudoku takes on a more-or-less mechanical character. Mathematicians often derogatorily refer to Sudoku as: 'Not really a puzzle, just an accounting exercise', and in many ways, they are right. But that does not seem to diminish the enthusiasm that is growing over Sudoku any more than knowing a Basketball is just a bouncy ball takes the pleasure out of shooting hoops. What I want to focus on now is the more-or-less mechanical, technical aspects of solving Sudoku.
In its most popular form, Sudoku is a game of numbers...randomly generated numbers. In our discussion, however, I will refer to these as Digits, rather than Numbers to stress that Sudoku really has nothing to do with mathematics or arithmetic...or even numbers. While numbers are the most popular and convenient form of symbol used, other symbols or qualities (like letters, colors, stippling, shapes, baseball logos, etc) can be used as well. But to sift through the random Digits and find the ones that create the Ultimate Solution for the Sudoku in question requires some 'ruminating', to say the least. But a systematic solution of Sudoku is not too unlike the systematic solution of Tic-Tac-Toe that I worked out when I was about 11 years old. You start with a 'given' mark (the mark of whomever goes first), you place a second mark, then a third and forth until the matrix is completed or someone wins. In Sudoku, you select a Digit and place it in a Cell, but only if it is a Unique Solution (US) for that cell. Unlike Tic-Tac-Toe, you do not play Sudoku 'against' someone...only with yourself. In that sense, it is much more like solitaire or crossword puzzles.
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Sudoku is inherently a 'game' of Logic. Whether you use 9 different digits, 9 different shapes, 9 different letters, 9 different colors...the goal is the same. Sudoku is now being used in some classrooms as an exercise in Logic and Reasoning. In my own experience, I have discovered a reawakening of these faculties, but that is just my subjective experience; not born out by experimental evidence. At times, the solution of Sudoku will seem completely deterministic. To be sure, one can solve Sudoku in many different ways, but there can only be a few, and in most cases, ONE, 'best' strategy. This does not deny, however, that there may be more than one effective strategy, as we will see later. But in general, the number of workable, definable strategies cannot be very large.
To understand how to find the most rapid Solution, you will be well served by following several Basic Strategies. This book is about such strategies and how to make them work for you. You will appreciate aspects of Sudoku that you may have never experienced before. You may discover your own unique strategy, or put them together in a unique and efficient manner. Their use can bring to light aspects of Sudoku that you may find surprising. Some strategies will not even be needed until you are ready to tackle the more difficult puzzles. It is a delightful and fun way in which to exercise ones counting skills and in doing so gain a more basic and accurate understanding of the world around you.
Sudoku is worked or 'played' on a Grid or Matrix. Matrices are extremely powerful tools for understanding and for manipulating our world. Every survey ever taken, every schedule ever developed, most fundamentally involves the manipulation of Matrices. Crosswords, Sudoku, Cryptics, Tic-Tac-Toe and so forth, are all Matrix-based puzzles. Understanding how Matrices work, you will see, can be somewhat exhilarating! While it may seem merely a mechanical accounting processes, it is unlocking the logical, rational solution centers of the brain and integrating them with other important areas of consciousness, programming your mental circuitry to think logically and rationally and unimpassioned. Unlike crossword puzzles, Sudoku is not bound by language. That means you can play Sudoku whether you speak Chinese, or Hindi or Pashtun. You can play whether I can speak at all. Regardless of the symbols used, some set of strategies must be applied in order to complete the Matrix correctly. While the strategy behind the Rational Sudoku and the standard strategy may someday be superceded by other methods, at least we will get there faster having some sort of system to start with. This book provides just that system. Anyway... 'Let the Games Begin!!'
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Matters of Style:
I want to make a comment about Digits and words that seem to express the same concept; most often: one (1), two (2), three (3) or four (4). To the extent possible within the confines of human error, I have attempted to use the words when they are descriptive and use the Digits only in the context of the Sudoku Solutions or the identification (Numbering) of Cells, Rows or Columns. In the phrase 'two Half-Pairs' (we shall learn about this later), for instance, the word two is descriptive of the term 'Half-Pairs'. A 6-Half-Pair, on the other hand, consists of two 6s in two different Cells. Here, the term 'two' is now descriptive of the Digit 6. Similarly, a '2-Half-Pair' consists of two 2s. The 2 (and the 6 before it) refers to a potential Solution of Sudoku. This will, hopefully, minimize any confusion between the descriptive use of 'two' and the Sudoku use of the Digit '2'.
Another matter that I want confront is my use of Capitalized Words in the text to stress terms that are unique to this discussion. While the Germans have done this for eons, it is shunned and spurned by writers in English and most probably will be strongly contested by my future publisher. I shall continue this practice, however, as I believe (with all my Heart) that this helps the reader identify and relate the terms to the core principles of this treatise.
Terminology:
Sudoku terminology is in flux: there is no uniform standard for Sudoku terms. The 3x3 Domains, for instance are called Boxes, Regions, Nonets, etc. Often the same word is used by different authors in different ways. Here I have taken what is reasonable and common (e.g. Box, Row, Pair, Solution, etc) and used these, but added other terms (e.g. Matrix, Discovery, Trio, Half-Pair, etc) where appropriate and descriptive. I refer to one who solves Sudoku as a Sudoku-ka (same plural form), in the Japanese martial arts tradition. A Sudoku-san is someone who is proficient at solving Sudoku and a Sudoku-sensai is a Sudoku Master. Vocabulary specific to the Rational Sudoku will be introduced and defined as needed.
Read the next two parts of this series: Intermediate and AdvancedOnly 5 distinct and Rational operations.
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